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APPENDICES F, G, AND H
TO
GENERATING STRAPDOWN SPECIFIC-FORCE/ANGULAR-RATE FOR SPECIFIED ATTITUDE/ POSITION VARIATION FROM A REFERENCE TRAJECTORY
Paul G. Savage
Strapdown Associates, Inc. Maple Plain, MN 55359 USA
WBN-14026a
www.strapdowassociates.com April 21, 2020
INTRODUCTION
This document is an addendum to [1], providing derivations in Appendices F, G, and H herein deemed too detailed for inclusion in [1]. Appendix F provides useful formulas applied in Appendices G and H, Appendix G provides derivations of specific-force/velocity/position solutions generated using the Method 1approach under [1] defined test example conditions, and Appendix H provides derivations of specific-force/velocity/position solutions generated using Method 2 under the same defined test example conditions. Definitions for analytical parameters appearing in these appendices are provided in [1] and repeated where used in this addendum.
NOTATION General Notation V = Arbitrary vector without specific coordinate frame component definition.
AV = Column matrix with elements equal to general vector V projections on general coordinate frame A axes.
( AV ×) = Cross-product (or skew symmetric) form of AV defined such that for the cross-product
of V with another arbitrary vector W in the general A frame: ( )A A AV W V W× = × A .
DAC = Generalized direction cosine matrix that transforms vectors from general coordinate
frame A to general coordinate frame D (i.e., D ADAV VC= ).
Coordinate Frames B = “Body” coordinate frame aligned with orthogonal strapdown inertial sensor axes fixed in the
rotating body. E = Earth frame fixed to the rotating earth.
1
E0 = Inertial non-rotating inertial frame aligned with E at trajectory start time t = 0. Trajectory Generator Update Cycle Indices m = Trajectory generator update cycle index ( at trajectory start time t = 0). 0m =n = Trajectory generator even (or alternate) update cycle index (i.e., ). 2m n=Important Note: Each cycle index subscript identifies the m cycle time instant value for that
parameter (e.g., subscript 2n indicates a parameter value as cycle , and 2n-1 indicates a parameter value at cycle .
2m = n2 1m n= −
Trajectory Type Subscripts ref = Parameter or coordinate frame identifier for the variation trajectory. var = Parameter or coordinate frame identifier for the variation trajectory. Parameter Definitions
Parameters are listed next in alphabetical order with Greek letters ordered using the English translation (i.e., Delta under D, muΔ μ under m, omega under o, phi ω φ under p, upsilon under u). Parameters used exclusively in the appendices are defined separately in the appendices where they appear.
υ
BSFa = Specific force acceleration vector of the rotating body (that would be measured by
strapdown accelerometers attached to the rotating body and aligned with body axes).
mDAC = Direction cosine matrix D
AC at the end of trajectory update cycle m. BmαΔ = Integral over an m cycle of B frame measured inertial angular rate Bω (that would be
measured by strapdown gyros attached to the rotating body and aligned with body axes,
i.e., 1
tB Bmm tm
dtα ω−
Δ = ∫ . BvarvarαΔ = Particular value of B
mαΔ defined for the sample to be constant for . 0 9m≥ > −
'BvarvarαΔ = Particular value of B
mαΔ defined for the sample to be constant for . 0m >BmυΔ = Integral over an m cycle of B frame measured specific-force (acceleration) B
SFa (i.e.,
1tB Bm
m SFtmdtaυ
−Δ = ∫ ).
g = Earth’s mass attraction gravity vector (relative to earth’s center) at trajectory position location R.
avgg = Constant average approximation of g to simplify the test example model.
I = Identity matrix. Brefl = Specified position displacement vector of varR relative to refR , a constant in the Bref
frame for the test example.
2
Bω = Rotating body angular rate vector relative to non-rotating inertial space that would be measured by strapdown gyros attached to the body and aligned with rotating body axes.
R = Position vector from earth’s center to the trajectory designated position location (“navigation center” ).
mR = Position vector R at the end of trajectory update cycle m. t = Elapsed time from the start of a trajectory.
mt = Time t at the end of trajectory update cycle m.
mT = Time interval from to (assumed constant for this article). 1mt − mtV = Velocity of trajectory position relative to non-rotating inertial space defined as the time rate
of change of position evaluated in inertially non-rotating 0E coordinates: 0
0E
E d RVdt
≡ .
mV = Velocity vector V at the end of trajectory update cycle m.
APPENDIX F
USEFUL FORMULAS
First order expansion approximations are employed in Appendix G and H for matrices of the form [ ]aM aI E= + where I is the identity matrix a is an arbitrary scalar and matrix aE is
much smaller than I. Additionally, note that and that[ ] [ ]1aa aa II EE
− ++ I=
[ ][ ]a aa aI IE E− +
aaI E+
2a aa E E+
] 1−
I=
]1aa
−
. Equating the previous two expressions and multiplying on
the right by [ yields the useful formula:
(F-1) [ ] [ ] [ ] [1 2a a aa a aa aI II IaE E E EE E
− = − + ≈ −+ + Another useful identity derived in [?, Sect. 3.1.1] is [?, Eq. (3.1.1-38)] is: ( ) ( )1C V C VC−× = ×⎡⎣ ⎤⎦ (F-2) where C is an arbitrary direction cosine matrix, V is an arbitrary vector, and (V ×) is the cross-product skew-symmetric form of V as defined in the Notation section of this article.
APPLICABLE EQUATIONS FROM THE MAIN ARTICLE
The following equations taken from the main article are referenced in Appendices G and H to follow, and repeated next for convenient referencing.
3
( )( ) ( )
( )1
0 00 0 011 1
2
2 2
1 cos sin1 1
12
m mm m
m m
mmm var
mmm m m m m
Bvar Bvarvar varBvar Bvar
Vvar var varBvar mBvar var varvarmtBvar Bvartvar SF
Bvar E EE E EmVvarBvarvar var var var var
IG
dta
g gV V C G T
α αα α
α αα
υ
υ
−
−− −
⎛ ⎞− Δ Δ⎜ ⎟≡ + × + − ×Δ Δ⎜ ⎟Δ Δ⎝ ⎠Δ
≡Δ ∫
= + + +Δ
(3)
( )( )
( ) ( )( )
0 0 0 011 1
2
2
2 2
sin1 1 12
1 cos1 12
mm m
m
m
mmm m m m
Bvarvar Bvar
Rvar varBvarBvar varvarm
Bvarvar Bvar
varmBvar Bvarvar varm m
BvarE E E Em mRvarBvarvar var var var
IG
VR R C GT
αα
αα
αα
α α
υ−− −
⎛ ⎞Δ⎜ ⎟≡ + − ×Δ⎜ ⎟Δ⎝ ⎠Δ
⎡ ⎤⎛ ⎞− Δ⎢ ⎥⎜ ⎟
+ − ×Δ⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟Δ Δ⎢ ⎥⎝ ⎠⎣ ⎦
= + + Δ ( )0 01
212
6 m mE E
mvar varg gT T
−+ +
(4)
( ) ( )0 0 0
1 1
0 01
10
1 2/1
26
m m m
m
m m
E E Emvar var varBvar E
mRvarBvarvar mm E Emvar var
VR R TC G T
g g Tυ
− −
−
−
−
⎡ ⎤− −⎢ ⎥
=Δ ⎢ ⎥− +⎢ ⎥
⎣ ⎦
(9)
4
( )( )
( )
022 1
0 02 1 2 1 2 1 2 12 2 2 2
0 0 0 0 02 1 2 2 2 2 2 2 2 2
102 1 22 1
2 1
2 1
2 1
2
12 6
nn
n n n nn n
n n n n n
EEn Rvar VvarBvar Bvar nn
E En mden Rvar Vvar VvarBvar Bvar
E E E E Em n mnum var var var var var
n
C G C GA
C G G C G
n
B A T
V V VR RB T A
A I
−
− − − −− −
− − − −
−− −
−
−
−
≡
⎡ ⎤≡ + −⎢ ⎥⎣ ⎦
≡ − − − −
⎛+ −⎜⎝
T
( )0 0 02 2 1 2
2 12 12 1
2 1 22 1
1
52 6n n n
nnn
E E Enn mvar var var
Bvarnumdenvar
Ag g gIIA T
B Bυ
− −
−−−
−−
−
⎡ ⎤⎞ ⎛ ⎞+ − + −⎟ ⎜ ⎟⎢ ⎥⎠ ⎝ ⎠⎣ ⎦
=Δ
2
(31)
( ) ( )( )
( )( )
02 1 2 12 2
0 02 2 22 1 2 1
0 0 0 0 02 2 2 2 2 2 2 2 2
02 2
102 2 12 2
2
2
22
2
12 6
n nn
n n nn n
n n n n n
n
EEn Rvar Vvar VvarBvar Bvar nn
E En mden Rvar VvarBvar Bvar
E E E E Em nnum var var var var var
Ennvar
C G G C GA
C G C GB A
V V VR Rn m
T
B T A T
A gI IA
− −−
− −
− − −
−
−−≡ +
≡ −
≡ − − − −
⎛ ⎞+ − + −⎜ ⎟⎝ ⎠
0 01 2
222
2 2
1
52 6n n
nnn
E Enmvar var
Bvarnumdenvar
Ag gI T
B Bυ
− −
−
⎡ ⎤⎛ ⎞+ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
=Δ
2
( ) ( ) ( )
( ) ( )
0 09
0 0 01 1
0 0 01 1
2
2
For 9 :
For 0 9 :
12
For 0 :
1' '
2
m
m m m
m m m
E EBvar Bvar
Bvar Bvar BvarE E Evar var varBvar Bvar Bvar
Bvar Bvar BvarE E Evar var varBvar Bvar Bvar
m C C
m
I IC C C
m
I IC C C
α α α
α α
−
− −
− −
≤ − =
≥ > −
⎡ ⎤ ⎡ ⎤≈ + × + × ≈ +Δ Δ Δ⎢ ⎥ ×⎢ ⎥⎣ ⎦⎣ ⎦>
⎡ ⎤≈ + × + × ≈ +Δ Δ Δ⎢ ⎥⎣ ⎦
( )'α⎡ ⎤×⎢ ⎥⎣ ⎦
(32)
( ) ( )( ) ( )
1For 9 :2
1 1 1For 0 9 :2 2 3
1 1 1For 0 : ' '2 2 3
m m
m m
m m
Vvar Rvar
Bvar BvarVvar var Rvar var
Bvar BvarVvar var Rvar var
m I IG G
m I IG G
m I IG G
α α
α α
≤ − = =
⎡ ⎤≥ > − ≈ + × ≈ + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤> ≈ + × ≈ +Δ Δ ×⎢ ⎥⎣ ⎦
(33)
0 0 0
0 mmBrefE E E
mref Bvarvar m VR CT= + l (37)
5
0 Constantm
Eavgvar
g ≈ ≡ 0Eg (38)
APPENDIX G
ANALYTICAL DETAIL FOR METHOD 1 TEST EXAMPLE SOLUTION
This appendix provides analytical detail leading to the Method 1 test example solutions of (9) for specific force and resulting velocity, position thereof. First order expansion approximations are employed in the development, facilitated by approximation methods derived in the previous Appendix F.
Under the (37) and (38) example conditions, Method 1 specific force in (9) becomes
( )
( )( )
00 0 01 1
0 0 00 0
00 01 1
120
1
10
1 2
10
1
1 /2
1/1
2
m mm m
m
m m
Bvar E EE E Em m mRvarBvar var var avgvar varmm
BrefE E Em mref Bvar refE
mRvarBvar mm Bref EE Em mBvar avgvar
EBvarm
gVR RC G T T T
m mV l VCT TC G T
gl VC T T
υ− −
− −
−
−
−
−
−
−
⎛ ⎞= − − −Δ ⎜ ⎟⎝ ⎠
⎛ ⎞+ − −⎜ ⎟= ⎜ ⎟
⎜ ⎟− − −⎝ ⎠
= ( ) ( ) ( ) 00 0 0 01 1 0
21 /2m m m
Bref EE E E Em mRvar Bvar Bvar ref avgvarm
gl V VC CC G T T− −⎡ ⎤− − − −⎢ ⎥⎣ ⎦
mT
(G-1)
Substituting (G-1) in (3) with (38) for gravity then obtains for velocity 0
mEvarV :
( ) ( )( )
( )
00 0 011
0 01 00 0
1100 0
1 0
01
10
1
11
12
mmm m m
m mmmm
m
mm
Bvar EE E EmVvarBvar avgvar var var
BrefE EBvar BvarE EE E m mVvar RvarBvar Bvar avgvar mm
EE Emrefvar
EVvar RvarBvar m
gV V C G T
lC C
T gV C G C G TgV V T
I C G G
υ−−
−
−−
−
−
−
−
−−
= + +Δ
⎡ ⎤−⎢ ⎥
⎢ ⎥= + +⎢ ⎥− − −⎢ ⎥⎣ ⎦
= − ( )
( ) ( ) ( )
01
0 01 00
100
0
01
11 01 1
2
m
m mmm
E EBvar varm
BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avgm m
EEmref
VC
lC C
T gC G G C TgV T
−
−
−
−
−−−
⎡ ⎤⎢ ⎥⎣ ⎦
⎡ ⎤−⎢ ⎥
⎢ ⎥+ +⎢ ⎥
+ −⎢ ⎥⎣ ⎦
(G-2)
6
With (32), the term in (G-1) and (G-2) is to first order: ( )0 01m m
E EBvar BvarC C −
−
( ) ( )
( )
0 01
0 0 0 0 01 1 1 1
0 0 01 1
For 9 :
0
For 0 9 :
For 0 :
'
m m
m m m m m
m m m
E EBvar Bvar
Bvar BvarE E E E Evar varBvar Bvar Bvar Bvar Bvar
BvarE E EvarBvar Bvar Bvar
m
C C
m
IC C C C C
m
C C C
α α
α
−
− − − −
− −
≤ −
− =
≥ > −
⎡ ⎤− ≈ + × − =Δ Δ⎢ ⎥⎣ ⎦>
− ≈ ×Δ
×
)
(G-3)
Applying (33) and (F-1) of Appendix F, the term in (G-2) is to first order
accuracy: ( 1
mVvar RvarmG G−
( )
( ) ( ) ( )( ) ( ) ( )
11
11
For 9 :
1 22
For 0 9 :
1 1 12 32
1 1 1 1 12 2 22 3 2 3 6
m
m
Vvar Rvarm
Bvar BvarVvar Rvar var varm
Bvar Bvar Bvar Bvavar var var var
m
I I IG G
m
I IG G
I I I I
α α
α α α
−−
−−
≤ −
⎛ ⎞= =⎜ ⎟⎝ ⎠
≥ > −
⎧ ⎫⎡ ⎤ ⎡ ⎤≈ + × + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎩ ⎭⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎛ ⎞≈ + × − × ≈ + − × = +Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎝ ⎠⎣ ⎦
( )
( ) ( )1
For 0 :12 '6m
r
BvarVvar Rvar varm
m
IG G
α
α−
⎡ ⎤×Δ⎢ ⎥⎣ ⎦>
⎡ ⎤≈ + ×Δ⎢ ⎥⎣ ⎦
(G-4)
With (G-4) and (F-2) of Appendix F, the term is given by ( ) ( )01
11 01mm
EEVvar RvarBvar Bvarm mC G G C−
−−−
7
( ) ( )
( ) ( ) ( ) ( )( )( )
01
0 01 1
0 01 1
11 01
1 11 0 01 1
10
1
For 9 :
2
For 0 9 :
126
1 12 23 3
mm
mm m
m m
EEVvar RvarBvar Bvarm m
BvarE EE EVvar Rvar varBvar BvarBvar Bvarm m m
Bvar BvarEE Evar varBvar BvarBvarm
m
IC G G C
m
IC G G CC C
I IC CC
α
α
−
− −
− −
−−−
− −−− −
−
−
≤ −
=
≥ > −
⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦
= + × = +Δ ( )
( ) ( ) ( )0 01 1
11 01
For 0 :
12 '3mm m
BvarEE EVvar Rvar varBvar BvarBvarm m
m
IC G G CC
α
α− −
−−−
⎡ ⎤×Δ⎢ ⎥⎣ ⎦>
⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦
(G-5)
Substituting (G-3) – (G-5) in (G-2) and retaining only first order terms then obtains velocity
0m
EvarV generated by Method 1 under the test example conditions:
( )
0 00
00 0 0 01 10
0 00
For 9 :
For 8, 6, 4, 2, 0 :
126
For 7, 5, 3, 1:
m
m mm
m
E Erefvar
BrefBvar Bvar EE E E E
mvar varref Bvar Bvar avgvarm
E Erefvar
m V V
m
l gV V C C TT
m V V
α α− −
≤ − =
= − − − −
⎛ ⎞≈ + × − ×Δ Δ⎜ ⎟⎜ ⎟
⎝ ⎠
= − − − − =
(G-6) For 1, 3, 5, :m =
( )( )
( )
0 0 010
001
0 0 0 01 10
'2
1'
6For 2, 4, 6, :
12
6
mm
m
m mm
BrefBvar BvarE E Evar varref Bvarvar
m
Bvar Bvar EEmvar varBvar avg
BrefBvar BvarE E E Evar varref Bvar Bvarvar
m
lV V CT
gC T
m
lV V C CT
α α
α α
α α
−
−
− −
⎡ ⎤= + − ×Δ Δ⎢ ⎥
⎢ ⎥⎣ ⎦
⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦=
⎡ ⎤= + × − ×Δ Δ⎢ ⎥
⎢ ⎥⎣ ⎦
0Emavgg T
Derivation steps that led to (G-6) are discussed at the end of this appendix.
Substituting (G-6) into (G-1) with (33) for yields, to first order accuracy, the
corresponding Method 1specific force profile that generated the (G-6) velocity response: RvarmG
8
( )
( ) ( )
( )
0
0
10
9
10
1
For 9 :
For 8, 6, 4, 2, 0 :
123
For 7, 5, 3, 1:
223
m
m
m
Bvar E EmBvar avgvar
BrefBvar Bvar Bvar E E
mvar var Bvar avgvar mm
BrefBvar Bvar Bvar
var varvarm
gm C T
m
l gI C TT
m
l IT
υ
υ α α
υ α α
−
−
−
−
−
≤ − = −Δ
= − − − −
⎡ ⎤≈ × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦= − − − −
⎡ ⎤= − × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦( ) 0
10
1E E
mBvar avgmgC T−
(G-7) For 1, 3, 5, :m =
( )( ) ( )
( )( )
01
01
1
1
2 2'
1'
3For 2, 4, 6, :
2 2'
1 2 '3
m
m
BrefBvar Bvar Bvar
var varvarm
Bvar Bvar E Emvar var Bvar avgm
BrefBvar Bvar Bvar
var varvarm
Bvar Bvarvar var B m
lT
gI C T
m
lT
I
υ α α
α α
υ α α
α α
−
−
−
−
= − ×Δ Δ Δ
⎧ ⎫⎡ ⎤− − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭=
= − − ×Δ Δ Δ
⎧ ⎫⎡ ⎤− − − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭( ) 00E Emvar avggC T
Derivation steps that led to (G-7) are discussed at the end of this appendix.
Velocity Derivation Steps Leading To (G-6): Applying (G-3) – (G-5) in (G-2) and retaining only first order terms obtains velocity 0
mEvarV
generated by Method 1 under test example conditions for m cycles and lower: 8−
9
( ) ( )
( ) ( ) ( )
0 09 0
0 0 0898 9
0 08 9 00
8900
0
11 08 9
11 08 9
For 8 :
For 8 :
12
E Erefvar
EE E EVvar RvarBvar Bvarvar var
BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avg
EEmref
m V V
m
IV VC G G C
lC C
T gC G G C TgV T
−
−−− −
− −−−
−−− −
−−− −
< − =
= −
⎡ ⎤= −⎢ ⎥⎣ ⎦
⎡ ⎤−⎢ ⎥
⎢ ⎥+ +⎢ ⎥
+ −⎢ ⎥⎣ ⎦
( )09 0
13
BvarEvarBvar refI VC α
−⎡ ⎡= − + ×Δ⎢ ⎢⎣ ⎦⎣ ⎦
0E⎤⎤⎥⎥ (G-8)
( )
( )
0 00 0 09 9 0
00 0 09 90
1 123 2
12
6
BrefBvar Bvar E EE E E
m mvar varBvar Bvar ref avg avgm
BrefBvar Bvar EE E E
mvar varref Bvar Bvar avgm
l g gI VC C T TT
l gV C C TT
α α
α α
− −
− −
⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞
≈ + × − ×Δ Δ⎜ ⎟⎜ ⎟⎝ ⎠
The next velocity calculations use the following first order approximation development based on
in (32) multiplied by 0m
EBvarC ( 1Bvar
varI α−
⎡ + ×Δ⎢⎣ )⎤⎥⎦ with (F-1) from Appendix F.
( ) ( )( ) ( ) ( ) ( )
( ) ( )
0 0 01
0 0 01
0 01
1
Similarly: ' '
m m m
m m m
m m
Bvar BvarE E Evar varBvar Bvar Bvar
Bvar Bvar Bvar BvarE E Evar var var varBvar Bvar Bvar
Bvar BvarE Evar varBvar Bvar
I IC C C
IC C C
C C
α α
α α α
α α
−
−
−
−⎡ ⎤ ⎡ ⎤= + × ≈ − ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
⎡ ⎤× = − × × ≈ ×Δ Δ Δ⎢ ⎥⎣ ⎦
× ≈ ×Δ Δ
∴ αΔ (G-9)
Eq. (G-9) is used frequently in this appendix for Method 1 velocity derivations and in Appendix H for the Method 2 solution under test example conditions.
Applying (G-8) - (G-9) and (G-3) – (G-5) in (G-2) finds for : 7m = −
10
( ) ( )
( ) ( ) ( )
0 0 07 78 87 8
0 07 8 00
7 78 800
0
11 0
11 0
For 7 :
12
EE E EVvar RvarBvar Bvarvar var
BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avg
EEmref avg
m
IV VC G G C
lC C
T gC G G C TgV T
− −− −− −
− −− −− −
−−
−−
= −
⎡ ⎤= −⎢ ⎥⎣ ⎦
⎡ ⎤−⎢ ⎥
⎢ ⎥+ +⎢ ⎥
+ −⎢ ⎥⎣ ⎦
( )( )
0 0900
800
9
213 1
6
BrefBvarE Evarref BvarBvarE m
varBvarBvar EE
mvarBvar avg
lV CTI C
gC T
αα
α
−
−
−
⎡ ⎤⎛ ⎞+ ×⎢ ⎥Δ⎜ ⎟⎜ ⎟⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎝= − + ×Δ⎢ ⎥⎢ ⎥
⎠⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥− ×Δ⎢ ⎥⎣ ⎦
(G-10)
( )
( )
0 00 0 08 9 0
00 0 08 80
08
1 123 2
12
6
13
BrefBvar Bvar E EE E E
m mvar varBvar Bvar ref avg avgm
BrefBvar Bvar EE E E
mvar varref Bvar Bvar avgm
BvaEvarBvar
l g gI VC C T TT
l gV C C TT
C
α α
α α
− −
− −
−
⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞
≈ − − × + ×Δ Δ⎜ ⎟⎜ ⎟⎝ ⎠
− ( )( ) ( )
0 0 080 0
0 0 00 0 08 80
00
2 2
1 13 6
Brefr BvarE E E
varref Bvar refm
Bvar BvarE EE E Em mvar varBvar ref Bvar avg
Eref
lV VCT
g gVC CT T
V
α α
α α
−
− −
⎛ ⎞× + × +Δ Δ⎜ ⎟⎜ ⎟
⎝ ⎠
− + × − × +Δ Δ
=
Emg T
Similarly, the velocity solution for through is as previously derived for and in (G-8) and (G-10).
6m = − 0m = 8m = −7−
Following the procedure leading to (G-10) then obtains for m cycles 1 and 2:
11
( ) ( )
( ) ( ) ( )
0 0 01 10 01 0
0 01 0 00
1 10 000
0
11 0
11 0
For 1:
12
EE E EVvar RvarBvar Bvarvar var
BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avg
EEmref avg
m
IV VC G G C
lC C
T gC G G C TgV T
−−
−−
=
⎡ ⎤= −⎢ ⎥⎣ ⎦
⎡ ⎤−⎢ ⎥
⎢ ⎥+ +⎢ ⎥
+ −⎢ ⎥⎣ ⎦
( )( )
0 0100
000
1
21'
3 16
BrefBvarE Evarref BvarBvarE m
varBvarBvar EE
mvarBvar avg
lV CTI C
gC T
αα
α
−
−
⎡ ⎤⎛ ⎞+ ×⎢ ⎥Δ⎜ ⎟⎜ ⎟⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎝= − + ×Δ⎢ ⎥⎢ ⎥
⎠⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥− ×Δ⎢ ⎥⎣ ⎦
(G-11)
( )
( )
0 00 0 00 0 0
00 0 00 00
00
1 12 ' '3 2
12
6
13
BrefBvar Bvar E EE E E
m mvar varBvar Bvar ref avg avgm
BrefBvar Bvar EE E E
mvar varref Bvar Bvar avgm
BvarEvarBvar
l g gI VC C T TT
l gV C C TT
C
α α
α α
α
⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞
≈ − − × + ×Δ Δ⎜ ⎟⎜ ⎟⎝ ⎠
− Δ( )( ) ( )( )
0 0 000 0
0 0 00 0 00 00
0 0 00 00
2 2' '
1 1' '
3 6
12 ' '6
BrefBvarE E Evarref Bvar ref
m
Bvar BvarE EE E Em mvar varBvar ref Bvar avg
BrefBvar Bvar BvarE E Evar var varref Bvar Bvar
m
lV VCT
g gVC CT T
lV C CT
α
α α
α α α
⎛ ⎞× + × +Δ⎜ ⎟⎜ ⎟
⎝ ⎠
− + × − × +Δ Δ
⎡ ⎤= + − × −Δ Δ Δ⎢ ⎥
⎢ ⎥⎣ ⎦
Emg T
( ) 0Bvar Emvar g Tα⎡ ⎤− ×Δ⎢ ⎥⎣ ⎦
12
( ) ( )
( ) ( ) ( )
0 0 02 21 12 1
0 02 1 00
2 21 100
0
11 0
11 0
For 2 :
12
EE E EVvar RvarBvar Bvarvar var
BrefE EBvar BvarE EE m mVvar RvarBvar Bvar avg
EEmref avg
m
IV VC G G C
lC C
T gC G G C TgV T
−−
−−
=
⎡ ⎤= −⎢ ⎥⎣ ⎦
⎡ ⎤−⎢ ⎥
⎢ ⎥+ +⎢ ⎥
+ −⎢ ⎥⎣ ⎦
( ) ( )( )
0 0000
100
0
2 '1
'3 1
'6
BrefBvar BvarE Evar varref BvarBvarE m
varBvarBvar Bvar EE
mvar varBvar avg
lV CTI C
gC T
α αα
α α
⎡ ⎤⎡ ⎤+ − ×⎢ ⎥Δ Δ⎢ ⎥
⎡ ⎤ ⎢ ⎥⎡ ⎤ ⎢ ⎥⎣ ⎦= − + ×Δ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎡ ⎤⎢ ⎥− − ×Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
(G-12)
( )
( ) ( )
0 00 0 01 1 0
00 0 01 10
1 12 ' '3 2
1' '2
6
BrefBvar Bvar E EE E E
m mvar varBvar Bvar ref avg avgm
BrefBvar Bvar Bvar Bvar EE E Evar var var varref Bvar Bvar avg
m
l g gI VC C T TT
l gV C CT
α α
α α α α
⎡ ⎤⎛ ⎞⎧ ⎫⎡ ⎤+ + × × + − +⎢ ⎥Δ Δ⎜ ⎟⎨ ⎬⎢ ⎥ ⎜ ⎟⎣ ⎦⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤
≈ − − − × + − ×Δ Δ Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦
( )( ) ( )
0 0 0 01 10 0
0 0 00 0 01 10
0 010
1 2 2' '3
1 1' '
3 6
2
m
BrefBvar BvarE E E Evar varBvar ref Bvar ref
m
Bvar BvarE EE E Em mvar varBvar ref Bvaravg avg
BrefBvarE Evarref Bvar
m
T
lV VC CT
g gVC CT T
lV CT
α α
α α
α
Emg T
⎡ ⎤⎢ ⎥⎣ ⎦
⎛ ⎞− × + × +Δ Δ⎜ ⎟⎜ ⎟
⎝ ⎠
− + × − × +Δ Δ
⎛ ⎞= + × −Δ⎜ ⎟⎜ ⎟
⎝ ⎠( )00
116
Bvar EEmvarBvar avggC Tα ×Δ
Similarly, the velocity solution for is as previously derived for and 2 in (G-11) and (G-12). The combined result for
2m > 1m =0
mEvarV velocity is then as shown in (G-6).
Specific Force Derivation Steps Leading To (G-7):
Included in the following development for specific force is the observation from (33) with
(H-2) that to first order accuracy, ( ) ( )1 123
BvarRvar varm IG α
− ⎡ ⎤≈ − ×Δ⎢ ⎥⎣ ⎦for 0 9 , and m≥ > −
( ) (1 12 '3
BvarRvar varm IG α
− ⎡≈ − ×Δ⎢⎣ ⎦)⎤⎥ for m > 0. Applying (G-3) – (G-6) in (G-1) and retaining
only first order terms then obtains specific force m
BvarvarυΔ generated by Method 1 under test
example conditions for eac m cycle:
13
( )( )
00 01 0
0 0
10
1
1 10 0
9 9
For 8 :
12
112 2
m mBvar E EE E
mRvar refBvar avgvar varmm
E E E Em mBvar Bvaravg avg
m
gV VC G T
g gIC CT T
υ−
−
−
− −
− −
< −
⎛ ⎞= − − +Δ ⎜ ⎟⎝ ⎠
⎛ ⎞= − = −⎜ ⎟⎝ ⎠
(G-13) For 8, 6, 4, 2, 0 :m = − − − −
( )
( ) ( )
00 0 01 1 0
00 0 01 0 0
10
1
10
1
12
1 123 2
mm m
m
BrefBvar BvarE EE E E
mvarRvar Bvar refBvar avgvar varmm m
BrefBvar BvarE EE E E
mvar varBvar ref refBvar avgm m
l gV VCC G TT
l gI V VCC TT
υ α
α α
− −
−
−
−
−
−
⎡ ⎤⎛ ⎞= × − +⎢ ⎥Δ Δ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎛ ⎞⎡ ⎤= − × × − + −⎢Δ Δ⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎝ ⎠⎣
−
( ) ( ) 01
01
123
BrefBvar Bvar E E
mvar var Bvar avgmm
l gI C TT
α α−
−
⎤⎥
⎢ ⎥⎦
⎡ ⎤≈ × − − ×Δ Δ⎢ ⎥⎣ ⎦
14
( )
( ) ( )
00 0 01 1 0
0 01 2
10
1
10
1
For 7, 5, 3, 1:
12
2123
mm m
m m
BrefBvar BvarE EE E E
mvarRvar Bvar refBvar avgvar varmm m
BrefBvar BvarE Evar varBvar BvarBvar E m
var Bvarm
m
l gV VCC G TT
lC C
TI C
υ α
αα
− −
− −
−
−
−
−
= − − − −
⎡ ⎤⎛ ⎞= × − +⎢ ⎥Δ Δ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
× −Δ Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦
−
( ) 0 002
1 16 2m
Bref
m
Bvar E EEm mvarBvar avg
lT
g gC T T
α
α−
⎧ ⎫⎛ ⎞×⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎨ ⎬
⎪ ⎪+ × −Δ⎪ ⎪⎩ ⎭
( ) 02
10
12 4
m
Bref BrefBvar BvarE Evar varBvarBvarmm m
l lCC
T Tα
−
−
−≈ × − ×Δ αΔ (G-14)
( ) ( ) ( ) ( )( ) ( )( )
( )( )
0 002
001
0
1 10 0
1 1
1 10 0
1 1
10
1
1 13 3
123
13
m
m
Bvar BvarE E EEm mvar varBvarBvar Bvaravg avgm m
BrefBvar BvarE EE
mvar varBvarBvar Bvar avgm mm
Bvar Evar Bvar avgm
g gICC CT T
l gCC C TT
C
α α
α α
α
−
−
− −
− −
− −
− −
−
−
⎡ ⎤⎡ ⎤+ × − − ×Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥
⎣ ⎦
+ ×Δ
E
E
( )( )( ) ( )
( ) ( )
0
0 0
0
10
1
1 10 0
1 1
10
1
223
223
E E Em mBvar avgm
BrefBvar Bvar E E E E
m mvar var Bvar Bvaravg avgm mmBref
Bvar Bvar E Emvar var Bvar avgmm
g gCT T
l g gC CT TT
l gI C TT
α α
α α
−
−
− −
− −
−
−
−
= − × + × −Δ Δ
⎡ ⎤= − × − − ×Δ Δ⎢ ⎥⎣ ⎦
15
( )
( ) ( )
00 0 01 1 0
0 01 2
10
1
10
1
For 1, 3, 5, :
1'
2
2'12 '3
mm m
m m
BrefBvar BvarE EE E E
mvarRvar Bvar refBvar avgvar varmm m
Bref BrefBvar BvarE Evar varBvar BvarBvar E m m
var Bvarm
m
l gV VCC G TT
l lC C
T TI C
υ α
α αα
− −
− −
−
−
−
−
=
⎛ ⎞= × − +Δ Δ⎜ ⎟⎜ ⎟
⎝ ⎠
× − ×Δ Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦
−
( ) 0 002
1 16 2m
Bvar E EEm mvarBvar avgg gC T Tα
−
⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎨ ⎬⎪ ⎪
+ × −Δ⎪ ⎪⎩ ⎭
( ) 02
10
12 4'
m
Bref BrefBvar BvarE Evar varBvarBvarmm m
lCC
T Tα
−
−
−≈ × − ×Δ lαΔ (G-15)
( ) ( ) ( ) ( )( ) ( ) ( )( )
( )
0 002
001
1 10 0
1 1
1 10 0
1 1
1
1 1'
3 3
12 2'3
1'
3
m
m
Bvar BvarE E EEm mvar varBvarBvar Bvaravg avgm m
BrefBvar Bvar BvarE EE
mvar var varBvarBvar Bvar avgm mm
Bvarvar m
g gICC CT T
l gCC C TT
α α
α α α
α
−
−
− −
− −
− −
− −
−
⎡ ⎤⎡ ⎤+ × − − ×Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦E
E⎡ ⎤≈ − × + ×Δ Δ Δ⎢ ⎥
⎣ ⎦
+ ×Δ ( ) ( )( ) ( ) ( )
( )( ) ( )
0 0
0
0
10 0
1 1
10
1
10
1
12 2' '3
12 2' '3
E E E Em mBvar Bvaravg avgm
BrefBvar Bvar Bvar Bvar E E
mvar var var var Bvar avgmm
E EmBvar avgm
BrefBvar Bvar Bvar Bvarvar var var var
m
g gC CT T
l gC TT
gC T
l IT
α α α α
α α α α
−
− −
−
−
−
−
−
⎡ ⎤= − × + + ×Δ Δ Δ Δ⎢ ⎥⎣ ⎦
−
⎡= − × − − + ×Δ Δ Δ Δ⎢⎣ ( ) 01
01
E EmBvar avgm
gC T−
−⎧ ⎫⎤⎨ ⎬⎥⎦⎩ ⎭
16
( )
( ) ( )
00 0 011 1 0
01
021
10
10
For 2, 4, 6, :
1'
2
'
12 2' '3
m mmm m
m
mm
BrefBvar BvarE EE E E
mvarRvar Bvar refBvar avgvar varm
BrefBvarEvarBvar
m
Bvar Bvar BvaE Evar var varBvarBvar
m
l gV VCC G TT
lC
T
I CC
υ α
α
α α
−− −
−
−−
−
−
=
⎛ ⎞= × − +Δ Δ⎜ ⎟⎜ ⎟
⎝ ⎠
×Δ
⎡ ⎤= − × − −Δ Δ⎢ ⎥⎣ ⎦
−
( )( )
( ) ( )
0 002
021
10
1 1'
6 2
2 4' '
m
mm
Brefr
m
Bvar Bvar E EEm mvar varBvar avg avg
Bref BrefBvar Bvar BvarE Evar var varBvarBvar
m m
lT
g gC T T
l lCC
T T
α
α α
α α α
−
−−
−
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎡ ⎤⎪ ⎪×Δ⎢ ⎥⎨ ⎬
⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪
⎡ ⎤⎪ ⎪+ − × −Δ Δ⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭⎡ ⎤
≈ × − − ×Δ Δ Δ⎢ ⎥⎢ ⎥⎣ ⎦
( ) ( ){ }0021
101
'3 mm
Bvar BvarE EEmvar varBvarBvar avggCC Tα α
−−
−⎡+ −Δ Δ⎢⎣
⎤×⎥⎦ (G-16)
( ) ( )( )
( ) ( ) ( )( )( ) ( )
01
0011 1
01
10
1 10 0
1 10 0
1
1'
3
2 2'
1'
3
1'
3
m
mm m
m
Bvar E Emvar Bvar avg
BrefBvar Bvarvar var
m
Bvar BvarE EEmvar varBvarBvar Bvar avg
Bvar E E Emvar Bvar Bvaravg am
gI C T
lT
gCC C T
gC CT
α
α α
α α
α
−
−− −
−
−
− −
− −
−
⎡ ⎤− − ×Δ⎢ ⎥⎣ ⎦
≈ − − ×Δ Δ
⎧ ⎫⎡ ⎤+ − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
+ × −Δ
E
( ) ( ) ( )( )
( ) ( ) ( )
0
01
01
0
10
10
10
1
12 2 2' '3
12 2 2' '3
m
m
Emvg
BrefBvar Bvar Bvar Bvar E E
mvar var var var Bvar avgm
E EmBvar avg
BrefBvar Bvar Bvar Bvar E Evar var var var Bvar avgmm
g T
l gC TT
gC T
l gI CT
α α α α
α α α α
−
−
−
−
−
−
⎡ ⎤= − − × + − ×Δ Δ Δ Δ⎢ ⎥⎣ ⎦
−
⎧ ⎫⎡ ⎤= − − × − − − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭mT
The combined (G-13) – (G-16) result for
mBvarvarυΔ specific force is then as shown in (G-7).
17
APPENDIX H
ANALYTICAL DETAIL FOR METHOD 2 TEST EXAMPLE SOLUTION
This appendix provides analytical detail leading to the Method 2 test example solution of Eqs. (31) for specific force and resulting velocity, position thereof. SPECIFIC FORCE
Under test example conditions, Method 2 performance is demonstrated with gravity
approximated to be constant 0Eg as in (38). Then (31) simplifies to
( )( )
( )( )
022 1
0 02 1 2 1 2 1 2 12 2 2 2
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
102 1 22 1
2 1
2 1
2 1
2
2
nn
n n n nn n
n n n n n
EEn Rvar VvarBvar Bvar nn
E En mden Rvar Vvar VvarBvar Bvar
E E E E Em n mnum var var var var var
En avg
C G C GA
C G G C G
n
B A T
V V VR RB T A
IA
−
− − − −− −
− − − −
−− −
−
−
−
≡
⎡ ⎤≡ + −⎢ ⎥⎣ ⎦
≡ − − − −
+ −
T
2 12 12 1
2
1nnn
m
Bvarnumdenvar
g T
B Bυ −−−−=Δ
(H-1)
( ) ( )( )
( )( )
02 1 2 12 2
0 02 2 22 1 2 1
0 0 0 0 02 2 2 2 2 2 2 2 2
0
22
102 2 12 2
2
2
22
2
2
n nn
n n nn n
n n n n n
nn
EEn Rvar Vvar VvarBvar Bvar nn
E En mden Rvar VvarBvar Bvar
E E E E Em nnum var var var var var
En mavg
Bvarvar
C G G C GA
C G C GB A
V V VR Rn m
T
B T A T
gIA T
υ
− −−
− −
− − −
−
−−≡ +
≡ −
≡ − − − −
+ −
=Δ 21
nnumdenB B−
The development sequence for deriving
2 1nBvarvarυ
−Δ and
2nBvarvarυΔ in (H-1) for the test example is,
based on (32) and (33), first finding , 2 1nA − 2nA , then , , then 2 1ndenB − 2ndenB 02n
EvarR , 0
2nEvarV ,
then , , and from those, 2nnumB 2 1nnumB − 2nBvarvarυ
1−Δ and
2nBvarvarυΔ .
Finding 2 1nA − And 2nA
To derive , 2 1nA − 2nA , first expand their equations in (H-1):
18
(H-2)
( )( ) ( )
( ) ( )( )
022 1
022 1
02 1 2 12 2
02 12 2
022 2
102 1 22 1
11 02 2 1
102 2 12 2
10
2 12 2
nn
nn
n nn
nn
nn
EEn Rvar VvarBvar Bvar nn
EERvar VvarBvar Bvarn n
EEn Rvar Vvar VvarBvar Bvar nn
EERvar VvarBvar Bvar nn
EBvar
C G C GA
C G G C
C G G C GA
IC G C G
C
−
−
− −−
−−
−
−− −
−−−
−
−−−
−−
≡
=
≡ +
= +
= ( ) ( )1
11 02 1 2 2
ERvar Vvar Bvarn n
IG G C−
−−− −
+
To obtain in (H-2), apply (33) for and , yielding to first
order accuracy: (2
12nRvar Vvar nG G
−) 2Vvar nG 2nRvarG
( )
( ) ( ) ( )( ) ( ) ( )
( )
2 2
2 2
2 2
1
11
1
1For 5 :2
For 0 4 :
1 1 122 3
1 1 1 1 12 3 2 2 6
1 1For 0 :2 6
n n
n n
n n
Rvar Vvar
Bvar BvarRvar var varVvar
Bvar Bvar Bvarvar var var
Rvar vaVvar
n IG G
n
I IG G
I I I
n IG G
α α
α α α
−
−−
−
≤ − =
≥ ≥ −
⎡ ⎤ ⎡ ⎤≈ + × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡= + × − × ≈ − ×Δ Δ Δ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣
> ≈ −
⎤⎥⎦
( )'Bvarrα⎡ ⎤×Δ⎢ ⎥⎣ ⎦
(H-3)
Substituting (H-3) in (H-2) and dropping higher order terms then obtains , 2 1nA − 2nA :
19
( ) ( )( )( ) ( )
( )
02 22 1 2 1
02 1 2 1
02 1 2 12 2 2 2
02 2 2 2
022 1
11 02 1
10
11 02
10
2 1
For 5 :
1 12 2
1 32 2For 4 :
n nn n
n n
n nn n
n n
nn
EEn Rvar VvarBvar Bvar
EEBvar Bvar
EEn Rvar VvarBvar Bvar
EEBvar Bvar
En RBvar
n
C G G CA
I IC C
IC G G CA
I I IC C
n
CA
− −
− −
− −− −
− −
−
−−−
−
−−
−
−
≤ −
=
= =
= +
= + =
= −
= ( ) ( )( ) ( )( )( )
2 1
02 1 2 1
02 1 2 1
11 02
10
10
1 12 6
1 12 6
n
n n
n n
Evar Vvar Bvarn
Bvar EEvarBvar Bvar
Bvar EEvarBvar Bvar
G G C
IC C
I C C
α
α
−
− −
− −
−−
−
−
⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦⎡ ⎤
= − ×Δ⎢ ⎥⎣ ⎦
( ) ( )02 1 2 12 2 2 2
11 0232n nn n
EEn Rvar VvarBvar Bvar I IC G G CA − −− −
−−= + = (H-4)
( )( )( ) ( )
( ) ( )( )
02 1 2 1
02 1 2 12 2 2 2
02 2 2 2
02 2 2 2
102 1
11 02
10
1
For 0 3 :
1 12 6
1 12 6
3 12 18
n n
n nn n
n n
n n
Bvar EEn varBvar Bvar
EEn Rvar VvarBvar Bvar
Bvar EEvarBvar Bvar
BvarEvarBvar Bva
n
I C CA
IC G G CA
I IC C
I C
α
α
α
− −
− −− −
− −
− −
−−
−−
−
−
≥ ≥ −
⎡ ⎤= − ×Δ⎢ ⎥
⎣ ⎦
= +
⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦
= − ×Δ ( )+
( )( )( )( )
02 1 2 1
02 2 2 2
0
102 1
102
For 0 :
1 1'
2 6
3 1'
2 18
n n
n n
Er
Bvar EEn varBvar Bvar
Bvar EEn varBvar Bvar
C
n
I C CA
I C CA
α
α
− −
− −
−−
−
⎡ ⎤⎢ ⎥⎣ ⎦
>
⎡ ⎤= − ×Δ⎢ ⎥
⎣ ⎦⎡ ⎤
= − ×Δ⎢ ⎥⎣ ⎦
Summary of (H-4) Results:
20
( )( )02 1 2 1
2 1 2
102 1 2
1 3For 5 :2 2
For 4 :
1 1 32 6 2n n
n n
Bvar EEn nvarBvar Bvar
n I IA A
n
I IC CA Aα− −
−
−−
≤ − = =
= −
⎡ ⎤= − × =Δ⎢ ⎥
⎣ ⎦
( )( )( )( )
02 1 2 1
02 2 2 2
102 1
10
2
For 0 3 :
1 12 6
3 12 18
n n
n n
Bvar EEn varBvar Bvar
Bvar EEn varBvar Bvar
n
I C CA
I C CA
α
α
− −
− −
−−
−
≥ ≥ −
⎡ ⎤= − ×Δ⎢ ⎥
⎣ ⎦⎡ ⎤
= − ×Δ⎢ ⎥⎣ ⎦
(H-5)
( )( )( )( )
02 1 2 1
02 2 2 2
102 1
10
2
For 0 :
1 1'
2 6
3 1'
2 18
n n
n n
Bvar EEn varBvar Bvar
Bvar EEn varBvar Bvar
n
I C CA
I C CA
α
α
− −
− −
−−
−
>
⎡ ⎤= − ×Δ⎢ ⎥
⎣ ⎦⎡ ⎤
= − ×Δ⎢ ⎥⎣ ⎦
Finding 2 1ndenB − And 2ndenB
To determine and , substitute (33) and (H-4) into the , expressions in (H-2), and drop higher order terms:
2 1ndenB − 2ndenB 2 1ndenB − 2ndenB
( )0 02 1 2 1 2 1 2 12 2 2 2
0 0 02 2 2 2 2 2 9
0 02 2 22 1 2 1
2 1
2
For 4 :
1 12 2
n n n nn n
n n n
n n nn n
E En mden Rvar Vvar VvarBvar Bvar
E E E Em mBvar Bvar Bvar Bvar
E Enden Rvar VvaBvar Bvar
n
C G G C G
0 m
B A T
I I I IC C C CT T
C G CB A
− − − −− −
− − −
− −
−
< −
⎡ ⎤= + −⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞= + − = =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
= −( )T−
0 0 0 02 1 2 1 2 1 9
1 32 2n n n
mr
E E E Em mBvar Bvar Bvar Bvar
G T
I I IC C C CT T− − − −⎛ ⎞= − = − = −⎜ ⎟⎝ ⎠
mT
(H-6)
21
( )
( )( )0 0
2 1 2 1 2 1 2 12 2 2 2
0 02 2 2 1 2 22 1
02 2 2 2
2 1
10
1
For 4 :
1 1 12 2 6
112
n n n nn n
n n n
n n
E En mden Rvar Vvar VvarBvar Bvar
Bvar EE E EmvarBvar Bvar BvarBvar
EBvar B
n
C G G C GB A
I I I IC C CC T
IC
α
− − − −− −
− − −
− −
−
−
−
= −
⎡ ⎤= + −⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= + − − ×Δ⎨ ⎬⎢ ⎥⎜ ⎟
⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭
= + ( )
0n
T
−
( )( )0 02 1 2 22 1
10 0
n nnBvarE EE E
mvarBvar Bvarvar BvarC CC C Tα− −−
−⎡ ⎤×Δ⎢ ⎥
⎣ ⎦
( )09
112
BvarEmvarBvar IC Tα
−⎡= + ×Δ⎢⎣ ⎦
⎤⎥ (H-7)
( )( ) ( )
( )
0 02 2 22 1 2 1
0 02 1 2 1
09
2
1 1 3 12 3 2 2
712
n n nn n
n n
E En mden Rvar VvarBvar Bvar
Bvar BvarE Emvar varBvar Bvar
BvarEmvarBvar
C G C GB A
I I IC C T
IC T
α α
α
− −
− −
−
= −
⎧ ⎫⎡ ⎤ ⎡= + × − + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭⎡ ⎤= − + ×Δ⎢ ⎥⎣ ⎦
T
⎤⎥⎦
22
( )
( ) ( )( )( )
0 02 1 2 1 2 1 2 12 2 2 2
02 2
02 1 2 22 1
2 1
10
For 0 4 :
1 1 12 3 2
1 12 6
n n n nn n
n
n nn
E En mden Rvar Vvar VvarBvar Bvar
Bvar BvarEvar varBvar
Bvar EEvarBvar BvarBvar
n
C G G C GB A
I IC
I C C
α α
α
− − − −− −
−
− −−
−
−
≥ > −
⎡ ⎤= + −⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤ ⎡ ⎤+ × + + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭=
⎡ ⎤− − ×Δ⎢ ⎥
⎣ ⎦
T
( )0 12
mBvarEvar
TIC α⎡ ⎤+ ×Δ⎢ ⎥⎣ ⎦
( )( )( )
( )
02 2
02 1 2 22 1
02 2
10
512
112
12
n
n nn
n
BvarEvarBvar
mBvar EEvarBvar BvarBvar
BvarEmvarBvar
ICT
C C
IC T
α
α
α
−
− −−
−
−
⎧ ⎫⎡ ⎤+ ×Δ⎪ ⎪⎢ ⎥⎪ ⎪⎣ ⎦= ⎨ ⎬⎪ ⎪+ ×Δ⎪ ⎪⎩ ⎭
⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦
0EC (H-8)
( )( )
( )( ) ( )
0 02 2 22 1 2 1
02 1
0 02 2 2 12 2
02 1 2 1
2
10
1 12 3
3 1 12 18 2
16
n n nn n
n
n nn
n n
E En mden Rvar VvarBvar Bvar
BvarEvarBvar
mBvar BvarEE Evar varBvar BvarBvar
EBvar Bv
C G C GB A T
IC
TI IC CC
C
α
α α
− −
−
− −−
− −
−
= −
⎧ ⎫⎡ ⎤+ ×Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪= ⎨ ⎬⎡ ⎤ ⎡ ⎤⎪ ⎪− − × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭
− +=
( ) ( )( )
( )
0 02 1
02 2
02 1
34
112
12
n
n
n
Bvar BvarE Evar varar Bvar
mBvarEvarBvar
BvarEmvarBvar
C CT
C
IC T
α α
α
α
−
−
−
⎧ ⎫× − ×Δ Δ⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪+ ×Δ⎪ ⎪⎩ ⎭
⎡ ⎤= − + ×Δ⎢ ⎥⎣ ⎦
23
( )
( ) ( )( )( )
0 02 1 2 1 2 1 2 12 2 2 2
02 2
02 1 2 22 1
2 1
10
For 0 :
1 1 1' '
2 3 2
1 1'
2 6
n n n nn n
n
n nn
E En mden Rvar Vvar VvarBvar Bvar
Bvar BvarEvar varBvar
Bvar EEvarBvar BvarBvar
n
C G G C GB A
I IC
I C C
α α
α
− − − −− −
−
− −−
−
−
>
⎡ ⎤= + −⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤ ⎡ ⎤+ × + + ×Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭=
⎡ ⎤− − ×Δ⎢ ⎥
⎣ ⎦
T
( )( ) ( )
( )( )
0
0 0 02 2 2 2 2 2
0 02 1 2 22 1
10
1'
2
2 1' '
3 41
'12
n n n
n nn
mBvarEvar
Bvar BvarE E Evar varBvar Bvar Bvar
mBvar EE EvarBvar BvarBvar
TIC
C C CT
C CC
α
α α
α
− − −
− −−
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥+ ×Δ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
⎡ ⎤+ × − ×Δ Δ⎢ ⎥⎢ ⎥=⎢ ⎥+ ×Δ⎢ ⎥⎣ ⎦
( )02 2
1'
2nBvarE
mvarBvar IC Tα−⎡= + ×Δ⎢⎣ ⎦
⎤⎥ (H-9)
( )( )
( )( ) ( )
0 02 2 22 1 2 1
02 1
0 02 2 2 12 2
02 1 2
2
10
1 1'
2 3
3 1 1' '
2 18 2
16
n n nn n
n
n nn
n n
E En mden Rvar VvarBvar Bvar
BvarEvarBvar
mBvar BvarEE Evar varBvar BvarBvar
EBvar
C G C GB A T
IC
TI IC CC
C
α
α α
− −
−
− −−
− −
−
= −
⎧ ⎫⎡ ⎤+ ×Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪= ⎨ ⎬⎡ ⎤ ⎡ ⎤⎪ ⎪− − × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭
− +=
( ) ( )( )
( )
0 01 2 1
02 2
02 1
3' '
41
'12
1'
2
n
n
n
Bvar BvarE Evar varBvar Bvar
mBvarEvarBvar
BvarEmvarBvar
C CT
C
IC T
α α
α
α
−
−
−
⎧ ⎫× − ×Δ Δ⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪+ ×Δ⎪ ⎪⎩ ⎭
⎡ ⎤= − + ×Δ⎢ ⎥⎣ ⎦
Summary of (H-6) to (H-9) Results:
24
( ) ( )
0 02 1 29 9
0 02 1 29 9
For 4 :
For 4 :1 7
12 12
n n
n n
E Em mden denBvar Bvar
Bvar BvarE Em mden denvar varBvar Bvar
n
C CB T B T
n
I IC CB T B Tα α
− − −
− − −
< −
= = −
= −
⎡ ⎤ ⎡ ⎤= + × = − + ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
(H-10) For 0 4 :n≥ > −
( ) ( )
( ) ( )
0 02 1 22 2 2 1
0 02 1 22 2 2 1
1 12 2
For 0 :1 1
' '2 2
n nn n
n nn n
Bvar BvarE Em mden denvar varBvar Bvar
Bvar BvarE Em mden denvar varBvar Bvar
I IC CB T B T
n
I IC CB T B T
α α
α α
− − −
− − −
⎡ ⎤ ⎡= + × = − + ×Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣>
⎡ ⎤ ⎡= + × = − + ×Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎤⎥⎦
⎤⎥⎦
Finding The 0
2nEvarR And 0
2nEvarV Terms
Eq. (37) for 0
mEvarR with is used as the basis for evaluating the 2m = n 0
2nEvarR , 0
2nEvarV terms
in (H-1) under the example conditions.
( ) ( )
( )
0 0 022 0
0 01 10 0
2
0 0 0 01 10 0
0 02 1 2 10
0
2
2
1 1
2
2
nn
m mn m
m m
n n
BrefE E Emref Bvarvar
E Evar varE E
var varm
Bref BrefE E E Em mref Bvar ref Bvar
mBrefE E
Bvar BvarEref
m
n V lR CT
R RV V
T
m mV l VC CT T
T
lC CV
T
+ −
+ −
+ −
= +
−= =
+ + − − −=
−= +
l (H-11)
From (H-11) with (32) for , the 0
mEBvarC 0 0
2 2nE Evar varR R
−−
2nposition change in (H-1) becomes:
25
( ) ( )( )
0 0 02 2 2 0
0 0 0 0 02 8 9 2
0 0 0 0 02 8 9 9
2
2
For 4 : 2
For 4 :
12
12
n n
n n
n n
E E Emrefvar var
Bvar BvarE E E E Evar varBvar Bvar Bvar Bvar Bvar
Bvar BvarE E E E Evar varBvar Bvar Bvar Bvar Bvar
n VR R T
n
IC C C C C
C C C C C
α α
α
−
− − −
− − − −
< − − =
= −
⎡ ⎤= = + × + × =Δ Δ⎢ ⎥⎣ ⎦
− = − = × +Δ Δ
9−
( )( )
( ) ( )
0 0 0 0 0 0 08 92 2 2 8 10 0
0 090
2
2
122
For 0 4 :
n nBrefE E E E E E E
mref Bvar Bvarvar var var var
Bvar Bvar BrefE Em var varref Bvar
V lR R R R C CT
V lCT
n
α
α α
− −− − −
−
⎡ ⎤×⎢ ⎥⎣ ⎦
− = − = + −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
≥ > −
( ) ( )( ) ( )
0 02 2 1
0 02 1 2 2
2
2
12
12
n n
n n
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
IC C
IC C
α α
α α
−
− −
⎡ ⎤= + × +Δ Δ ×⎢ ⎥⎣ ⎦⎡ ⎤= + × +Δ Δ ×⎢ ⎥⎣ ⎦
(H-12)
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( )
0 02 2 2
02 2
02 2
2 2
2 2
2
1 12 2
1 12 2
2 2
n n
n
n
Bvar Bvar Bvar BvarE Evar var var varBvar Bvar
Bvar Bvar Bvar Bvar BvarEvar var var var varBvar
Bvar BvaEvar varBvar
I IC C
IC
IC
α α α α
α α α α α
α
−
−
−
⎡ ⎤ ⎡ ⎤= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
⎡ ⎤= + × + × + × + × + ×Δ Δ Δ Δ Δ⎢ ⎥⎣ ⎦
= + × +Δ
2
( )( ) ( )
( ) ( )( ) ( )
0 0 02 2 2 2 2
02 2
0 0 0 02 22 2 2 0
2
2
2
2 2
2
2 2
Fo
n n n
n
nn n
r
Bvar BvarE E Evar varBvar Bvar Bvar
Bvar BvarEvar varBvar
Bvar Bvar BrefE E E Em var varref Bvarvar var
I IC C C
C
V lR R CT
α
α α
α α
α α
− −
−
−−
⎡ ⎤× +Δ⎢ ⎥⎣ ⎦
⎡ ⎤− = + × + × + −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− ≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦
( ) ( )0 0 0 02 22 2 2 0
2
r 0 :
2 2 ' 'nn n
Bvar Bvar BrefE E E Em var varref Bvarvar var
n
V lR R CT α α−−
>
⎡ ⎤− ≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦
Summary of (H-12) Results:
26
( ) ( )
( ) ( )
0 0 02 2 2 0
0 0 0 098 10 0
0 0 0 02 22 2 2 0
2
2
For 4 : 2
For 4 :
122
For 0 4 :
2 2
n n
nn n
E E Emrefvar var
Bvar Bvar BrefE E E Em var varref Bvarvar var
Bvar BvarE E E Em var varref Bvarvar var
n VR R T
n
V lR R CT
n
VR R CT
α α
α α
−
−− −
−−
< − − =
= −
⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
≥ > −
⎡− ≈ + × + ×Δ Δ⎣
( ) ( )0 0 0 02 22 2 2 0
2
For 0 :
2 2 ' 'nn n
Bref
Bvar Bvar BrefE E E Em var varref Bvarvar var
l
n
V lR R CT α α−−
⎤⎢ ⎥
⎦>
⎡ ⎤− ≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦
(H-13)
The 0
2 2nEvarV
− and 0 0
2 2nE Evar varV V
−−
2nvelocity terms in (H-1) are obtained from (H-11) with
(32) for . First, 0m
EBvarC 0
2nEvarV is obtained. Then what follows is modified to get 0
2 2nEvarV
− and
0 02 2n n
EvarV V
−−
2Evar .
27
( )( )
( )
0 0 0 0 02 1 2 12 1 2 1 0
0 0 0 09 90 0
0 02 1 2 10 0
10 0
0 08 9
For 4 :
2
2 2
2For 4 :
n nn n
n n
BrefE E E E Emref Bvar Bvarvar var
BrefE E E Em mref Bvar Bvar ref
E Evar varE E
refvarm
BvarE EvarBvar Bvar
n
V lR R C CT
V l VC CT T
R RV V
Tn
IC C α
+ −+ −
− −
+ −−
− −
< −
− = + −
= + − =
−= =
= −
= + × +Δ ( )( ) ( )
( ) ( ) ( ) ( )
0 07 8
09
2
2
2 2
12
12
1 12 2
Bvarvar
Bvar BvarE Evar varBvar Bvar
Bvar Bvar Bvar BvarEvar var var varBvar
IC C
I IC
α
α α
α α α α
− −
−
⎡ ⎤×Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡ ⎤= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
( ) ( )09
22 2Bvar BvarE
var varBvar IC α α−⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
+ (H-14)
( ) ( )( ) ( )
( )( )
0 0 07 9 9
09
0 0 0 0 07 97 9 0
0 090
2
2
2
2 2
2
2
2 2
Bvar BvarE E Evar varBvar Bvar Bvar
Bvar BvarEvar varBvar
BrefE E E E Emref Bvar Bvarvar var
Bvar BvarE Em var varref Bvar
I IC C C
C
V lR R C CT
V CT
α α
α α
α α
− − −
−
− −− −
−
⎡ ⎤− = + × + × + −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ × + ×Δ Δ⎢ ⎥⎣ ⎦
− = + −
= + × +Δ Δ( )
( ) ( )0 0
7 90 0 098 0
2
2
Bref
E E Brefvar var Bvar BvarE E E
var varref Bvarvarm m
l
R R lV V CT T
α α− −−−
⎡ ⎤×⎢ ⎥⎣ ⎦
− ⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦
28
( ) ( )( ) ( )( ) ( )
( ) ( )
0 07 8
0 06 7
0 05 6
07
2
2
2
2
For 3 :
12
12
12
12
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
n
IC C
IC C
IC C
IC
α α
α α
α α
α α
− −
− −
− −
−
= −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡= + × + ×Δ Δ⎣
( ) ( )( ) ( ) ( ) ( )0
8
2
2 2
12
1 2 22
Bvar Bvarvar var
Bvar Bvar Bvar BvarEvar var var varBvar
I
I IC
α α
α α α α−
⎤ ⎡ ⎤+ × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦
⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎤+ ⎥⎦
( ) ( )08
2932
Bvar BvarEvar varBvar IC α α
−⎡= + × + ×Δ Δ⎢⎣ ⎦
⎤⎥ (H-15)
( ) ( )( )
( ) ( )
0 0 05 7 8
0 0 0 0 05 75 7 0
0 080
0 05 70
6 0
2
2
2 2
2
2 2 2
2
Bvar BvarE E Evar varBvar Bvar Bvar
BrefE E E E Emref Bvar Bvarvar var
Bvar Bvar BrefE Em var varref Bvar
E Evar varE
rvarm
C C C
V lR R C CT
V lCT
R RV
T
α α
α α
− − −
− −− −
−
− −−
⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦
− = + −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
−= = ( ) ( )0 0
8
22
BrefBvar BvarE Evar varef Bvar
m
lV CT
α α−⎡ ⎤+ × + ×Δ Δ⎢ ⎥⎣ ⎦
29
( ) ( )( ) ( )( ) ( )
( ) ( )
0 05 6
0 04 5
0 03 4
05
2
2
2
2
For 2 :
12
12
12
12
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
n
IC C
IC C
IC C
IC
α α
α α
α α
α α
− −
− −
− −
−
= −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡= + × + ×Δ Δ⎣
( ) ( )( ) ( ) ( ) ( )0
6
2
2 2
12
1 2 22
Bvar Bvarvar var
Bvar Bvar Bvar BvarEvar var var varBvar
I
I IC
α α
α α α α−
⎤ ⎡ ⎤+ × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦
⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎤+ ⎥⎦
( ) ( )06
2932
Bvar BvarEvar varBvar IC α α
−⎡= + × + ×Δ Δ⎢⎣ ⎦
⎤⎥ (H-16)
( ) ( )( ) ( )
( )
0 0 03 5 6
0 0 0 063 5 0
0 03 50 0 0
64 0
2
2
2
2 2
2 2 2
22
Bvar BvarE E Evar varBvar Bvar Bvar
Bvar Bvar BrefE E E Em var varref Bvarvar var
E Evar var Bvar BvarE E E
var varref Bvarvarm
C C C
V lR R CT
R RV V C
T
α α
α α
α
− − −
−− −
− −−−
⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
−= = + × +Δ Δ( )
Bref
m
lT
α⎡ ⎤×⎢ ⎥⎣ ⎦
30
( ) ( )( ) ( )( ) ( )
( ) ( )
0 03 4
0 02 3
0 01 2
03
2
2
2
2
For 1:
12
12
12
12
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
n
IC C
IC C
IC C
IC
α α
α α
α α
α α
− −
− −
− −
−
= −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡= + × + ×Δ Δ⎣
( ) ( )( ) ( ) ( ) ( )0
4
2
2 2
12
1 2 22
Bvar Bvarvar var
Bvar Bvar Bvar BvarEvar var var varBvar
I
I IC
α α
α α α α−
⎤ ⎡ ⎤+ × + ×Δ Δ⎢ ⎥ ⎢ ⎥⎦ ⎣ ⎦
⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎤+ ⎥⎦
( ) ( )04
2932
Bvar BvarEvar varBvar IC α α
−⎡= + × + ×Δ Δ⎢⎣ ⎦
⎤⎥ (H-17)
( ) ( )( ) ( )
( )
0 0 01 3 4
0 0 0 041 3 0
0 01 30 0 0
42 0
2
2
2
2 2
2 2 2
22
Bvar BvarE E Evar varBvar Bvar Bvar
Bvar Bvar BrefE E E Em var varref Bvarvar var
E Evar var Bvar BvarE E E
var varref Bvarvarm
C C C
V lR R CT
R RV V C
T
α α
α α
α
− − −
−− −
− −−−
⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
−= = + × +Δ Δ( )
Bref
m
lT
α⎡ ⎤×⎢ ⎥⎣ ⎦
31
( ) ( )( ) ( )
( ) ( )( ) ( )
0 01 2
0 00 1
0 01 0
01
2
2
2
2
For 0 :
12
12
1' '
2
12
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
n
IC C
IC C
IC C
IC
α α
α α
α α
α α
− −
−
−
=
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤= + × + ×Δ Δ⎢⎣
( ) ( )( ) ( )
( ) ( )( )
01
02
2
2
22
1' '
2
1' '
2
'1
12 '2
Bvar Bvarvar var
Bvar Bvar Bvar BvarEvar var var varBvar
Bvar Bvarvar var
Bvar BvarEvar varBvar Bvar
var va
I
IC
IIC
α α
α α α α
α αα α
α
−
−
⎡ ⎤+ × +Δ Δ ×⎥ ⎢ ⎥⎦ ⎣ ⎦
⎧ ⎫⎡ ⎤ ⎡= + + × + + ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭⎤⎥⎦
⎡ ⎤+ + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦ + +Δ( )Bvar
rα
⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬
⎡ ⎤⎪ ⎪×Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
( ) (02
212 2' '2
Bvar Bvar Bvar BvarEvar var var varBvar IC α α α α
−
⎧ ⎫⎡ ⎤ ⎡= + + × + +Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭) ⎤×⎥⎦ (H-18)
( )( ) ( )
( )( )
0 0 01 1 2
0 0 0 0 01 11 1 0
0 020
2 2
'
1 2'2
2
'2
Bvar Bvarvar var
E E EBvar Bvar Bvar Bvar Bvar Bvar
var var var
BrefE E E E Emref Bvar Bvarvar var
Bvar Bvarvar var
E Emref Bvar
C C C
V lR R C CT
V CT
α α
α α α
α α
− −
−−
−
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦− ≈ ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
− = + −
⎡ + ×Δ Δ⎣
= +( ) ( )
( )( ) ( )
0 01 10
0
0 020
2 2
2 2
1 2'2
2
'1
12 2'2
BrefBvar Bvar Bvarvar var var
E Evar varE
varm
Bvar Bvarvar var Bref
E Eref Bvar Bvar Bvar Bvar mvar var var
l
R RV
T
lV CT
α α α
α α
α α α
−
−
⎤⎢ ⎥⎦
⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
−=
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
32
( ) ( )( ) ( )( ) ( )
( ) ( )
0 01 0
0 02 1
0 03 2
01
2
2
2
2
For 1:
1' '
2
1' '
2
1' '
2
1' '
2
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
n
IC C
IC C
IC C
IC
α α
α α
α α
α α
=
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡= + × + ×Δ Δ⎣
( ) ( )21' '
2Bvar Bvarvar varI α α⎤ ⎡ ⎤+ × +Δ Δ⎢ ⎥ ⎢ ⎥
⎦ ⎣ ⎦×
( ) (01
22 2' 'Bvar BvarE
var varBvar IC α α⎡= + × + ×Δ Δ⎢⎣ ⎦
) ⎤⎥ (H-19)
( ) ( ) ( ) ( )( ) ( )
( ) ( )
00
00
0 0 03 1 0
0 03 1 0
2 2
2
2
1 2 2' ' ' '2
93 ' '2
2 2' '
2
Bvar Bvar Bvar BvarEvar var var varBvar
Bvar BvarEvar varBvar
Bvar BvarE E Evar varBvar Bvar Bvar
E Erefvar var
I IC
IC
C C C
R R
α α α α
α α
α α
⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦
− =
⎤⎥⎦
( )( ) ( )
( ) ( )
0 0 03 1
0 000
0 03 10 0 0
02 0
2
2
2 2 2' '
2' '2
BrefE E Em Bvar Bvar
Bvar Bvar BrefE Em var varref Bvar
E E Brefvar var Bvar BvarE E E
var varref Bvarvarm m
V lC CT
V lCT
R R lV V CT T
α α
α α
+ −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
− ⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦
33
( ) ( )( ) ( )( ) ( )
( ) ( )
0 03 2
0 04 3
0 05 4
03
2
2
2
2
For 2 :
1' '
2
1' '
2
1' '
2
1' '
2
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
n
IC C
IC C
IC C
IC
α α
α α
α α
α α
=
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡= + × + ×Δ Δ⎣
( ) ( )21' '
2Bvar Bvarvar varI α α⎤ ⎡ ⎤+ × +Δ Δ⎢ ⎥ ⎢ ⎥
⎦ ⎣ ⎦×
( ) (03
22 2' 'Bvar BvarE
var varBvar IC α α⎡= + × +Δ Δ⎢⎣ ⎦
) ⎤× ⎥ (H-20)
( ) ( ) ( ) ( )( ) ( )
( ) ( )
02
02
0 0 05 3 2
0 05 3 0
2 2
2
2
1 2 2' ' ' '2
93 ' '2
2 2' '
2
Bvar Bvar Bvar BvarEvar var var varBvar
Bvar BvarEvar varBvar
Bvar BvarE E Evar varBvar Bvar Bvar
E Erefvar var
I IC
IC
C C C
R R
α α α α
α α
α α
⎡ ⎤ ⎡= + × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− ≈ × + ×Δ Δ⎢ ⎥⎣ ⎦
− =
⎤⎥⎦
( )( ) ( )
( ) ( )
0 0 05 3
0 020
0 05 30 0 0
24 0
2
2
2 2 2' '
2' '2
BrefE E Em Bvar Bvar
Bvar Bvar BrefE Em var varref Bvar
E Evar var Bvar Bvar BrefE E E
var varref Bvarvarm
V lC CT
V lCT
R RV V C
T
α α
α α
+ −
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
− ⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦
l
Summary of (H-14) to (H-20) Results:
34
( ) ( )
( ) ( )
( ) ( )
0 010 0
0 0 098 0
0 0 086 0
0 0 064 0
2
2
2
2
2
E Erefvar
BrefBvar BvarE E Evar varref Bvarvar
mBref
Bvar BvarE E Evar varref Bvarvar
mBref
Bvar BvarE E Evar varref Bvarvar
V V
lV V CT
lV V CT
lV V C
α α
α α
α α
−
−−
−−
−−
=
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦ mT
( ) ( )0 0 042 0
22
BrefBvar BvarE E Evar varref Bvarvar
m
lV V CT
α α−−⎡= + × + ×Δ Δ⎢⎣ ⎦
⎤⎥ (H-21)
( )( ) ( )
( ) ( )
0 0 020 0
0 0 002 0
0 0 024 0
2 2
2
'1
12 2'2
2' '
Bvar Bvarvar var Bref
E E Eref Bvarvar Bvar Bvar Bvar m
var var var
BrefBvar BvarE E Evar varref Bvarvar
m
E E Evarref Bvarvar
lV V CT
lV V CT
V V C
α α
α α α
α α
−
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤= + × + ×Δ Δ⎢ ⎥⎣ ⎦
= + ( ) ( )22' 'Bvar Bvar Bref
var lα α⎡ ⎤× + ×Δ Δ⎢ ⎥⎣ ⎦
To obtain 0
2 2nEvarV
− in (H-1) as a function of as is 0
2 2nEBvarC −
02n
EvarV in (H-21), find
as a function of . First note that
02n
EBvarC
02 2n
EBvarC −
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
2 2
2 2
2 3 4
4
1 12 2
1 12 2
1 1 12 2 4
14
Bvar Bvar Bvar Bvarvar var var var
Bvar Bvar Bvar Bvar Bvarvar var var var var
Bvar Bvar Bvarvar var var
Bvarvar
I I
I
I
I
α α α α
α α α α α
α α α
α
⎡ ⎤ ⎡+ × + × − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎡ ⎤− × + × + × − × + ×Δ Δ Δ Δ Δ⎢ ⎥⎢ ⎥=⎢ ⎥+ × − × + ×Δ Δ Δ⎢ ⎥⎣ ⎦
= + ×Δ
+
3
⎤⎥⎦
( ) ( ) ( ) ( ) ( )2 21 1' ' ' ' '
2 2Bvar Bvar Bvar Bvar Bvarvar var var var varI Iα α α α α⎡ ⎤ ⎡ ⎤× + × − × + × = + ×Δ Δ Δ Δ Δ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
414
(H-22)
But
35
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
12 2
12 2
1 122
1 1' ' ' '22
Bvar Bvar Bvar Bvarvar var var var
Bvar Bvar Bvar Bvarvar var var var
I I I
I I I
α α α α
α α α α
−
−
⎡ ⎤ ⎡+ × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎡ ⎤ ⎡+ × + × + × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎤ =⎥⎦
⎤ =⎥⎦
(H-23)
Thus, to third order accuracy:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
12 2
12 2
112 2
11' ' ' '2 2
Bvar Bvar Bvar Bvarvar var var var
Bvar Bvar Bvar Bvarvar var var var
I I
I I
α α α α
α α α α
−
−
⎡ ⎤ ⎡+ × + × ≈ − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎡ ⎤ ⎡+ × + × ≈ − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
⎤⎥⎦
⎤⎥⎦
(H-24)
Applying (H-24) to (32) obtains:
( ) ( )( ) ( )
( ) ( )
0 01 9
0 01
0
0 01
12
2
2
For 9 :
For 0 9 :
12
12
1For 0 : ' '2
m
m m
m
m m
E EBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarEvar varBvar
Bvar BvarE Evar varBvar Bvar
m C C
m
IC C
IC
m IC C
α α
α α
α α
− −
−
−
−
≤ − =
≥ > −
⎡ ⎤≈ + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤> ≈ − × +Δ Δ ×⎢ ⎥⎣ ⎦
(H-25)
Successive application of (H-23) then finds as a function of : 0
2mEBvarC −
0m
EBvarC
36
( ) ( )
( ) ( )( ) ( ) ( )
0 01 9
0 09 8
0 08 7
06
2
2
2
For 8 :
12
For 2 8 :
12
1 12 2
mE EBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar BvarE Evar varBvar Bvar
Bvar Bvar BvarEvar var varBvar
m C C
IC C
m
IC C
I IC
α α
α α
α α α
− −
− −
− −
−
< − =
⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦
− ≥ > −
⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤= − × + × − × +Δ Δ Δ⎢ ⎥⎣ ⎦
( )( ) ( ) ( ) ( ) ( )
( ) ( )
06
06
2
2 2
2
1 12 2
2 2
Bvarvar
Bvar Bvar Bvar Bvar BvarEvar var var var varBvar
Bvar BvarEvar varBvar
IC
IC
α
α α α α α
α α
−
−
⎡ ⎤×Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ − × + × − × + × + ×Δ Δ Δ Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤= − × + ×Δ Δ⎢ ⎥⎣ ⎦
2
( ) (0 02
22 2
m mBvar BvarE Evar varBvar Bvar IC C α α
+ )⎡ ⎤= − × +Δ Δ ×⎢ ⎥⎣ ⎦
(H-26)
( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( )( )
0 01 0
01
01
2
2 2
2
For 1:
12
1 1' '
2 2
1'
2
'
Bvar BvarE Evar varBvar Bvar
Bvar Bvar Bvar BvarEvar var var varBvar
Bvar Bvar Bvarvar var varE
BvarBvar Bvarvar var
m
IC C
I IC
IC
α α
α α α α
α α α
α
−
= −
⎡ ⎤≈ − × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡= − × + × − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
− × + × − ×Δ Δ Δ=
+ ×Δ
⎤⎥⎦
( ) ( )( ) ( )
( ) ( )
01
0 02
2
2
2
1'
21
' '2
For 0 :
2 2' 'm m
Bvarvar
Bvar Bvar Bvar BvarEvar var var varBvar
Bvar BvarE Evar varBvar Bvar
IC
m
IC C
α α
α α α α
α α+
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥× + ×Δ Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤ ⎡= − + × + + ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭≥
⎡ ⎤= − × + ×Δ Δ⎢ ⎥⎣ ⎦
⎤⎥⎦
With (H-26) and (H-21), 0
2 2nEvarV
− can be found from (H-1) as a function of .0
2 2nEBvarC −
37
( ) ( )
( ) ( ) ( ) ( )
0 02 2 0
0 0 0 092 2 8 0
0 080
00
2
2 2
For 4 :
For 3 :
12
n
n
E Erefvar
BrefBvar BvarE E E Evar varref Bvarvar var
mBref
Bvar Bvar Bvar BvarE Evar var var varref Bvar
m
Eref
n V V
n
lV V V CT
lIV CT
V
α α
α α α α
−
−− −
−
≤ − =
= −
⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
≈ +
⎤⎥⎦
( )08
higher than second order termsBref
BvarEvarBvar
m
lC
Tα
−⎡ ⎤× +Δ⎢ ⎥⎣ ⎦
(H-27) For 2 :n = −
( ) ( )
( ) ( ) ( ) ( )
( )
0 0 0 082 2 6 0
0 060
0 060
2
2 2
2
2 2 2
higher than
n
BrefBvar BvarE E E Evar varref Bvarvar var
mBref
Bvar Bvar Bvar BvarE Evar var var varref Bvar
m
BvarE Evarref Bvar
lV V V CT
lIV CT
V C
α α
α α α α
α
−− −
−
−
⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
≈ + × +Δ
⎤⎥⎦
( )0 0 0 042 2 4 0
second order terms
For 1:
higher than second order termsn
Bref
m
BrefBvarE E E Evarref Bvarvar var
m
lT
n
lV V V CT
α−− −
⎡ ⎤⎢ ⎥⎣ ⎦
= −
⎡ ⎤= ≈ + × +Δ⎢ ⎥⎣ ⎦
(Continued)
38
(H-27) Continued
( )( ) ( )
( )( )
( )
0 0 020 0
0 000
2 2
2
For 0 :
'1
12 2'2
'212 2
Bvar Bvarvar var Bref
E E Eref Bvarvar Bvar Bvar Bvar m
var var var
Bvar BvarBvar var varvarE Eref Bvar Bvar
var
n
lV V CT
IV C
α α
α α α
α αα
α
−
=
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤+ ×⎡ ⎤ Δ Δ− ×Δ ⎢ ⎥⎣ ⎦⎢ ⎥= + ⎢ ⎥
⎢ ⎥+ ×Δ⎣ ⎦ ( ) ( )
( )( ) ( )( ) ( )
0 000
2 2
2 2
1 2'2
'
1 2'12
22 '
+ higher
Bref
Bvar Bvar Bvar mvar var var
Bvar Bvarvar var
Bvar Bvar BvarE E var var varref Bvar
Bvar Bvar Bvarvar var var
lT
V C
α α α
α α
α α α
α α α
⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦= + ⎩ ⎭
⎡ ⎤− × + ×Δ Δ Δ⎢ ⎥⎣ ⎦
( )( ) ( )( ) ( )
( ) ( ) ( )0 0
00
2 2
2
than second order terms
'
4 4' '1 12 2 4 '
+ highe
Bref
m
Bvar Bvarvar var
Bvar Bvar Bvar Bvarvar var var varE E
ref BvarBvar Bvar Bvar Bvarvar var var var
lT
V C
α α
α α α α
α α α α
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫× + × × + ×Δ Δ Δ Δ⎪ ⎪⎪ ⎪= + + ⎨ ⎬⎪ ⎪⎡ ⎤− × − × + ×Δ Δ Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
( ) ( ) ( )0 000
2 2
r than second order terms
1' '1
22
+ higher than second order terms
Bref
m
Bvar Bvar Bvar BvarE E var var var varref Bvar
lT
V Cα α α α
⎧ ⎫⎡ ⎤⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎪ ⎪⎢ ⎥⎢ ⎥⎣ ⎦= + ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭
(Continued)
39
(H-27) Continued
( )( ) ( )
( )( )
0 0 0 022 2 0 0
0 000
2 2
2
For 1:
'1
12 2'2
'212 2
n
Bvar Bvarvar var Bref
E E E Eref Bvarvar var Bvar Bvar Bvar m
var var var
Bvar BvBvar var varvarE Eref Bvar Bvar
var
n
lV V V CT
IV C
α α
α α α
αα
α
−−
=
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦= = + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
+⎡ ⎤ Δ− ×Δ⎢ ⎥
= + ⎢ ⎥⎢ ⎥+ ×Δ⎣ ⎦
( )( ) ( )
( ) ( ) ( )( ) ( )
0 000
2 2
2 2
1 2'2
1 2' '1 22
2 '
arBref
Bvar Bvar Bvar mvar var var
Bvar Bvar Bvar Bvar Bvarvar var var var varE E
ref BvarBvar Bvar Bvarvar var var
lT
V C
α
α α α
α α α α α
α α α
⎡ ⎤×Δ⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎧ ⎫⎡ ⎤ ⎡ ⎤+ × + + × − ×Δ Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭= +⎡− × + ×Δ Δ Δ⎢⎣
( )( ) ( )( ) ( )
( ) ( ) ( )0 0
00
2 2
2
+ higher than second order terms
'
4 4' '1 12 2 4 '
Bref
m
Bvar Bvarvar var
Bvar Bvar Bvar Bvarvar var var varE E
ref BvarBvar Bvar Bvar Bvarvar var var var
lT
V C
α α
α α α α
α α α α
⎤⎥⎦
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫× + × × + ×Δ Δ Δ Δ⎪ ⎪⎪ ⎪= + + ⎨ ⎬⎪ ⎡ ⎤− × − × + ×Δ Δ Δ Δ⎢ ⎥⎪ ⎣ ⎦⎩
( ) ( ) ( )0 000
0 0 0 002 2 2 0
2 2
+higher than second order terms
1' '1
22
+ higher than second order terms
For 2 :
n
Bref
m
Bvar Bvar Bvar Bvar BrefE E var var var varref Bvar
m
E E E Eref Bvarvar var
lT
lV CT
n
V V V C
α α α α
−
⎪⎪⎭
⎧ ⎫⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦= + ⎩ ⎭
=
= = + ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
0 020
0 020
2
2 2
2 2
2' '
2 2 2' ' ' '
2 2 + higher than sec' ' '
BrefBvar Bvarvar var
mBref
Bvar Bvar Bvar BvarE Evar var var varref Bvar
m
Bvar Bvar BvarE Evar var varref Bvar
lT
lIV CT
V C
α α
α α α α
α α α
⎡ ⎤× + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
= + × + × − ×Δ Δ Δ
⎤⎥⎦
( )0 020
ond order terms
+ higher than second order terms'
Bref
mBref
BvarE Evarref Bvar
m
lT
lV CT
α
⎡ ⎤⎢ ⎥⎣ ⎦
⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦ (Continued)
40
(H-27) Concluded
( ) ( )
( ) ( ) ( ) ( )
( )
0 0 0 022 2 4 0
0 040
0 040
2
2 2
For 3 :
2' '
2 2 2' ' ' '
hi'
n
BrefBvar BvarE E E Evar varref Bvarvar var
mBref
Bvar Bvar Bvar BvarE Evar var var varref Bvar
m
BvarE Evarref Bvar
n
lV V V CT
lIV CT
V C
α α
α α α α
α
−
=
⎡ ⎤= = + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡= + − × + × × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
≈ + × +Δ
⎤⎥⎦
( )0 0 02 22 2 0
gher than second order terms
For 3 :
higher than second order terms'nn
Bref
m
BrefBvarE E Evarref Bvarvar
m
lT
n
lV V CT
α−−
⎡ ⎤⎢ ⎥⎣ ⎦
>
⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦
Summary of (H-27) Results:
( )
( )
0 02 2 0
0 0 088 0
0 0 066 0
0 04 0
For 4 :
higher than second order terms
higher than second order terms
nE E
refvar
BrefBvarE E Evarref Bvarvar
mBref
BvarE E Evarref Bvarvar
m
Erefvar
n V V
lV V CT
lV V CT
V
α
α
−
−−
−−
−
≤ − =
⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦
≈ ( )04
higher than second order termsBref
BvarE EvarBvar
m
lV CT
α−⎡ ⎤+ × +Δ⎢ ⎥⎣ ⎦
( )0 0 022 0
higher than second order termsBref
BvarE E Evarref Bvarvar
m
lV V CT
α−−⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦
(H-28)
( ) ( ) ( )
( )
0 0 000 0
0 0 022 0
2 21' '1
22
+ higher than second order terms
+ higher than second order term'
Bvar Bvar Bvar Bvar BrefE E E var var var var
ref Bvarvarm
BvarE E Evarref Bvarvar
lV V CT
V V C
α α α α
α
⎧ ⎫⎡ ⎤⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎪ ⎪⎢ ⎥⎢ ⎥⎣ ⎦= + ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭
= + ×Δ
( )
( )
0 0 044 0
0 0 02 22 2 0
s
higher than second order terms'
For 3 :
higher than second order terms'nn
Bref
mBref
BvarE E Evarref Bvarvar
m
BrefBvarE E Evarref Bvarvar
m
lT
lV V CT
n
lV V CT
α
α−−
⎡ ⎤⎢ ⎥⎣ ⎦
⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦
>
⎡ ⎤≈ + × +Δ⎢ ⎥⎣ ⎦
To find 0 02 2n
E Evar varV V
−−
2n for (H-1), subtract (H-28) from (H-21):
41
( ) ( )
( ) ( )
0 0 0 010 12 0 0
0 0 0 098 10 0 0
09
0 0 0 086 8 0
2
2
0E E E Eref refvar var
BrefBvar BvarE E E E Evar varref Bvar refvar var
mBref
Bvar BvarEvar varBvar
m
BE E E Evarref Bvarvar var
V V V V
lV V V VCT
lC
T
V V V C
α α
α α
− −
−− −
−
−− −
− = − =
⎡ ⎤− = + × + × −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤= × + ×Δ Δ⎢ ⎥⎣ ⎦
− = +
0
( ) ( )
( )
( )
0 080
08
0 0 064 6 0
2
2
2
higher than second order terms
2 higher than second order terms
Brefvar Bvar
varm
BrefBvarE Evarref Bvar
mBref
BvarEvarBvar
m
E E Eref Bvvar var
lT
lV CT
lC
T
V V V
α α
α
α
−
−
−− −
⎡ ⎤× + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− − × +Δ⎢ ⎥⎣ ⎦
⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦
− = + ( ) ( )02
2Bref
Bvar BvarEvar varar
m
lC
Tα α⎡ ⎤× + ×Δ Δ⎢ ⎥
⎣ ⎦
( )0 060
higher than second order termsBref
BvarE Evarref Bvar
m
lV CT
α−⎡ ⎤− − × +Δ⎢ ⎥⎣ ⎦
(H-29)
( )
( ) ( )
( )
06
0 0 0 042 4 0
0 040
2
2
2 higher than second order term
2
higher than second order terms
BrefBvarEvarBvar
mBref
Bvar BvarE E E Evar varref Bvarvar var
m
BvarE Evarref Bvar
lC
T
lV V V CT
V C
α
α α
α
−
−− −
−
⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦
⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− − × +Δ⎢⎣ ⎦
( )( )
( ) ( )
04
0 0 0 020 2 0
00
2
2 2
2 higher than second order terms
'1
12 2'2
Bref
mBref
BvarEvarBvar
m
Bvar Bvarvar var Bref
E E E Eref Bvarvar var Bvar Bvar Bvar m
var var var
ref
lT
lC
T
lV V V CT
α
α α
α α α
−
−−
⎥
⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦
⎡ ⎤+ ×Δ Δ⎢ ⎥⎣ ⎦− = + ⎧ ⎫⎡ ⎤+ + × − ×Δ Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
− ( )
( )
02
02
higher than second order terms
1 higher than second order term'2
BrefBvarE EvarBvar
mBref
Bvar BvarEvar varBvar
m
lV CT
lC
T
α
α α
−
−
⎡ ⎤− × +Δ⎢ ⎥⎣ ⎦
⎡ ⎤= − × −Δ Δ⎢ ⎥⎣ ⎦
(Continued)
42
(H-29) Concluded
( ) ( )( ) ( ) ( )
0 0 0 002 0 0
0 000
0
2
2 2
2' '
1' '1
22
+ higher than second order terms
BrefBvar BvarE E E Evar varref Bvarvar var
m
Bvar Bvar Bvar Bvar BrefE E var var var varref Bvar
m
B
lV V V CT
lV CT
α α
α α α α
⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤⎡ ⎤+ × + × − ×Δ Δ Δ Δ⎪ ⎪⎢ ⎥⎢ ⎥⎣ ⎦− − ⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭
=
( ) ( )( ) ( ) ( )
( )
0
00
2
2 2
2
2' '
1 1' '
2 4higher than second order terms
1 7'
2 4
Bvar Bvarvar var
BrefBvar Bvar Bvar BvarEvar var var varvar
m
Bvar Bvar BvarE var var varBvar
lC
T
C
α α
α α α α
α α
⎧ ⎫× + ×Δ Δ⎪ ⎪⎪ ⎪
⎡ ⎤⎪ ⎪⎡ ⎤− + × − × − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪⎪ ⎪−⎪ ⎪⎩ ⎭
⎡ ⎤− × +Δ Δ Δ⎢ ⎥⎣ ⎦= ( ) ( )
( ) ( )
( )
0 0 0 024 2 0
0 020
2
2
1'
4higher than second order term
2' '
+ higher than second order terms'
Bvar Brefvar
m
BrefBvar BvarE E E Evar varref Bvarvar var
m
BvarE Evarref Bvar
lT
lV V V CT
V C
α α
α α
α
⎡ ⎤× + ×Δ⎢ ⎥⎢ ⎥
−⎢ ⎥⎣ ⎦
⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− − ×Δ⎢ ⎥⎣ ⎦
( )
( )
( ) ( )
( )
02
02
0 0 0 046 4 0
0 040
2
2
2
2 higher than second order term'
2 '
2' '
'
Bref
mBref
BvarEvarBvar
mBref
BvarEvarBvar
mBref
Bvar BvarE E E Evar varref Bvarvar var
m
BvarE Evarref Bvar
lT
lC
T
lC
T
lV V V CT
V C
α
α
α α
α
⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦
≈ ×Δ
⎡ ⎤− = + × + ×Δ Δ⎢ ⎥⎣ ⎦
− − × +Δ
( )04
2
higher than second order terms
2 '
Bref
mBref
BvarEvarBvar
m
lT
lC
Tα
⎡ ⎤⎢ ⎥⎣ ⎦
⎡ ⎤≈ ×Δ⎢ ⎥⎣ ⎦
Summary of (H-29) Results:
43
( ) ( )
( )
0 0 0 010 12 0 0
0 0 098 10
0 0 086 8
0 04 6
2
2
0
2 higher than second order terms
E E E Eref refvar var
BrefBvar BvarE E Evar varBvarvar var
mBref
BvarE E EvarBvarvar var
m
E Evar var
V V V V
lV V CT
lV V CT
V V
α α
α
− −
−− −
−− −
− −
− = − =
⎡ ⎤− = × + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦
− ( )06
22 higher than second order term
BrefBvarEvarBvar
m
lC
Tα
−⎡ ⎤= × −Δ⎢ ⎥⎣ ⎦
( )0 0 042 4
22 higher than second order terms
BrefBvarE E EvarBvarvar var
m
lV V CT
α−− −⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦
(H-30)
( )( ) ( ) ( )
0 0 020 2
0 0 002 0
2 2
1 higher than second order term'2
1 7 1' '
2 4 4higher than second order
BrefBvar BvarE E Evar varBvarvar var
m
Bvar Bvar Bvar BvarE E E var var var var
Bvarvar var
lV V CT
V V C
α α
α α α α
−−⎡ ⎤− = − × −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− × + × + ×Δ Δ Δ Δ⎢ ⎥⎣ ⎦− =−
( )
( )
0 0 024 2
0 0 046 4
2
2
term
2 higher than second order term'
2 higher than second order terms'
Bref
m
BrefBvarE E EvarBvarvar var
mBref
BvarE E EvarBvarvar var
m
lT
lV V CT
lV V CT
α
α
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦
⎡ ⎤− = × −Δ⎢ ⎥⎣ ⎦
Finding 2 1nnumB − And 2nnumB
To find and , substitute , 2 1nnumB − 2nnumB 2 1nA − 2nA from (H-5), 0 02 2n
E Evar varR R
−−
2n from (H-
13), 02 2n−
EvarV from (H-28), and 0 0
2nEar −2n
Evar v−
2V V from (H-30) into the (H-1) and
equations: 2 1nnumB − 2nnumB
( )( )
( )( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0 0
0 0 0 0 02 2 2 2 2 2 2 2 2
0 0
2 1
2 22 1
2
2 22
For 4 :
2
2
2
2
n n n n n
n n n n m n
E E E E Em nnum var var var var var
E En m mavg
E E E E Em n mnum var var var var var
E En m mavg
n
V V VR Rn mB T A T
g gIA T T
V V VR RB T A
g gIA T T
− − − −
− − −
−
−
< −
≡ − − − −
+ − = −
≡ − − − −
+ − =
T
(H-31)
44
( )( )
( ) ( )( )( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
0 0 090 0
09 9
2 1
22 1
2
10
For 4 :
2
2
2 2
1 12 6
n n n n n nE E E E E
m n mnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar EEvarBvar Bvar
n
V V VR RB T A
gIA T
V lCT T
I C C
α α
α
− − − −
−
− −
−
−
−
= −
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
⎡− − ×Δ
⎣
T
V
( ) ( )
( )( )( ) ( )
( )
09
009 9
0 09 9
009
2
120
2
16
12
16
Bvar Bvar BrefEvar varBvar
Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar BrefE Evar varBvar Bvar
Bvar EEmvarBvar avg
lC
gI C C T
l lC C
gI C T
α α
α
α α
α
−
− −
− −
−
−
⎤ ⎡ ⎤× + ×Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦⎦
⎡ ⎤− + ×Δ⎢ ⎥⎣ ⎦
≈ × − ×Δ Δ
⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
( ) ( ) 00 09 9
21 12 6
Bvar Bref Bvar EE Emvar varBvar Bvar avggIlC C Tα α
− −⎧ ⎡ ⎤= × − + ×Δ Δ⎨ ⎢ ⎥⎣ ⎦⎩ ⎭
⎫⎬ (H-32)
( )( )
( ) ( ) ( ) ( )
( )
0 0 0 0 02 2 2 2 2 2 2 2 2
0
0 09 9
0
09
2
22
2 2
2
2
2
32
n n n n m nE E E E E
m n mnum var var var var var
En mavg
Bvar Bvar Bref Bvar Bvar BrefE Evar var var varBvar Bvar
Emavg
Bvar BEvarBvar
V V VR RB T A
gIA T
l lC C
g T
C
α α α α
α
− − −
− −
−
≡ − − − −
+ −
⎡ ⎤ ⎡= × + × − × + ×Δ Δ Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
+
≈ ×Δ
T
⎤⎥⎦
( )( )
009
009
2
2
32
12
ref Bvar Bref EEmvarBvar avg
Bvar Bref EEmvarBvar avg
gl lC T
glC T
α
α
−
−
− × +Δ
= − × +Δ
45
( )( )
( ) ( )
( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
0 0 080 0
08
7
2 1
22 1
2
For 3 :
2
2
2 2 2
2
16
n n n n nE E E E E
m nnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
BrefBvarEvarBvar
m
Bva
n
V V VR Rn mB T A T
gIA T
V lCT T
lC
T
I
α α
α
− − − −
−
−
−
−
−
= −
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− −
V
( )( ) ( )
( )( )( ) ( )
0 087
007 7
00 08 8
1 20
120
2 2
16
16
Bvar Bvar BrefEE Evar varr BvarBvar
Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar EE Emvar varBvar Bvar avg
lC CC
gI C C T
gIlC C T
α α
α
α α
−−
− −
− −
−
−
⎡ ⎤× ×Δ Δ⎢ ⎥
⎣ ⎦⎡ ⎤
− + ×Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤= × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ (H-33)
( )( )
( ) ( )( )
( )
0 0 0 0 02 2 2 2 2 2 2 2 2
0
0 0 080 0
08
08 2
2
22
2
1
2
2
2 2 2
2
1318
n n n n m
n
E E E E Em nnum var var var var var
En m
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar BrefEvarBvar
BvarEvarBvar
V V VR Rn mB T A T
gIA T
V lCT T
lC
I C
α α
α
α
− − −
−
−
−
−
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− − ×Δ ( )
V
( )
( )( )( ) ( )
082
008 8
00 08 8
20
120
2 2
16
16
Bvar BrefE EvarBvarBvar
Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar EE Emvar varBvar Bvar avg
lCC
gI C C T
gIlC C T
α
α
α α
−−
− −
− −
−
⎡ ⎤×Δ⎢ ⎥
⎣ ⎦⎡ ⎤
+ − ×Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
46
( )( )
( ) ( )
( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
0 0 02 20 0
02 2
2 1
22 1
2
For 0 > 3 :
2
2
2 2 2
2
1
n n n n n
n
n
E E E E Em nnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
BrefBvarEvarBvar
m
n
V V VR Rn mB T A T
gIA T
V lCT T
lC
T
I
α α
α
− − − −
−
−
−
−
> −
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− −
V
( )( ) ( )
( )( )( ) ( )
0 02 1 2 22 1
002 1 2 1
00 02 2 2 2
1 20
120
2 2
6
16
16
n nn
n n
n n
Bvar Bvar BrefEE Evar varBvar BvarBvar
Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar EE Emvar varBvar Bvar avg
lC CC
gI C C T
gIlC C T
α α
α
α α
− −−
− −
− −
−
−
⎡ ⎤× ×Δ Δ⎢ ⎥
⎣ ⎦⎡ ⎤
− + ×Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤≈ × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ (H-34)
( )( )
( ) ( )( )
0 0 0 0 02 2 2 2 2 2 2 2 2
0
0 0 02 20 0
02 2
02 2
2
22
2
2
2
2 2 2
2
1318
n n n n m
n
n
n
E E E E Em nnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar BrefEvarBvar
BvEvarBvar
V V VR Rn mB T A T
gIA T
V lCT T
lC
I C
α α
α
− − −
−
−
−
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− −
V
( )( ) ( )
( )( )( ) ( )
02 22 2
002 2 2 2
00 02 2 2 2
1 20
120
2 2
16
16
nn
n n
n n
ar Bvar BrefE EvarBvarBvar
Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar EE Emvar varBvar Bvar avg
lCC
gI C C T
gIlC C T
α α
α
α α
−−
− −
− −
−
−
⎡ ⎤× ×Δ Δ⎢ ⎥
⎣ ⎦⎡ ⎤
+ − ×Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
47
( )( )
( ) ( )( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
0 0 020 0
02
1
2 1
22 1
2
For 0 :
2
2
2 2 2
2
1 14 6
n n n n n nE E E E E
m n mnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar BrefEvarBvar
Bvar
n
V V VR RB T A
gIA T
V lCT T
lC
I
α α
α
− − − −
−
−
−
−
−
=
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− −
T
V
( )( ) ( )( )
( ) ( )
0 021
001
0 02 2
10
2
'
16
1 1'
4 6
BrefBvar Bvar BvarEE Evar var varBvarBvar
m
Bvar EEmvarBvar avg
Bvar Bvar Bref BvarE Evar var varBvar Bvar a
lC CC
T
gI C T
IlC C
α α α
α
α α α
−−
−
− −
−⎡ ⎤ ⎡ ⎤× −Δ Δ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎧ ⎫⎡ ⎤⎡ ⎤≈ − − × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
×
0 2Emvgg T
(H-35)
( )( )
( ) ( )( )
( )
0 0 0 0 02 2 2 2 2 2 2 2 2
0
0 0 020 0
02
02
2
22
2
2
2
2 2 2
2
3 14 18
n n n n m nE E E E E
m n mnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar BrefEvarBvar
BvarEvarBvar
V V VR RB T A
gIA T
V lCT T
lC
I C
α α
α
α
− − −
−
−
−
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− − ×Δ ( )
T
V
( )( )
( )( )( )
022
002 2
02
1 20
2
120
'
12'12
2
16
3'
4
Bvar Bvarvar var
BrefE E Bvar BvarBvarBvar var var
Bvarvar
Bvar E EEmvarBvar Bvar avg
Bvar BvarEvar varBvar
lCC
gI C C T
C
α α
α α
α
α
α
−−
− −
−
−
−
⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤ ⎧ ⎫⎡ ⎤+ ×Δ Δ⎢ ⎥ ⎪ ⎪⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦ + ⎨ ⎬
⎪ ⎪− ×Δ⎪ ⎪⎩ ⎭
⎡ ⎤+ − ×Δ⎢ ⎥⎣ ⎦
≈ − −Δ Δ( ) ( ) 002
216
Bref Bvar EEmvarBvar avggIl C Tα α
−⎧ ⎫⎡ ⎤⎡ ⎤× + − ×Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
Note: Second order terms in (H-35) are neglected. If second order terms were included,
( )( 'Bvar Bvarvar varα α×Δ Δ )× products would appear which are too difficult to explain. For simplification, this
article only carries the highest order ( )Bvarvarα ×Δ or ( 'Bvar
varα ×Δ ) products.
48
( )( )
( ) ( )( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
0 0 000 0
00
2 1
22 1
2
For 1:
2
2
2 2 2' '
'
n n n n nE E E E E
m nnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar Bvarvar var v
EBvar
n
V V VR Rn mB T A T
gIA T
V lCT T
C
α α
α α
− − − −−
−
=
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤+ × −Δ Δ⎢ ⎥⎣ ⎦−
V
( ) ( )( ) 2
'
1'
2
Bvar Bvar Bvarar var var
Bref
Bvar Bvarvar var
lα α α
α α
⎧ ⎫⎡ ⎤× + ×Δ Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎨ ⎬
⎡ ⎤⎪ ⎪+ + ×Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
( )( )( )
( )( )
0 01 01
1 20
2
'
1 1 7' '
4 6 212
Bvar Bvarvar var
Bvar Bvar BrefEE Evar varBvar BvarBvar
Bvarvar
I lC CC
α α
α α
α
−
⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤ ⎪ ⎪− − × + ×Δ Δ⎨ ⎬⎢ ⎥
⎣ ⎦ ⎪ ⎪⎪ ⎪
+ ×Δ⎪ ⎪⎩ ⎭
(H-36)
( )( )( ) ( )
( ) ( )
001 1
0 00 0
00 00 1
120
2
1'
6
2 ' '
1 1' '
4 6
Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar Bvar BrefE Evar var varBvar Bvar
Bvar Bvar Bref Bvar EE Emvar var varBvar Bvar
gI C C T
l lC C
gIlC C
α
α α α
α α α
−⎡ ⎤− + ×Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ × − + ×Δ Δ Δ⎢ ⎥⎣ ⎦⎧ ⎫⎡ ⎤⎡ ⎤− − × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
( ) ( ) 00 00 0
23 1' '
4 6Bvar Bvar Bref Bvar EE E
mvar var varBvar Bvar
T
gIlC C Tα α α⎧ ⎫⎡ ⎤⎡ ⎤= − × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
(Continued)
49
(H-36) Concluded
( )( )
( ) ( )( ) ( )
0 0 0 0 02 2 2 2 2 2 2 2 2
0
0 0 000 0
00
2
22
2
2
2
2 2 2' '
'
n n n n m nE E E E E
m n mnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar Bvar Bvar Bvvar var var var
EBvar
V V VR RB T A
gIA T
V lCT T
C
α α
α α α
− − −≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤+ × − ×Δ Δ Δ⎢ ⎥⎣ ⎦−
T
V
( )( )
( )( )( )
( )( )
0 00 00
2
1 20
2
'
1'
2
'
3 1 1 7' '
2 18 2 212
ar Bvarvar
Bref
Bvar Bvarvar var
Bvar Bvarvar var
Bvar Bvar BEE Evar varBvar BvarBvar
Bvarvar
l
I C CC
α α
α α
α α
α α
α
−
⎧ ⎫⎡ ⎤+ ×Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎪ ⎪⎨ ⎬
⎡ ⎤⎪ ⎪+ + ×Δ Δ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤ ⎪ ⎪− − × + ×Δ Δ⎨ ⎬⎢ ⎥
⎣ ⎦ ⎪ ⎪⎪ ⎪
+ ×Δ⎪ ⎪⎩ ⎭
( )( ) ( )
( ) ( )
000
0 00 0
00 00 0
21'
6
2 ' '
3 1' '
4 6
ref
Bvar EEmvarBvar avg
Bvar Bref Bvar Bvar BrefE Evar var varBvar Bvar
Bvar Bvar Bref Bvar EE Evar var varBvar Bvar avg
l
gI C T
l lC C
IlC C
α
α α α
α α α
⎧ ⎫⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎡ ⎤≈ × − + ×Δ Δ Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤⎡ ⎤− − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
( ) ( ) 00 00 0
2
21 1' '
4 6
m
Bvar Bvar Bref Bvar EE Emvar var varBvar Bvar avg
g T
gIlC C Tα α α⎧ ⎫⎡ ⎤⎡ ⎤= − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
50
( )( )
( ) ( )( )
0 0 0 0 02 1 2 2 2 2 2 2 2 2
0
0 0 02 20 0
02 2
2 1
22 1
2
For 1:
2
2
2 2 2' '
2 '
1 12
n n n n n
n
n
E E E E Em nnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar BrefEvarBvar
n
V V VR Rn mB T A T
gIA T
V lCT T
lC
I
α α
α
− − − −
−
−
−
−
>
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− −
V
( )( ) ( )( )
( ) ( )
0 02 1 2 22 1
002 1
00 02 2 2 1
2
1 20
2
2 2
2' '6
1'
61
' '6
n nn
n
n n
n
Bvar Bvar BrefEE Evar varBvar BvarBvar
Bvar EEmvarBvar avg
Bvar Bref Bvar EE Emvar varBvar Bvar avg
lC CC
gI C T
gIlC C T
α α
α
α α
− −−
−
− −
−
−⎡ ⎤× ×Δ Δ⎢ ⎥
⎣ ⎦⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎧ ⎫⎡ ⎤≈ × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
≈ ( ) ( ) 00 02 2 2
2 21' '
6 nBvar Bref Bvar EE E
mvar varBvar Bvar avggIlC C Tα α−
⎧ ⎡ ⎤× − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎫
(H-37)
( )( )
( ) ( )( )
0 0 0 0 02 2 2 2 2 2 2 2 2
0
0 0 02 20 0
02 2
02 2
2
22
2
2
2
2 2 2' '
2 '
3 12 18
n n n n m n
n
n
n
E E E E Em n mnum var var var var var
En mavg
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar BrefEvarBvar
EvBvar
V V VR RB T A
gIA T
V lCT T
lC
I C
α α
α
− − −
−
−
−
≡ − − − −
+ −
⎡ ⎤= + × + × −Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
− −
T
V
( )( ) ( )( )
( ) ( )
02 22 2
002 2
00 02 2 2 2
1 20
2
2 2
2' '
1'
61
' '6
nn
n
n n
Bvar Bvar BrefE Ear varBvarBvar
Bvar EEmvarBvar avg
Bvar Bref Bvar EE Emvar varBvar Bvar avg
lCC
gI C T
gIlC C T
α α
α
α α
−−
−
− −
−⎡ ⎤× ×Δ Δ⎢ ⎥
⎣ ⎦⎧ ⎫⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎧ ⎫⎡ ⎤≈ − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
Summary of (H-31) – (H-37) Results:
51
( ) ( )( )
0 02 1 2
00 02 1 9 9
002 9
02 1 8
2 2
2
2
2
For 4 :
For 4 :1 12 6
12
For 3 :
n n
n
n
n
E Em mnum num
Bvar Bref Bvar EE Emnum var varBvar Bvar avg
Bvar Bref EEmnum varBvar avg
Enum vBvar
n
g gB T B Tn
gIlC CB T
glCB T
n
CB
α α
α
−
− − −
−
− −
< −
= − =
= −
⎧ ⎫⎡ ⎤= × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
= − × +Δ
= −
= ( ) ( )( ) ( )
008
00 02 8 8
2
2 2
1616n
Bvar Bref Bvar EEmar varBvar avg
Bvar Bref Bvar EE Emnum var varBvar Bvar avg
gIl C T
gIlC CB T
α α
α α
−
− −
⎧ ⎫⎡ ⎤× − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
(H-38) For 0 > 3 :n > −
( ) ( )( ) ( )
00 02 1 2 2 2 2
00 02 2 2 2 2
02 1 2
2 2
2 2
1616
For 0 :14
n n n
n n n
n
Bvar Bref Bvar EE Emnum var varBvar Bvar avg
Bvar Bref Bvar EE Emnum var varBvar Bvar avg
BvaEnum varBvar
gIlC CB T
gIlC CB T
n
CB
α α
α α
− − −
− −
− −
⎧ ⎫⎡ ⎤= × − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤= − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭=
≈ − ( ) ( )( ) ( )
002
00 02 2 2
02 1 0
2
2
1'
63 1
'4 6
For 1:34
n
n
r Bvar Bref Bvar EEmvar varBvar avg
Bvar Bvar Bref Bvar EE Emnum var var varBvar Bvar avg
BvaEnum varBvar
gIl C T
gIlC CB T
n
CB
α α α
α α α
−
− −
−
⎧ ⎫⎡ ⎤⎡ ⎤− × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤⎡ ⎤≈ − − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
=
= ( ) ( )( ) ( )
000
00 02 0 0
02 1 2 2
2
2
2
1' '
61 1
' '4 6
For 1:
n
n n
r Bvar Bref Bvar EEmvar varBvar avg
Bvar Bvar Bref Bvar EE Emnum var var varBvar Bvar avg
BvaEnum varBvar
gIl C T
gIlC CB T
n
CB
α α α
α α α
− −
⎧ ⎫⎡ ⎤⎡ ⎤− × − + ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤⎡ ⎤= − × + − ×Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎩ ⎭
>
≈ ( ) ( )( ) ( )
002 2
00 02 2 2 2 2
2
2 2
1' '
61
' '6
n
n n n
r Bref Bvar EEmvarBvar avg
Bvar Bref Bvar EE Emnum var varBvar Bvar avg
gIl C T
gIlC CB T
α α
α α
−
− −
⎧ ⎫⎡ ⎤× − + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤≈ − × + − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
52
Finding Specific Force 2 1n
Bvarvarυ
−Δ And
2nBvarvarυΔ
Specific force
2 1nBvarvarυ
−Δ and
2nBvarvarυΔ are obtained by substituting , from (H-10)
and , from (H-38) into the (H-1) 2 1
1ndenB −
−2
1ndenB−
2 1nnumB − 2nnumB 2 1nBvarvarυ
−Δ ,
2nBvarvarυΔ formulas:
( )( )
02 12 1 92 1
022 92
11 0
11 0
For 4 :
nnn
nnn
Bvar E Emnumden Bvar avgvar
Bvar E Emnumden Bvar avgvar
n
gCB B T
gCB B T
υ
υ
−− −−
−
−−
−−
< −
= = −Δ
= = −Δ
(H-39)
53
( ) ( ) ( )( )
( ) ( ) ( )
2 12 12 1
09
900
9
099
1
10
10
For 4 :
11 2
12 16
1 12 6
nnnBvar
numdenvarBref
BvarEvarBvarBvar E m
var BvarBvar EE
mvarBvar avg
BrefBvar BvarE Evar varBvarBvar
m
n
B B
lC
TI CgI C T
l I CCT
υ
αα
α
α α
−−−
−
−
−
−−
−
−
−
= −
=Δ
×Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦ ⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
≈ × − +Δ Δ
( )( )( ) ( ) ( ) ( )( )
( )( )( )
0
09
0 0099 9 9
09
10
1 10 0 0
10
1
112
1 12 6
112
12
Emavg
Bvar E Emvar Bvar avg
BrefBvar BvarE E E E EE
m mvar varBvarBvar Bvar Bvar avgm
Bvar E Emvar Bvar avg
BrefBvarvar
m
g T
gC T
l g gCC C CT TT
gC T
lT
α
α α
α
α
−
−− − −
−
−
−
− −
−
−
⎧ ⎫⎡ ⎤×⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
+ ×Δ
= × − − ×Δ Δ
+ ×Δ
= × −Δ ( )
1−
( )( )( ) ( ) ( )
( ) ( ) ( )
0 09 9
09
09
10 0
10
10
112
1 12 12
1 1 12 4 3
BvarE E E Em mvarBvar Bvaravg avg
BrefBvar Bvar E E
mvar var Bvar avgm
BrefBvar Bvar E E
mvar var Bvar avgm
g gC CT T
l gI C TT
l gI C TT
α
α α
α α
−
−
−
−
−
−
− ×Δ
⎡ ⎤= × − + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞= × − − − ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ (H-40)
( ) ( ) ( )( ) ( ) ( )( )
( )
222
0099
0 09 9
1
10
1 10 0
7 112 2
1 72 12
1 72 12
nnnBvar
numdenvar
Bvar Bvar BrefE EEvar varBvarBvar avg
BrefBvar BvarE E E E
mvar varBvar Bvaravg avgm
BrefBvarvar var
m
B B
gI lCC
l g gC C TT
l IT
υ
α α
α α
α
−−
− −
−
−
− −
=Δ
⎡ ⎤ ⎡= − − × − × +Δ Δ⎢ ⎥ ⎢⎣ ⎦ ⎣
≈ × − + ×Δ Δ
= × − −Δ
⎤⎥⎦
( ) ( )( ) ( ) ( )
09
09
10
101 1 1
2 4 3
Bvar E EmBvar avg
BrefBvar Bvar E E
mvar var Bvar avgm
gC T
l gI C TT
α
α α
−
−
−
−
⎡ ⎤×Δ⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞= × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
54
( ) ( ) ( )( )
( ) ( )
2 12 12 1
02 2
2 200
2 2
02 22 2
1
21
0
12 0
For 0 4 :
12 1
6
16
nnn
n
n
n
nn
Bvarnumdenvar
BrefBvarEvarBvarBvar E m
var BvarBvar EE
mvarBvar avg
BrefBvar E Evar BvarBvar
m
n
B B
lC
TI CgI C T
l I CCT
υ
αα
α
α
−−−
−
−
−
−−
−
−
−
> > −
=Δ
×Δ⎡ ⎤= − ×Δ⎢ ⎥⎣ ⎦ ⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
≈ × − +Δ ( )( )( )
( ) ( ) ( )( )( ) ( )
0
02 2
0 02 2 2 2
2 2
10
1 12 0 0
12
12
1 12 6
1 12 6
n
n n
n
Bvar Emvar avg
Bvar E Emvar Bvar avg
BrefBvar BvarE E E E
m mvar varBvar Bvar avgm
BrefBvar Bvarvar var
m
g T
gC T
l g gC CT TT
l IT
α
α
α α
α α
−
− −
−
−
− −
−
⎧ ⎫⎡ ⎤×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
+ ×Δ
⎛ ⎞= × − + − ×Δ Δ⎜ ⎟⎝ ⎠
⎡ ⎤⎛ ⎞= × − − − ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( ) 00E E
mBvar avggC T
(H-41)
( ) ( ) ( )( )
( ) ( ) ( )
222
02 2
2 100
2 2
02 22 1
1
21
0
12 0
12 1
6
16
nnn
n
n
n
nn
Bvarnumdenvar
BrefBvarEvarBvarBvar E m
var BvarBvar EE
mvarBvar
BrefBvar BvarE Evar varBvarBvar
m
B B
lC
TI CgI C T
l I CCT
υ
αα
α
α α
−
−
−
−−
−
−
−
=Δ
− ×Δ⎡ ⎤= − − ×Δ⎢ ⎥⎣ ⎦ ⎧ ⎫⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎧ ⎡ ⎤≈ × − − ×Δ Δ⎨ ⎢ ⎥⎣ ⎦
( )( )( ) ( ) ( )( )
( ) ( ) ( )
0
02 1
0 02 1 2 1
02 1
10
1 12 0 0
12 0
12
1 12 6
1 12 6
n
n n
n
Emavg
Bvar E Emvar Bvar avg
BrefBvar BvarE E E E
m mvar varBvar Bvaravg avgm
BrefBvar Bvar E E
mvar var Bvar avgm
g T
gC T
l g gC CT TT
l gI C TT
α
α α
α α
−
− −
−
−
− −
−
⎫⎬
⎩ ⎭
+ ×Δ
⎛ ⎞≈ × − + + ×Δ Δ⎜ ⎟⎝ ⎠
⎡ ⎤⎛ ⎞= × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
55
( ) ( )( )
( )( )
2 12 12 1
02
2 002
1
10
2
For 0 :
1'
1 4 /126
1'
4
nnnBvar
numdenvar
Bvar Bvar BrefEvar varBvar
Bvar Emvar Bvar Bvar EE
mvarBvar avg
BrefBvar Bvarvar var
n
B B
lCI C T
gI C T
l
υ
α αα
α
α α
−−−
−
−
−
−
−
=
=Δ
⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤≈ − − ×Δ Δ⎢ ⎥⎣ ⎦
( ) ( ) ( )( )0 0022 2
1 10 01 1
6 2
m
Bvar BvarE EEm mvar varBvarBvar Bvaravg avg
T
g gI CC CT Tα α−− −
− −⎧ ⎫⎡ ⎤− + × + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭E E
( ) ( )( ) ( ) ( )( )
02
0 0022 2
10
1 10 0
1'
4
1 16 2
BrefBvar Bvar E E
mvar var Bvar avgm
Bvar BvarE E EEm mvar varBvarBvar Bvar avg
l gC TT
g gCC CT T
α α
α α
−
−− −
−
− −
⎡ ⎤≈ − − × −Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− × + ×Δ Δ⎢ ⎥⎣ ⎦E
(H-42)
( ) ( )( ) ( )( ) ( )( )
( ) ( )
02
0 0022 2
02
10
1 10 0 0
10
1'
4
1 16 2
1'
4
16
BrefBvar Bvar E E
mvar var Bvar avgm
Bvar BvarE E E EEm mvar varBvarBvar Bvar Bvaravg avg
BrefBvar Bvar E E
mvar var Bvarm
l gC TT
g gCC C CT T
l gC TT
α α
α α
α α
−
−− −
−
−
− −
−
⎡ ⎤≈ − − × −Δ Δ⎢ ⎥⎣ ⎦
− × + ×Δ Δ
⎡ ⎤≈ − − × −Δ Δ⎢ ⎥⎣ ⎦
−
2
1 E−
−
( )( ) ( )( )( ) ( ) ( )
0 02 2
02
1 10 0
10
12
1 1 1'
4 2 6
Bvar BvarE E E Em mvar varBvar Bvaravg avg
BrefBvar Bvar Bvar E E
mvar var var Bvar avgm
g gC CT T
l gI C TT
α α
α α α
− −
−
− −
−
× + ×Δ Δ
⎡ ⎤⎛ ⎞⎡ ⎤= − − × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
(Continued)
56
(H-42) Concluded
( ) ( )( )
( )( ) ( )
222
02
1 002
021
1
10
2
10
3'
1 4 /126
3'
4
nnnBvar
numdenvar
Bvar Bvar BrefEvar varBvar
Bvar Emvar Bvar Bvar EE
mvarBvar
BrBvar BvarE Evar varBvarBvar
B B
lCI C T
gI C T
CC
υ
α αα
α
α α
−
−
−
−−
−
−
−
=Δ
⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤= − ×Δ Δ⎢ ⎥⎣ ⎦
( ) ( ) ( )( )( ) ( )
( )( ) ( )
0 0021 1
01
01 1
1 10 0
10
1 10
1 16 2
3'
4
1 16 2
ef
m
Bvar BvarE EEm mvar varBvarBvar Bvaravg avg
BrefBvar Bvar E E
mvar var Bvar avgm
Bvar BvarE Emvar varBvar Bvavg
lT
g gI CC CT T
l gC TT
gC T
α α
α α
α α
−− −
−
− −
− −
−
− −
⎧ ⎫⎡ ⎤− − × + ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤≈ − × −Δ Δ⎢ ⎥⎣ ⎦
+ × + ×Δ Δ ( )
E E
( ) ( ) ( )
0
01
0
103 1 1
'4 2 6
E Emar avg
BrefBvar Bvar Bvar E E
mvar var var Bvar avgm
gC T
l gI C TT
α α α−
−⎡ ⎤⎛ ⎞⎡ ⎤= − × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
57
( ) ( )( )
( )( )
2 12 12 1
00
0 000
1
10
2
For 1:
3'
1 4 /'12 '6
3'
4
nnnBvar
numdenvar
Bvar Bvar BrefEvar varBvar
Bvar Emvar Bvar Bvar EE
mvarBvar avg
BrefBvar Bvarvar var
m
n
B B
lCI C T
gI C T
lT
υ
α αα
α
α α
−−−−
−
=
=Δ
⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤≈ − × +Δ Δ⎢ ⎥⎣ ⎦ ( )( )( ) ( )
( ) ( )( )( ) ( )( )
00
0000
00
00 0
10
10
10
1 10 0
1'
2
1'
6
3 1' '
4 2
1'
6
Bvar E Emvar Bvar avg
BvarE EEmvarBvarBvar avg
BrefBvar Bvar Bvar E E
mvar var var Bvar avgm
BvarE E Em varBvar Bvaravg avg
gC T
gI CC T
l gC TT
gC CT
α
α
α α α
α
−
−
−
− −
×Δ
⎧ ⎫⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤≈ − × + ×Δ Δ Δ⎢ ⎥⎣ ⎦
− − ×Δ
( ) ( ) ( )
0
00
103 1 1
' '4 2 6
Em
BrefBvar Bvar Bvar E E
mvar var var Bvar avgm
g T
l gI C TT
α α α−⎡ ⎤⎛ ⎞⎡ ⎤= − × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
(H-43)
( ) ( )( )
( )( )
222
00
1 000
1
10
2
1'
1 4 /'12 '6
1 1' '
4 2
nnnBvar
numdenvar
Bvar Bvar BrefEvar varBvar
Bvar Emvar Bvar Bvar EE
mvarBvar avg
BrefBvar Bvar Bvarvar var var
m
B B
lCI C T
gI C T
lT
υ
α αα
α
α α α
−
−
=Δ
⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦⎡ ⎤= − − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤≈ − − × +Δ Δ Δ⎢ ⎥⎣ ⎦ ( )( )( ) ( )
( ) ( ) ( )
01
0001
01
10
10
10
1'
6
1 1 1' '
4 2 6
E EmBvar avg
BvarE EEmvarBvarBvar avg
BrefBvar Bvar Bvar E E
mvar var var Bvar avgm
gC T
gI CC T
l gI C TT
α
α α α
−
−
−
×
⎧ ⎫⎡ ⎤− − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤⎛ ⎞⎡ ⎤≈ − − × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
58
( ) ( ) ( )( )
( ) ( )
2 12 12 1
02 2
2 2 002 1
2 12 2
1
21
02
12 0
For 1:
'1 /' 12 '6
1'
6
nnn
n
n
n
nn
Bvarnumdenvar
Bvar BrefEvarBvarBvar E
mvar Bvar Bvar EEmvarBvar avg
BrefBvar Evar BvarBvar
m
n
B B
lCI C T
gI C T
l ICT
υ
αα
α
α
−−−
−
−
−
−−
−
−
−
>
=Δ
×Δ⎡ ⎤= − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤− + ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
≈ × − +Δ ( )( )( )
( ) ( ) ( )
00
02 2
02 2
10
12 0
'
1'
21 1 /' '2 6
n
n
Bvar EEmvar avg
Bvar E Emvar Bvar avg
BrefBvar Bvar E E
mvar var Bvar avgm
gC T
gC T
l gI C TT
α
α
α α
−
−
−
−
⎧ ⎫⎡ ⎤×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
+ ×Δ
⎡ ⎤⎛ ⎞≈ × − − − ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(H-44)
( ) ( ) ( )( )
( ) ( )
222
02 2
2 1 002 2
02 22 1
1
21
02
12 0
'1 /' 12 '6
1'
6
nnn
n
n
n
nn
Bvarnumdenvar
Bvar BrefEvarBvarBvar E
mvar Bvar Bvar EEmvarBvar avg
BrefBvar BvarE Evar varBvarBvar
m
B B
lCI C T
gI C T
l I CCT
υ
αα
α
α α
−
−
−
−−
−
−
−
=Δ
− ×Δ⎡ ⎤= − − ×Δ⎢ ⎥ ⎧ ⎫⎣ ⎦ ⎡ ⎤+ − ×Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
≈ × − −Δ Δ( )( )( )
( ) ( ) ( )
0
02 1
02 1
10
12 0
'
1'
21 1
' '2 6
n
n
Emavg
Bvar E Emvar Bvar avg
BrefBvar Bvar E E
mvar var Bvar avgm
g T
gC T
l gI C TT
α
α α
−
−
−
−
⎧ ⎫⎡ ⎤×⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
+ ×Δ
⎡ ⎤⎛ ⎞≈ × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Summary of (H-39) – (H-44) Specific Force Results:
59
( ) ( )
( ) ( ) ( )( )
0 09 92 1 2
092 1
2
1 10 0
10
For 4 :
For 4 :
1 1 12 4 3
12
n n
n
n
Bvar BvarE E E Em mBvar Bvaravg avgvar var
BrefBvar Bvar Bvar E E
mvar var Bvar avgvarm
BrefBvar Bvar
varvarm
n
g gC CT T
n
l gI C TT
l IT
υ υ
υ α α
υ α
− −−
−−
− −
−
< −
= − = −Δ Δ
= −
⎡ ⎤⎛ ⎞= × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
= × − −Δ Δ ( ) ( )
( ) ( ) ( )( ) ( )
09
02 22 1
2
10
12 0
2
1 14 3
For 0 4 :
1 12 6
1 12 6
nn
n
Bvar E Emvar Bvar avg
BrefBvar Bvar Bvar E E
mvar var Bvar avgvarm
BrefBvar Bvar Bvar
var varvarm
gC T
n
l gI C TT
l IT
α
υ α α
υ α α
−
−−
−
−
⎡ ⎤⎛ ⎞+ ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦> > −
⎡ ⎤⎛ ⎞= × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞= × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )
( ) ( ) ( )
02 1
022 1
10
10
For 0 :
1 1 1'
4 2 6
n
n
E EmBvar avg
BrefBvar Bvar Bvar Bvar E E
mvar var var Bvar avgvarm
gC T
n
l gI C TT
υ α α α
−
−−
−
−
=
⎡ ⎤⎛ ⎞⎡ ⎤= − − × − − − ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
( ) ( ) ( ) 012
103 1 1
'4 2 6n
BrefBvar Bvar Bvar Bvar E E
mvar var var Bvar avgvarm
l gI C TT
υ α α α−
−⎡ ⎤⎛ ⎞⎡ ⎤= − × − − + ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ (H-45)
( ) ( ) ( )( ) ( ) ( )
002 1
12
10
10
For 1:
3 1 1' '
4 2 6
1 1 1' '
4 2 6
n
n
BrefBvar Bvar Bvar Bvar E E
mvar var var Bvar avgvarm
BrefBvar Bvar Bvar Bvar E
var var var Bvar avgvarm
n
l gI C TT
l I CT
υ α α α
υ α α α
−
−
−
=
⎡ ⎤⎛ ⎞⎡ ⎤= − × − − − ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞⎡ ⎤= − − × − − + ×Δ Δ Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
( ) ( ) ( )( ) ( ) ( )
0
02 22 1
02 12
12 0
12 0
For 1:
1 1' '
2 6
1 1' '
2 6
nn
nn
Em
BrefBvar Bvar Bvar E E
mvar var Bvar avgvarm
BrefBvar Bvar Bvar E E
mvar var Bvar avgvarm
g T
n
l gI C TT
l gI C TT
υ α α
υ α α
−−
−
−
−
>
⎡ ⎤⎛ ⎞= × − − − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞= × − − + ×Δ Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Recognizing that the inverse of a direction cosine matrix equals its transpose, the (
terms in (H-45) become , generating (49) in [1].
) 10
nEBvarC
−
0nBvarEC
60
VELOCITY DETERMINATION
Approximating gravity for test example conditions as the constant 0Eavgg , velocity from (3)
with becomes at time instants m-1 and m: 2m = n
00 0 02 12 22 1 2 2 2 1
0 0 0 022 12 2 1 2
nnn n n
nnn n n
EBvarE E EmVvar avgBvarvar var var
BvarE E E EmVvarBvarvar var var avg
gV V C G T
gV V C G T
υ
υ
−−− − −
−−
= + +Δ
= + +Δ (H-46)
With initial velocity at 00
ErefV , (H-45) for Bvar
varυΔ specific force, (32) for attitude, and
(33) for , (H-46) becomes to first order accuracy for velocity:
0EBvarC
VvarG
( )00 0 0
2 12 22 1 2 2 2 1
0 00 0 09 90 0
00 0 022 12 2 1 2
0 02 10
10
For 4 :
nnn n n
nnn n n
n
Bvar EE E EmVvarBvar avgvar var var
E E EE E Em mref Bvar refBvar
Bvar EE E EmVvarBvar avgvar var var
E Eref Bvar
n
gV V C G T
g gIV VC C T T
gV V C G T
V C
υ
υ
−−− − −
− −
−−
−
−
< −
= + +Δ
= − + =
= + +Δ
= − ( ) 0 0 09 0
10E E E E
m m refBvar avgg gI VC T T−
−+ =
(H-47)
61
( )( )
00 0 02 12 22 1 2 2 2 1
00 0 09109 10 9
0 090 1
For 4 :
12
1 14 3
nnn n nBvar EE E E
mVvarBvar avgvar var var
Bvar EE E EmVvarBvar avgvar var var
BrefBvarvar
mE Eref Bvar
Bvarvar
n
gV V C G T
gV V C G T
lTIV C
I
υ
υ
α
α
−−− − −
−−− − −
− −
= −
= + +Δ
= = + +Δ
×Δ= +
⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )
( ) ( )( )( ) ( )( )
0
09
00 0 09 9 90
00 0 09 9 90
0
10
10
1 1 12 4 3
1 12 12
Emavg
E EmBvar
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgmBref
Bvar Bvar E EE E Evar varref Bvar Bvar Bvar avg
m
g TgC T
l gV C C C TT
lV C C CT
α α
α α
−
− − −
− − −
−
−
⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭
⎛ ⎞= + × + − ×Δ Δ⎜ ⎟⎝ ⎠
= + × − ×Δ Δ mg T
(H-48)
( ) ( )( )
( )( )
00 0 0 0892 8 9 2
00 0 09 9 90
09
101 1
2 12
121
2 1 14 3
n nBvar EE E E E
mVvarBvar avgvar var var varBref
Bvar Bvar E EE E Emvar varref Bvar Bvar Bvar avg
mBref
Bvarvar
mBvarEvarBvar
gV V V C G T
l gV C C C TT
lTIC
I
υ
α α
αα
−−− −
− − −
−
−
= = + +Δ
= + × − ×Δ Δ
×Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦ ⎛− − + ( ) ( )
( ) ( )( )( ) ( )
0
09
00 0 09 9 90
09
10
10
1
1 12 12
1 1 12 4 3
Emavg
Bvar E Emvar Bvar
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
BrefBvar BvarEvar varBvar
m
g TgC T
l gV C C C TT
l ICT
α
α α
α α
−
− − −
−
−
−
−
⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬
⎡ ⎤⎞⎪ ⎪×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
= + × − ×Δ Δ
⎡ ⎤⎛ ⎞+ × − − + ×Δ Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )
( )( )( )
09
0 009 9
0 090
0
101
2
E EmBvar avg
Bvar E E EEm mvarBvar Bvar avg
BrefBvarE Evarref Bvar
m
gC T
g gC C T T
lV CT
α
α
−
− −
−
−
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
− × +Δ
= + ×Δ
62
( )( ) ( ) ( )
00 0 02 12 22 1 2 2 2 1
0 02 30
0 002 2 2 2
10
For 0 4 :
1 1 12 2 6
nnn n n
n
n n
Bvar EE E EmVvarBvar avgvar var var
BrefBvarE Evarref Bvar
m
Bvar Bvar E E EEmvar varBvar Bvar avg avg
n
gV V C G T
lV CT
gI IC C T
υ
α
α α
−−− − −
−
− −
−
> > −
= + +Δ
= + ×Δ
⎡ ⎤⎡ ⎤ ⎛ ⎞− + × − − × +Δ Δ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦
( ) ( )( )( ) ( )
( )
00 0 02 2 2 2 2 20
002 2 2 2
0 0 02 2 2 20
10
10
12
1 12 6
16
n n n
n n
n n
m
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvarm
Bvar E EEmvarBvar Bvar avg
BrefBvar BvaE E Evar varref Bvar Bvar
m
g T
l gV C C C TT
gC C T
lV C CT
α α
α
α
− − −
− −
− −
−
−
≈ + × − ×Δ Δ
⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
= + × −Δ ( )( ) 02 2
10
nr E E
mBvar avggC Tα−
−×Δ
(H-49)
( ) ( )( )( ) ( )
00 0 022 12 2 1 2
00 0 02 2 2 2 2 20
02 1
101
6
1 1 12 2 6
nnn n n
n n n
n
Bvar EE E EmVvarBvar avgvar var var
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
Bvar BvarEvar varBvar
gV V C G T
l gV C C C TT
I IC
υ
α α
α α
−−
− − −
−
−
= + +Δ
≈ + × − ×Δ Δ
⎡ ⎤⎡ ⎤ ⎛ ⎞− + × − + ×Δ Δ⎜ ⎟⎢⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦( )
( ) ( )( )( )( )
( )
0 02 1
00 0 02 1 2 1 2 10
002 1 2 1
02 1 2
10
10
10
1
16
12
1 12 6
n
n n n
n n
n n
E E Em mBvar avg
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
Bvar E EEmvarBvar Bvar avg
BvarEvarBvar
g gC T T
l gV C C C TT
gC C T
C
α α
α
α
−
− − −
− −
− −
−
−
−
−
+⎥
≈ + × − ×Δ Δ
− ×Δ
⎡ ⎤⎛ ⎞− − + ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦( )
( )
01
0 02 10
0
n
E EmBvar avg
BrefBvarE Evarref Bvar
m
gC T
lV CT
α−
= + ×Δ
63
( )
( )
00 0 0 02 12 22 1 1 2 2 2 1
00 0122 1
0 030
02 2
For 0 :
141
2
nnn n n
n
Bvar EE E E EmVvarBvar avgvar var var var
Bvar EE EmVvarBvar avgvar var
BrefBvarE Evarref Bvar
m
varBvarEvarBvar
n
gV V V C G T
gV C G T
lV CT
IC
υ
υ
α
α
−−− − − −
−−− −
−
−
=
= = + +Δ
= + +Δ
= + ×Δ
−⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦
( )( )( )
( ) ( )( )
0
02
00 0 02 2 20
02
10
120
'
1 12 6
12
1'
4
Bvar Bvar Brefvar
EmavgBvar E E
mvar Bvar avg
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
BvarEvarBvar
lg T
gI C T
l gV C C C TT
C
α α
α
α α
α
−
− − −
−
−
−
⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪ +⎨ ⎬⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ + × − ×Δ Δ
− Δ( ) ( )( )( ) ( )( )
02
00 0 02 2 20
10
10
1 12 6
1 15 '4 6
Bvar Bref Bvar E Emvar var Bvar avg
BrefBvar Bvar Bvar E EE E E
mvar var varref Bvar Bvar Bvar avgm
gl C T
l gV C C C TT
α α
α α α
−
− − −
−
−
⎧ ⎫⎛ ⎞⎡ ⎤− × − − ×Δ Δ⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎩ ⎭
⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦
(H-50)
( ) ( )( )
00 0 0 022 12 0 2 1 2
00 0011 0
00 0 02 2 20
101 15 '
4 6
nnn n nBvar EE E E E
mVvarBvar avgvar var var var
Bvar EE EmVvarBvar avgvar var
BrefBvar Bvar Bvar E EE E Evar var varref Bvar Bvar Bvar
m
gV V V C G T
gV C G T
lV C CT
υ
υ
α α α
−−
−−
− − −
−
= = + +Δ
= + +Δ
⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦ C
( )( )
( ) ( )( )
001 0
1
0 01 10
2
120
3'
411 122 6
1 15 '4 6
m
Bvar Bvar Brefvar var
Bvar EEmvarBvar avgBvar E E
mvar Bvar
BrefBvar BvarE Evar varref Bvar Bv
m
g T
lgIC T
gI C T
lV CT
α αα
α
α α
−
−
− −
−
⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥ ⎡ ⎤⎛ ⎞⎣ ⎦ ⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
⎡ ⎤≈ + − × −Δ Δ⎢ ⎥⎣ ⎦ ( )( )( )( ) ( )
( )( )
001
00 01 11
001 1
0 010
120
120
120
1 3'
2 41 12 6
12
Bvar E EEmvarar Bvar avg
Bvar Bvar Bvar BrefE EE Emvar var varBvar BvarBvar
Bvar E EEmvarBvar Bvar avg
BvarE Evar vref Bvar
gC C T
g lC CC T
gC C T
V C
α
α α
α
α
−
− −−
− −
−
−
−
−
×Δ
⎡ ⎤− × + − ×Δ Δ Δ⎢ ⎥⎣ ⎦
⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠
= + +Δ
α
( )'Bref
Bvarar
m
lT
α⎡ ⎤×Δ⎢ ⎥⎣ ⎦
64
( )
( )
00 0 0 02 12 22 1 1 2 2 2 1
00 0100 1
0 010
00
For 1:
1'
2
1'
2
nnn n nBvar EE E E E
mVvarBvar avgvar var var var
Bvar EE EmVvarBvar avgvar varBref
Bvar BvarE Evar varref Bvar
m
BvarEvarBvar
n
gV V V C G T
gV C G T
lV CT
IC
υ
υ
α α
α
−−− − −
−
=
= = + +Δ
= + +Δ
⎡ ⎤= + + ×Δ Δ⎢ ⎥⎣ ⎦
+ + ×Δ( )
( )( )( ) ( )( )
0
00
00 0 00 0 00
10
10
3'
4
1 1'
2 6
1 1' '
2 2
BrefBvar Bvarvar var
m Emavg
Bvar E Emvar Bvar
BrefBvar Bvar Bvar EE E Evar var varref Bvar Bvar Bvar avg
m
lT g T
gI C T
lV C CT
α α
α
α α α
−
−
⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤ +⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
⎡ ⎤≈ + + × − ×Δ Δ Δ⎢ ⎥⎣ ⎦ C
( ) ( )( )( ) ( )( )
00 00 0 0
00 0 00 0 00
10
10
3 1 1' '
4 2 6
1 15 ' '4 6
Em
BrefBvar Bvar Bvar E EE E
mvar var varBvar Bvar Bvar avgm
BrefBvar Bvar Bvar E EE E E
mvar var varref Bvar Bvar Bvar avgm
g T
l gC C C TT
l gV C C TT
α α α
α α α
−
−
⎛ ⎞⎡ ⎤+ − × + − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦ C
(H-51)
( ) ( )( )
00 0 0 022 12 2 2 1 2
00 0211 2
00 0 00 0 00
101 15 ' '
4 6
nnn n nBvar EE E E E
mVvarBvar avgvar var var var
Bvar EE EmVvarBvar avgvar var
BrefBvar Bvar Bvar E EE E Evar var varref Bvar Bvar Bvar avg
m
gV V V C G T
gV C G T
l gV C C CT
υ
υ
α α α
−−
−
= = + +Δ
= + +Δ
⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦
( )( )
( )( )( )
001
01
0 01 10
10
1'
41'
2 1 1'
2 6
1 15 '4 6
m
BrefBvar Bvarvar var
mBvar EEmvarBvar avg
Bvar E Emvar Bvar
BrefBvar BvarE Evar varref Bvar Bvar
m
T
lT gIC T
gI C T
lV CT
α αα
α
α α
−
⎧ ⎫⎡ ⎤− − ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
⎡ ⎤≈ + − × −Δ Δ⎢ ⎥⎣ ⎦ ( )( )( )( ) ( )
( )( )( )
001
00 01 11
001 2 1
0 010
10
10
10
'
1 1' '
2 4
1 1'
2 6
'
n
Bvar E EEmvar Bvar avg
BrefBvar Bvar BvarE EE E
mvar var varBvar BvarBvar avgm
Bvar E EEmvarBvar Bvar avg
BreBvarE Evarref Bvar
gC C T
lgC CC TT
gC C T
V C
α
α α α
α
α
−
−
−
−
×Δ
⎡ ⎤− × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦
⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠
= + ×Δf
m
lT
65
( )
( )( )
( )
00 0 02 12 22 1 2 2 2 1
0 02 30
02 2
2 2
2
1
For 1:
'
'1
'2 1 1
'2 6
nnn n n
n
n
n
Bvar EE E EmVvarBvar avgvar var var
BrefBvarE Evarref Bvar
mBref
Bvarvar
mBvarEvarBvar
Bvarvar B
n
gV V C G T
lV CT
lTIC
I
υ
α
αα
α
−−− − −
−
−
−
−
>
= + +Δ
= + ×Δ
×Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( )( ) ( )( )
0
0
00 0 02 2 2 2 2 20
0
101
' '6n n n
Emavg
E Emvar
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
g TgC T
l gV C C C TT
α α− − −
−
⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭
≈ + × − ×Δ Δ
(H-52)
( ) ( )( )
( )( )
00 0 022 12 2 1 2
00 0 02 2 2 2 2 20
02 1
10
2
1' '
6
'1
'2 1
2
nnn n n
n n n
n
Bvar EE E EmVvarBvar avgvar var var
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
BrefBvarvar
mBvarEvarBvar
gV V C G T
l gV C C C TT
lTIC
I
υ
α α
αα
−−
− − −
−
−
= + +Δ
= + × − ×Δ Δ
×Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦
− − + ( ) ( )( ) ( )( )
( )( )
0
02 1
00 0 02 1 2 1 2 10
002 1 2 1
2
10
10
10
1'
6
1' '
6
1'
2
n
n n n
n n
n
Emavg
Bvar E Emvar Bvar
BrefBvar Bvar E EE E E
mvar varref Bvar Bvar Bvar avgm
Bvar E EEmvarBvar Bvar avg
g TgC T
l gV C C C TT
gC C T
α
α α
α
−
− − −
− −
−
−
−
−
⎧ ⎫⎪ ⎪⎪ ⎪ +⎨ ⎬
⎡ ⎤⎛ ⎞⎪ ⎪×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ + × − ×Δ Δ
− ×Δ
+ ( )( )( )
001 2 1
0 02 10
101 1
'2 6
'
n
n
Bvar E EEmvarBvar Bvar avg
BrefBvarE Evarref Bvar
m
gC C T
lV CT
α
α
−
−
−⎛ ⎞+ ×Δ⎜ ⎟⎝ ⎠
= + ×Δ
Summary of (H-47) to (H-52) Velocity Results:
66
( ) ( )( )( )
0 0 0 02 1 20 0
00 0 0 09 9 99 0
0 0 098 0
2 1
10
For 4 :
For 4 :
1 12 12
For 0 4 :
n n
n
E E E Eref refvar var
BrefBvar Bvar E EE E E E
mvar varref Bvar Bvar Bvar avgvarm
BrefBvarE E Evarref Bvarvar
m
var
n V V V V
n
l gV V C C C TT
lV V CT
n
α α
α
−
− − −−
−−
−
−
< − = =
= −
= + × − ×Δ Δ
= + ×Δ
> > −
( ) ( )( )( )
00 0 0 02 2 2 2 2 20
0 0 02 12 0
101
6n n n
nn
BrefBvar Bvar E EE E E E
mvar varref Bvar Bvar Bvar avgm
BrefBvarE E Evarref Bvarvar
m
l gV V C C C TT
lV V CT
α α
α
− − −
−
−= + × − ×Δ Δ
= + ×Δ
( )( )( )
( )
0 0 021 0
002 2
0 0 010 0
10
For 0 :
1 5 '4
16
1'
2
BrefBvar BvarE E Evar varref Bvarvar
m
Bvar E EEmvarBvar Bvar avg
BrefBvar BvarE E Evar varref Bvarvar
m
n
lV V CT
gC C T
lV V CT
α α
α
α α
−−
− −
−
−
=
⎡ ⎤= + − ×Δ Δ⎢ ⎥⎣ ⎦
− ×Δ
⎡ ⎤= + + ×Δ Δ⎢ ⎥⎣ ⎦
(H-53)
( ) ( )( )( )
( )
00 0 0 00 0 01 0
0 0 012 0
0 0 02 22 1 0
10
For 1:
1 15 ' '4 6
'
For 1:
'nn
BrefBvar Bvar Bvar E EE E E E
mvar var varref Bvar Bvar Bvar avgvarm
BrefBvarE E Evarref Bvarvar
m
BvarE E Evarref Bvarvar
n
l gV V C C C TT
lV V CT
n
V V C
α α α
α
α−−
−
=
⎡ ⎤= + − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦
= + ×Δ
>
= + ×Δ ( )( )( )
002 2 2 2
0 0 02 12 0
101
'6
'
n n
nn
BrefBvar E EE
mvarBvar Bvar avgm
BrefBvarE E Evarref Bvarvar
m
l gC C TT
lV V CT
α
α
− −
−
−− ×Δ
= + ×Δ
Recognizing that the inverse of a direction cosine matrix equals its transpose, the (
terms in (H-53) become , generating (50) in [1].
) 10
nEBvarC
−
0nBvarEC
67
POSITION DETERMINATION
Approximating gravity for test example conditions as the constant 0Eavgg , position from (4)
with becomes at time instants m-1 and m: 2m = n
00 0 0 02 12 22 1 2 2 2 2 2 1
00 0 0 022 12 2 1 2 1 2
2
2
12
12
nnn n n n
nnn n n n
EBvarE E E Em mRvar avgBvarvar var var var
Bvar EE E E Em mRvarBvarvar var avgvar var
gVR R C GT T
gVR R C GT T
υ
υ
−−− − − −
−− −
= + + +Δ
= + + +Δ
m
m
T
T
(H-54)
Initializing 0
2 1nEvarR
− position (i.e., 0
2 2nEvarR
−) for derives directly from (37) with
, and from (32):
4n < −
2m n= − 2 0 09m
E EBvar BvarC C −
=
( )0 0 0
92 2 0For 4 : 2 2
nE E E Bref
mref Bvarvarn n VR C lT −−< − = − + (H-55)
With 0
2 2nEvarR
− from (H-55) for , (H-53) for 4n < − 0E
varV velocity, (H-45) for BvarvarυΔ
specific force, (32) for attitude, and (33) for , (H-54) becomes to first order accuracy for position:
0EBvarC RvarG
( ) ( )00 0 0 0
2 12 22 1 2 2 2 2 2 1
0 00 0 0 09 2 2 90 0
2
12 20
For 4 :12
1 12 22 2
nnn n n n
n
Bvar EE E E Em mRvarBvarvar var avgvar var
E E EE E E EBrefm m mref Bvar ref Bvar Bvar avg avg
n
gVR R C GT T
g gn V VC l C CT T T
υ−−− − − −
− − −
−
< −
= + + +Δ
= − + + − +
m
m
T
T
( ) 0 0
90
00 0 0 022 12 2 1 2 1 2
2
2 1
12nnn n n n
E E Brefmref Bvar
Bvar EE E E Em mRvarBvarvar var avgvar var
n V C lT
gVR R C GT Tυ
−
−− −
= − +
= + + +Δ mT (H-56)
( ) ( ) 0 00 0 0 09 9 90 0
0 090
1201 12 1
2 2
2
E E EE E E EBrefm m m mref Bvar ref Bvar Bvar avg avg
E E Brefmref Bvar
g gn V VC l C CT T T T
n V C lT
− − −
−
−= − + + − +
= +
mT
68
( )( ) ( )
00 0 0 09109 10 10 9
0 0 090 0
09
09
2
10
For 4 :12
10
121
2 1 14 3
Bvar EE E E Em mRvarBvarvar var avgvar var
E E EBrefm mref Bvar ref
BrefBvarvar
mEBvar
Bvar E Emvar Bvar
n
gVR R C GT T
V VC lT T
lT
CgI C T
υ
α
α
−−− − − −
−
−
−
−
= −
= + + +Δ
= − + +
⎧×Δ⎪
+ ⎨⎡ ⎤⎛ ⎞− − − ×Δ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
mT
( )( )( )
0
0 090
009 9
2
120
12
194
1 1 12 4 3
Em avg
Bvar BrefE Em varref Bvar
Bvar E EEmvarBvar Bvar avg
gT
IV lCT
gC C T
α
α
−
− −
−
⎫⎪
⎪ ⎪ +⎬⎪ ⎪⎪ ⎪⎩ ⎭
⎡ ⎤= − + + ×Δ⎢ ⎥⎣ ⎦
⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠
mT (H-57)
(Continued)
69
(H-57) Concluded
( )( )( )
00 0 0 0898 9 9 8
0 0 090 0
009 9
09
2
120
12
194
1 1 12 4 3
12
Bvar EE E E Em m mRvarBvarvar var avgvar var
Bvar BrefE E Em mvarref Bvar ref
Bvar E EEmvarBvar Bvar avg
BvarEvarBvar
gVR R C GT T T
IV l VCT T
gC C T
C
υ
α
α
−−− − − −
−
− −
−
−
= + + +Δ
⎡ ⎤= − + + × +Δ⎢ ⎥⎣ ⎦
⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠
+ Δ( ) ( )( )
( )( )
( ) ( )
009 9
009
09
0 090
120
21
0
112
121 1 1
2 3 21 14 3
18
Bref Bvar E EEmvarBvar Bvar avg
BrefBvarvar
mBvar EEm mvarBvar avg
Bvar E Emvar Bvar
E Emref Bvar
gl C C T
lT gIC T T
gI C T
IV CT
α α
αα
α
− −
−
−
−
−
−
× − ×Δ
⎧ ⎫×Δ⎪ ⎪
⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
= − + + ( ) ( )( )( ) ( )( )
( )( )
009 9
00 09 9 9
009 9
09
120
120
10
1 1 14 2 4 3
1 12 12
16
14
Bvar Bref Bvar E EEmvar varBvar Bvar avg
Bvar Bref Bvar E EE Emvar varBvar Bvar Bvar avg
Bvar E EEvarBvar Bvar avg
BEvarBvar
gl C C T
glC C C T
gC C
C
α α
α α
α
− −
− − −
− −
−
−
−
−
⎡ ⎤ ⎛ ⎞× + − ×Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
+ × − ×Δ Δ
− ×Δ
+ ( ) ( )( )( ) ( )( )
( )( )
009 9
00 0 09 9 90
009 9
120
120
10
1 1 12 4 3
1 1 182 4 3
112
var Bref Bvar E EEmvarBvar Bvar avg
Bvar Bref Bvar E EE E Em mvar varref Bvar Bvar Bvar
Bvar E EEmvarBvar Bvar avg
gl C C T
gIV lC C CT T
gC C
α α
α α
α
− −
− − −
− −
−
−
−
⎛ ⎞× + + ×Δ Δ⎜ ⎟⎝ ⎠
⎛ ⎞⎡ ⎤= − + + × + − ×Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
− ×Δ ( )( )( )( )
( )
009 9
009 9
0 090
0 080
12 0
120
16
1 1 12 4 3
8
8
Bvar E EEvarBvar Bvar avg
Bvar E EEmvarBvar Bvar avg
Bvar BrefE Em varref Bvar
BrefE Emref Bvar
gC CT
gC C T
IV lCT
V lCT
α
α
α
− −
− −
−
−
−
−
− ×Δ
⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠
⎡ ⎤= − + + ×Δ⎢ ⎥⎣ ⎦
= − +
70
( ) ( )
( )
00 0 0 02 12 22 1 2 2 2 2 2 1
0 0 0 02 2 2 30 0
02 2
2
2
For 0 4 :12
2 2
1 12 3
nnn n n n
n n
n
Bvar EE E E Em mRvarBvarvar var avgvar var
Bref Bvar BrefE E E Em m varref Bvar ref Bvar
Bvar
BvarEvarBvar
n
gVR R C GT T
n V l V lC CT T
IC
υ
α
α
−−− − − −
− −
−
> > −
= + + +Δ
= − + + + ×Δ
⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦
mT
( )( ) ( ) 0
2 2
101 1
2 6 n
Brefvar
mm
Bvar E Emvar Bvar
lT
TgI C T
α
α−
−
⎧ ⎫×Δ⎪ ⎪
⎪ ⎪⎨ ⎬
⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
0 212
Emavgg T+ (H-58)
( ) ( )( )( )
( )( )( ) ( )
0 0 02 2 2 20
002 2 2 2
002 2 2 2
0 02 20
120
120
2 1
16
1 1 12 2 6
2 1
n n
n n
n n
n
Bref Bvar BrefE E Em varref Bvar Bvar
Bvar E EEmvarBvar Bvar avg
Bvar E EEmvarBvar Bvar avg
BvarE Em varref Bvar
n V lC CT
gC C T
gC C T
n IV CT
α
α
α
α
− −
− −
− −
−
−
−
≈ − + + ×Δ
− ×Δ
⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠
⎡= − + + ×Δ
l
( ) 0 02 10
2 1n
Bref
BrefE Emref Bvar
l
n V lCT −
⎤⎢ ⎥⎣ ⎦
= − +
(Continued)
71
(H-58) Concluded
( )
( )( )( )
00 0 0 022 12 2 1 2 1 2
0 02 10
02 2
00
002 2 2 2
2
10
12
2 1
16
nnn n n n
n
n
n n
Bvar EE E E Em mRvarBvarvar var avgvar var
BrefE Emref Bvar
BrefBvarEvarBvarE m
mrefBvar E EE
mvarBvar Bvar avg
gVR R C GT T
n V lCT
lC
TV TgC C T
υ
α
α
−− −
−
−
− −
−
= + + +Δ
= − +
⎧ ⎫×Δ⎪
⎪+ + ⎨ ⎬⎪− ×Δ⎪⎩
mT
( )( )
( ) ( )( )
002 1
02 1
0 0 02 1 2 10
2
2
21
0
1 1 12 3 21 1
2 6
2
16
n
n
n n
n
m
BrefBvarvar
mBvar EEm mvarBvar avg
Bvar E Emvar Bvar avg
Bref Bvar BrefE E Em varref Bvar Bvar
T
lT gIC T T
gI C T
n V l lC CT
αα
α
α
−
−
− −
−
−
⎪⎪
⎪⎪⎭
⎧ ⎫×Δ⎪ ⎪
⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ + + ×Δ
− ( )( )( )( )
( )( )( )
002 2 2
002 1 2 1
0 002 1 2 1
0 02 10
120
120
12 20
16
1 1 1 12 2 6 2
2
n
n n
n n
n
Bvar E EEmvarBvar Bvar avg
Bvar E EEmvarBvar Bvar avg
Bvar E E EEm mvarBvar Bvar avg avg
Bvar BreE Em varref Bvar
gC C T
gC C T
g gC C T T
n IV CT
α
α
α
α
−
− −
− −
−
−
−
−
×Δ
− ×Δ
⎛ ⎞+ + × +Δ⎜ ⎟⎝ ⎠
⎡ ⎤= + + ×Δ⎢ ⎥⎣ ⎦0 0
202
n
f
BrefE Emref Bvar
l
n V lCT= +
72
( )
( )( )
00 0 0 0121 2 2 1
0 0 0 02 30 0
02
2
For 0 :12
2
1'
41 12 3
Bvar EE E E Em mRvarBvarvar var avgvar var
BrefBref BvarE E E E
m m varref Bvar ref Bvarm
Bvar Bvarvar var
BvarEvarBvar
n
gVR R C GT T
lV l VC CT TT
IC
υ
α
α αα
−−− − − −
− −
−
=
= + + +Δ
= − + + + ×Δ
− − ×Δ Δ⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦
m
m
T
T
( )( ) 02
101 1
2 6
Bref
mm
Bvar E Emvar Bvar avg
lT
TgI C Tα
−
−
⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎨ ⎬
⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
0 212
Emavgg T+ (H-59)
( )( )( )
( ) ( )( )
0 0 02 20
002 2
00 02 2 2
00
120
120
16
1 1 1'
8 4 12
Bref Bvar BrefE E Em varref Bvar Bvar
Bvar E EEmvarBvar Bvar avg
Bvar Bvar Bref Bvar E EE Emvar var varBvar Bvar Bvar avg
Emref
V l lC CT
gC C T
glC C C T
V T
α
α
α α α
− −
− −
− − −
−
−
≈ − + + ×Δ
− ×Δ
⎛ ⎞⎡ ⎤− − × + − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
= − ( ) ( )( )
( )
0 0 02 2 2
0 0 01 10
0 010
1'
81
'8
1'
8
Bref Bvar Bref Bvar Bvar BrefE E Evar var varBvar Bvar Bvar
Bref Bvar Bvar BrefE E Em var varref Bvar Bvar
Bvar BvarE Em var varref Bvar
l lC C C
V l lC CT
IV CT
α α α
α α
α α
− − −
− −
−
⎡ ⎤+ + × − − ×Δ Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤≈ − + − − ×Δ Δ⎢ ⎥⎣ ⎦
= − + − − ×Δ Δ
l
Brefl⎧ ⎫⎡ ⎤⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭ (Continued)
73
(H-59) Concluded
( )
( )
00 0 0 0010 1 1 0
0 010
0 0 02 20
212
1'
8
1 15 '4 6
Bvar EE E E Em m mRvarBvarvar var avgvar var
Bvar Bvar BrefE Em var varref Bvar
BrefBvar BvarE E E
m var var vref Bvar Bvarm
gVR R C GT T T
IV lCT
lV C CTT
υ
α α
α α
−− −
−
− −
= + + +Δ
⎧ ⎫⎡ ⎤= − + − − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤+ + − × −Δ Δ⎢ ⎥⎣ ⎦ ( )( )
( )( )
( ) ( )
02
001
01
01
10
21
0
3'
41 1 12 3 21 1
2 6
18
Bvar E Emar Bvar avg
BrefBvar Bvarvar var
mBvar EEm mvarBvar avg
Bvar E Emvar Bvar
BvaEvarBvar
gC T
lT gIC T T
gI C T
IC
α
α αα
α
−
−
−
−
−
−
⎧ ⎫⎪ ⎪×Δ⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎡ ⎤− ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ − ( )( ) ( )( )( )( ) ( )
00 01 1 1
00 01 11
10
10
'
1 15 '4 61 3
'6 8
r Bvar Brefvar
Bvar Bvar Bref Bvar E EE Emvar var varBvar Bvar Bvar avg
Bvar Bvar Bvar BrefE EE Emvar var varBvar BvarBvar
l
glC C C T
g lC CC T
α α
α α α
α α α
− − −
− −−
−
−
⎧ ⎫⎡ ⎤− ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤+ − × − ×Δ Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− × + − ×Δ Δ Δ⎢ ⎥⎣ ⎦
( )( )( )
001 1
01
00
1201 1 1
2 2 6Bvar E EE
mvarBvar Bvar avg
Bvar BrefEvarBvar
BrefEBvar
gC C T
I lC
lC
α
α
− −
−
−⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠
⎡ ⎤= + ×Δ⎢ ⎥⎣ ⎦
=
74
( )
( )( )
00 0 0 0101 0 0 1
0 0 00 10
00
2
For 1:12
1'
23
'41 1
'2 3
Bvar EE E E Em m mRvarBvarvar var avgvar var
Bref Bvar Bvar BrefE E Em var varref Bvar Bvar
Bvar Bvarvar var
BvarEvarBvar
n
gVR R C GT T T
V l lC CT
IC
υ
α α
α αα
−
=
= + + +Δ
⎡ ⎤= + + + ×Δ Δ⎢ ⎥⎣ ⎦
⎡ ⎤− ×Δ Δ⎢⎣ ⎦⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦ ( )( )0
00
21
20
11 1 2'2 6
Bref
EmavgBvar E E
mvar Bvar
lg T
gI C Tα−
⎧ ⎫⎪ ⎪⎥⎪ ⎪ +⎨ ⎬⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
( )0 0 00 00
1'
2Bref Bvar Bvar BrefE E E
m var varref Bvar BvarV lC CT α α⎡≈ + + + ×Δ Δ⎢⎣l⎤⎥⎦
(H-60)
( )( )( ) ( )( )
( )
000 0
00 00 0 2 2
0 0 00 00
120
120
1'
63 1 1 1
' '8 2 2 6
1'
8
n
Bvar E EEmvarBvar Bvar avg
Bvar Bvar Bref Bvar E EE Emvar var varBvar Bvar Bvar avg
Bvar Bref BvarE E Em var varref Bvar Bvar
gC C T
glC C T
IV lC CT
α
α α α
α
−
−
−
− ×Δ
⎛ ⎞⎡ ⎤+ − × + − ×Δ Δ Δ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
⎡ ⎤= + + × −Δ⎢ ⎥⎣ ⎦
C
( )( )0 0
10
'
1'
8
Bvar Brefvar
Bvar Bvar BrefE Em var varref Bvar
l
IV lCT
α α
α α
⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤≈ + − − ×Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
(Continued)
75
(H-60) Concluded
( )( )
00 0 0 0212 1 1 2
0 0 010 0
00
00
212
1'
81 5 '41
'6
Bvar EE E E Em m mRvarBvarvar var avgvar var
Bvar Bvar BrefE E Em mvar varref Bvar ref
Bvar Bvar BrefEvar varBvar
BvarEvarBvar
gVR R C GT T T
IV l VCT T
lC
C
υ
α α
α α
α
= + + +Δ
⎧ ⎫⎡ ⎤= + − − × +Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
⎡ ⎤− ×Δ Δ⎢ ⎥⎣ ⎦+
− Δ( )( )
( )( )
( )( )
00
001 0
1
0 010
120
21
20
1'
41 1 1'
1 12 3 2'2 6
128
E EmBvar
Bvar Bvar Brefvar var
Bvar EEmvarBvar avgBvar E E
mvar Bvar avg
E Em varef Bvar
gC T
lgIC T
gI C T
IV CT
α αα
α
−
−
⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪×⎪ ⎪⎩ ⎭
⎧ ⎫⎡ ⎤− − ×Δ Δ⎪ ⎪⎢ ⎥⎣ ⎦⎪ ⎪⎡ ⎤+ + × +Δ ⎨ ⎬⎢ ⎥ ⎡ ⎤⎛ ⎞⎣ ⎦ ⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ + − ( ) ( )( )( ) ( )( )
( )
01
0 00 01 11 1
01
1 12 20 0
1 5' '4
1 1' '
6 61
'8
Bvar Bvar Bref Bvar Bvar BrefEr var var varBvar
Bvar BvarE E E EE Em mvar varBvar BvarBvar Bvaravg avg
Bvar BvarEvar varBvar
l lC
g gC CC CT T
C
α α α α
α α
α α
− −
⎧ ⎫⎡ ⎤ ⎡− × + − ×Δ Δ Δ Δ⎨ ⎬⎢ ⎥ ⎢⎣ ⎦ ⎣⎩ ⎭
− × − ×Δ Δ
⎡ ⎤− − ×Δ Δ⎢ ⎥⎣ ⎦
⎤⎥⎦
( )( )( )
001 1
0 010
0 020
1201 1 1
'2 2 6
2 '
2
Bref
Bvar E EEmvarBvar Bvar avg
Bvar BrefE Em varref Bvar
BrefE Emref Bvar
l
gC C T
IV lCT
V lCT
α
α
−⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠
⎡ ⎤= + + ×Δ⎢ ⎥⎣ ⎦
= +
76
( ) ( )
( )
00 0 0 02 12 22 1 2 2 2 2 2 1
0 0 0 02 2 30 0
02 2
2
2
For 1:12
2 2 '
1 1'
2 3
nnn n n n
n n
n
Bvar EE E E Em mRvarBvarvar var avgvar var
Bref Bvar BrefE E E Em m varref Bvar ref Bvar
Bvavar
BvarEvarBvar
n
gVR R C GT T
n V l V lC CT T
IC
υ
α
α
−−− − − −
− −
−
>
= + + +Δ
= − + + + ×Δ
⎡ ⎤+ + ×Δ⎢ ⎥⎣ ⎦
mT
( )( ) ( )
( ) ( )( )
0
02 2
0 0 02 2 20
0 02 10
21
20
'1
1 1 2'2 6
2 1 '
2 1
n
n n
n
r Bref
EmavgBvar E E
mvar Bvar
Bref Bvar BrefE E Em varref Bvar Bvar
BrefE Emref Bvar
lg T
gI C T
n V l lC CT
n V lCT
α
α
α
−
− −
−
−
⎧ ⎫×Δ⎪ ⎪⎪ ⎪ +⎨ ⎬⎡ ⎤⎛ ⎞⎪ ⎪− − − ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ − + + ×Δ
= − +
(H-61)
( ) ( )( )( )
00 0 0 022 12 2 1 2 1 2
0 0 0 02 1 2 20 0
002 2 2 2
2 1
2
120
12
2 1 '
1'
6
nnn n n n
n n
n n
n
Bvar EE E E Em mRvarBvarvar var avgvar var
Bref Bvar BrefE E E Em m varref Bvar ref Bvar
Bvar E EEmvarBvar Bvar avg
Bvar
gVR R C GT T
n V l V lC CT T
gC C T
υ
α
α
−−
− −
− −
−
−
= + + +Δ
= − + + + ×Δ
− ×Δ
+
mT
( )( )
( ) ( )( )
00
02 1
0 0 02 1 2 10
02 1
2
21
20
'1 1 1
'1 12 3 2'2 6
2 '
16
n
n n
n
Bvar Brefvar
Bvar EEmvar avgBvar E E
mvar Bvar avg
Bref Bvar BrefE E Em varref Bvar Bvar
BvarEvarBvar
lgIC T
gI C T
n V l lC CT
C
αα
α
α
−
− −
−
−
⎧ ⎫×Δ⎪ ⎪⎪ ⎪⎡ ⎤+ × +Δ ⎨ ⎬⎢ ⎥ ⎡ ⎤⎛ ⎞⎣ ⎦ ⎪ ⎪− − + ×Δ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
≈ + + ×Δ
− ( )( )( )( )
( )( )( )
02 1
002 1 2 1
002 1 2 1
0 0 02 1 2 10
00
120
120
120
'
1'
61 1 1
'2 2 6
2 '
2
n
n n
n n
n n
E EmBvar avg
Bvar E EEmvarBvar Bvar avg
Bvar E EEmvarBvar Bvar avg
Bref Bvar BrefE E Em varref Bvar Bvar
Emref
gC T
gC C T
gC C T
n V l lC CT
n V T
α
α
α
α
−
− −
− −
− −
−
−
−
×Δ
− ×Δ
⎛ ⎞+ + ×Δ⎜ ⎟⎝ ⎠
= + + ×Δ
= 02n
BrefEBvar lC+
Summary of (H-56) of (H-61) Position Results:
77
( )
( )( )( )
0 0 0 0 0 09 92 1 20 0
0 0 099 0
009 9
120
For 4 :
2 1 2
For 4 :194
1 1 12 4 3
n nE E E E E EBref Bref
m mref Bvar ref Bvarvar var
Bvar BrefE E Em varref Bvarvar
Bvar E EEmvarBvar Bvar avg
n
n nV VR RC l C lT T
n
IV lR CT
gC C T
α
α
− −−
−−
− −
−
< −
= − + = +
= −
⎡ ⎤= − + + ×Δ⎢ ⎥⎣ ⎦
⎛ ⎞+ − ×Δ⎜ ⎟⎝ ⎠
0 0 088 0
8
For 0 4 :
BrefE E Emref Bvarvar V lR CT
n−−
= − +
> > −
( )0 0 02 12 1 0
2 1nn
BrefE E Emref Bvarvar n VR CT −−
= − + l (H-62)
( )
( )
0 0 022 0
0 0 0 0 01 01 00
0 0 011 0
2
For 0 :1
'8
For 1:1
'8
nnBrefE E E
mref Bvarvar
Bvar Bvar Bref BrefE E E E Em var varref Bvar Bvarvar var
Bvar BvarE E Em var varref Bvarvar
n V lR CT
n
IV lR RC CT
n
IVR CT
α α
α α
−−
= +
=
⎧ ⎫⎡ ⎤= − + − − × =Δ Δ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭=
⎡ ⎤= + − − ×Δ Δ⎢ ⎥⎣ ⎦
l
( )
0 0 022 0
0 0 0 0 0 02 1 22 1 20 0
2
For 1:
2 1 2n nn n
Bref
BrefE E Emref Bvarvar
Bref BrefE E E E E Em mref Bvar ref Bvarvar var
l
V lR CT
n
n nV l VR RC CT T−−
⎧ ⎫⎨ ⎬⎩ ⎭
= +
>
= − + = + l
Recognizing that the inverse of a direction cosine matrix equals its transpose, the
terms in (H-62) become , generating (51) in [1].
( ) 10
nEBvarC
−
0nBvarEC
REFERENCES
[1] Savage, Paul G., “Generating Strapdown Specific-Force/Angular-Rate For Specified Attitude/ Position Variation From A Reference Trajectory”, SAI WBN-14026, www.strapdownassociates.com, April 21, 2020. http://www.strapdownassociates.com/Variation%20Trajectory%20Generator.pdf
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