Post on 08-Aug-2015
transcript
Dr. Rakhesh Singh Kshetrimayum
7. Transmission line analysis
Dr. Rakhesh Singh Kshetrimayum
5/20/20131 Electromagnetic Field Theory by R. S. Kshetrimayum
7.1 Introduction Transmission line analysis
Introduction
Wave equation
Lossless line
High reactance effect
Telegrapher’s equations
Lossy line Smith chart
Ideal
Terminated
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Line junction
Distributed element concept Fig. 7.1 Transmission line
λ/4 transformerLine impedance
Terminated
IdealImpedance
Admittance
Radiation effect
Sizeeffect
7.1 IntroductionHigh reactance effect
Consider a 10-V ac source is connected to a 50 Ω load by a small copper wire of 1 mm radius
Assume that the dc resistance of the wire is R=1m Ω and inductance of L=0.1 µH
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inductance of L=0.1 µH
At 10 GHz, inductive reactance is jXL=jωL≈6283 Ω and hence all the ac signal will die out in the wire itself
The load will not receive any signal
Hence we need special devices which will take these signals from the source to the load
7.1 IntroductionRadiation effect
An accelerating or decelerating charge radiates electromagnetic energy
Besides the energy radiated from a current carrying conductor depends on the frequency of current flowing
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conductor depends on the frequency of current flowing
You might have observed this when you study Herz dipole (an infinitesimally current carrying element)
( )232
0 /ˆ4
sin
2
1mWattr
r
dlISavg
ωε
β
π
θ
=
7.1 Introduction Hence the radiation power loss is directly proportional to the square of the frequency of the ac current flowing
So there will be high loss of power
We definitely cannot use open wires for transferring energy or signal
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or signal
7.1 Introduction7.1.1 Introduction
What is a transmission line?
A structure, which can guide electrical energy from one point to another
Generally, a transmission line is a two parallel conductor system
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a two parallel conductor system one end of which is connected to a source and the other end is connected to a load
Examples: coaxial cables waveguides microstrip lines
7.1 Introduction
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Fig. 7.2 (a) Transmission lines examples (b) General transmission line structure
7.1 Introduction Two conductor systems could support transverse electromagnetic (TEM) waves
Both electric and magnetic fields are perpendicular (transverse) to the direction of the propagation
It is guided wave between these two conductors
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It is guided wave between these two conductors
Hence the radiation losses are minimized
What is microwave frequency?
300 MHz to 30 GHz (λ=1m to 10 cm)
Nowadays it is meant for frequency up to 300 GHz (λ=1cm)
7.1 IntroductionSize effect
Size of commonly used lump elements like capacitor, inductor and resistor are of the order of cm
Now this size is comparable to the microwave wavelength
Hence the phase βl=(2πf/c)l of the electrical signal might
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Hence the phase βl=(2πf/c)l of the electrical signal might vary along the length of the device
For instance, consider a parallel plate capacitor
We assume capacitor conductor plate is an equipotentialsurface
7.1 Introduction But this is not true at microwave frequencies
Besides radiation also increases the problem
So we cannot use such capacitors at high frequencies
So we will see later that a section of a transmission line could be used as an inductor or resistor or series/shunt RLC
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be used as an inductor or resistor or series/shunt RLC resonators
If we increase the frequency of operation of a circuit, Usually we require temporal analysis at low frequency
we can’t neglect ‘space’ in the circuit analysis due to size effect
7.1 Introduction7.1.2 Causal effect
What is causal effect?
EM wave requires a finite time to travel along an electrical circuit
Since no EM wave can travel with infinite velocity
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Since no EM wave can travel with infinite velocity (What is the maximum speed?)
A finite time delay between the 'cause' and the ‘effect’ Also known as the causal effect in physics
When is this effect important?
7.1 Introduction If the time period of the EM wave or signal
(T=1/f) >> the transit time (tr), we may ignore this effect
1r
l vT t l l
f v fλ>> ⇒ >> ⇒ >> ⇒ >>
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Causal effect becomes important when the length of the line (l) becomes comparable to the wavelength (λ)
As the frequency increases, the wavelength reduces, and
the Causal effect becomes more evident
7.1 Introduction7.1.3 Distributed vs lumped elements To overcome the effect of transit time or causality or size effect (more appropriate to use), one can chop off the transmission line into small sections
such that for each section, this causality effect is minuscule
At high frequencies,
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At high frequencies, the circuit elements cannot be defined for the whole transmission line instead it has to be defined for a unit length of the line
The circuit elements are not located at a point of the line but are distributed all along the length
7.1 Introduction Analysis of a transmission lines must be carried out using
the concept of distributed elements not as
lumped elements
as we used to do from our previous circuit analysis at the low frequencies
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frequencies
But, we can still employ lump element analysis of transmission lines by chopping off small sections of the line
so that the Causal effect is negligible in the chopped off sections
7.2 Telegrapher’s equations 7.2.1 Lumped element circuit model
Per unit length parameters: L=Series inductance per unit length
C=Shunt capacitance per unit length
R=Series resistance per unit length
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R=Series resistance per unit length
G= Shunt conductance per unit length
7.2 Telegrapher’s equations
z∆
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zR∆ zL∆
zG∆zC∆)t,z(I),t,z(V
)t,zz(I),t,zz(V ∆+∆+
Fig. 7.3 (a) Sub-section of length ∆z of a general transmission line and its (b) lumped element equivalent circuit
7.2 Telegrapher’s equations L represents the self-inductance of the two conductors (magnetic energy storage)
C is due to the close proximity of two conductors (electric energy storage)
R is due to the finite conductivity of the two conductors
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R is due to the finite conductivity of the two conductors (power loss due to finite conductivity of metallic conductors)
G is due to dielectric loss in the material between the conductors (power dissipation in lossy dielectric)
7.2 Telegrapher’s equations 7.2.2 Telegrapher’s equations
Let the voltage at the input be V and current at the input be I
Due to voltage drop in the series arm, the output voltage will be different from the input voltage, say V+∆V
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V+∆V
Due to current through the capacitance and the conductance, the output current will be different from the input current, say I+∆I
7.2 Telegrapher’s equations Applying Kirchoff’s voltage law (KVL) and Kirchoff’s current law (KCL)
i(z, t)v(z, t) R zi(z, t) L z v(z z, t) 0
t
δ− ∆ − ∆ − + ∆ =
δ
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v(z z, t)i(z, t) G zv(z z, t) C z i(z z, t) 0
t
δ + ∆− ∆ + ∆ − ∆ − + ∆ =
δ
7.2 Telegrapher’s equations Dividing the above two equations by Δz and taking the limit
Δz 0 (What is its implications?)
v(z, t) i(z, t)Ri(z, t) L
z t
δ δ= − −
δ δ
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z tδ δ
i(z, t) v(z, t)Gv(z, t) C
z t
δ δ= − −
δ δ
Telegrapher’s Equations
7.2 Telegrapher’s equations 7.2.3 Wave propagation
For time-harmonic signals, telegrapher’s equation reduces to
dV(z)(R j L)I(z)
dz= − + ω
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dz
( )-( ) ( )
dI zG j C V z
dzω= +
7.2 Telegrapher’s equations It is similar to Maxwell’s curl equations, hence, we can get wave equations
0V(z)γdz
V(z)d 2
2
2
=−
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0I(z)γdz
I(z)d 2
2
2
=−
( )( )j R j L G j Cγ α β ω ω= + = + +
Transmission line analysis Traveling wave solutions for the above two equations are
z z
0 0V(z) V e V e+ −γ − γ= +
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0 0( ) z zI z I e I e
γ γ+ − −= +
Point to be noted: current or voltage is wave which is a function of both space and time unlike the low frequency counter-parts (where is the time dependence?)
7.2 Telegrapher’s equations Physical interpretations
Wave phase has two components: time phase (ωt) and
space phase (βz)
Since βz is the phase of the wave as function of z,
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Since βz is the phase of the wave as function of z, β represents phase change per unit length of the transmission line for a traveling wave
phase constant (unit is radians per meter)
)φzβtωcos(eVeeeVReeeVRe zαzβjtωjzαφjzβjtωjzα +−== −+−−+−−+
7.2 Telegrapher’s equations For a positive α, the amplitude exponentially decreases as a function of z
α represents attenuation of the wave on the transmission line
attenuation constant of the line (unit is Nepers per meter, 1 Neper= 8.68dB)
zV e
α+ −
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1 Neper= 8.68dB)z z
0 0I(z) V e V eR j L
+ −γ − γγ = − + ω
0
R j L R j LZ
G j C
+ ω + ω= =
γ + ω
0 00
0 0
V VZ
I I
+ −
+ −= = −
7.2 Telegrapher’s equations The characteristic impedance Z0 of a transmission line is
defined as the ratio of positively traveling voltage wave to current wave at any point on the line
Now for a wave the distance over which the phase changes by 2H is called the wavelength 'λ’
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2H is called the wavelength 'λ’
phase change per unit length β=2H/λ
p
2 fv f
π ω= λ = =
β β
7.3 Lossless line7.3.1 Ideal lossless line
zL∆
zC∆)t,z(I),t,z(V)t,zz(I),t,zz(V ∆+∆+
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)t,z(I),t,z(V
Fig. 7.4 Lumped element equivalent circuit of a sub-section oflength ∆z of a lossless transmission line (R=G=0)
7.3 Lossless line
j j LCγ = α + β = ω LCβ = ω 0α =
LZ =
j z j zV(z) V e V e+ − β − β= +
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0ZC
= 0 0V(z) V e V e= +
j z j z0 0
0 0
V VI(z) e e
Z Z
+ −
− β β= −2 2
LC
π πλ = =
β ω
p
1v
LC LC
ω ω= = =
β ω
7.3 Lossless line7.3.2 Terminated lossless lines
β,Z
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β,Z0
Fig. 7.4 (b) A lossless transmission line terminated with load impedance ZL
7.3 Lossless line7.3.3 Reflection coefficient
At the load, z=0,
( )( ) 0
ooL Z
VV0VZ
−+
−+ +== 0L0 ZZV −
=Γ=+
−
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( ) 0
oo
ooL Z
VV0IZ
−+ −==
0L
0L
0
0
ZZV +=Γ=
+
( ) ( )zβjzβj0 eeVzV Γ+= −+ ( ) ( )zβjzβj
0
0 eeZ
VzI Γ−= −
+
z = −l ( ) ( ) lj
lj
lj
eeV
eVl
β
β
β2
0
0 0 −
++
−−
Γ==Γ ( )0
00ZZ
ZZ
L
L
+
−=Γ
7.3 Lossless line7.3.4 Power flow and return loss
Time average power flow along the line at the point z,
( ) ( ) 2
* 20 * 2 * 21 1Re Re 1
2 2
z z
avg
VS V z I z e e
Z
β β
+
− − = = − Γ + Γ − Γ
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02 2
avgZ
2
20
0
11
2avg
VS
Z
+
= − Γ
7.3 Lossless line When the load is mismatched,
not all of the available power from the generator is delivered to the load,
this “loss” is called Return loss (RL) and is defined in dB as
7.3.5 Standing wave ratio (SWR)
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7.3.5 Standing wave ratio (SWR)
Γ−= log20RL
( ) 2
0 1 j zV z V e
β+= + Γ2 j ze 1β = Γ+= + 10max VV
2 j ze 1β = − Γ−= + 10min VVΓ−
Γ+==
1
1
min
max
V
VVSWR
7.3 Lossless lineWhat is BW?
For acceptable value of VSWR = 2 within the operating frequency region of a device also known as bandwidth (BW)
54.93
1log20;
3
1
12
12
1
1=
−==
+
−=
+
−=Γ RL
VSWR
VSWR
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Return loss (RL) should be higher than 9.54, which is approximately 10 dB
RL ≥10 dB has become an acceptable definition for BW of many devices
33121 ++VSWR
7.3 Lossless line
max
min
IISWR
I=
( )2max maxV I
PSWR PSWR ISWR VSWRV I
= = × =
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For VSWR=2, only 89% of the incident power reaches the load
min minV I
( )( )
22
2
1 41 1
1 1
i rL
i i
P PP VSWR VSWR
P P VSWR VSWR
− − = = − Γ = − =
+ +
7.3 Lossless line
Point to be noted:
z= −l ( ) 2
01 j l
V l V eβ+ −− = + Γ
2
λ+l ( )
222
0 01 12
j lj l
V l V e V e V l
λβ
βλ
− + + + − − − = + Γ = + Γ = −
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Point to be noted: shortest distance between two successive maxima (or minima) is not λ but λ/2,
it is very important to realize this since in your experiment on Frequency and Wavelength measurements, this is a major mistake most of you make
7.3 Lossless line
the distance between adjacent maximum and minimum is λ/4
7.3.6 Transmission line impedance equation
4
λ+l
224
0 01 14
j lj l
V l V e V e
λβ
βλ
− + + + − − − = + Γ = − Γ
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7.3.6 Transmission line impedance equation
A certain value of load impedance at the end of a particular transmission line is transformed into another value of impedance at the input of the line
impedance transformer
7.3 Lossless line Transmission line impedance equation
( )( )
0
0 0
0 0
0 0
j l j lL
j l j l
L
in j l j l
j l j lL
Z Ze e
V e e Z ZV lZ Z Z
I l V e e Z Ze e
β ββ β
β ββ β
−
+ −
+ −
−
−+ + Γ +− = = =
− − Γ − −
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( ) ( )
( ) ( )
( ) ( )
0
0
00 0
0 0
0 0
j l j lL
L
j l j l j l j lj l j lLL L
j l j l
L L
e eZ Z
Z e e Z e eZ Z e Z Z eZ Z
Z Z e Z Z e Z
β β β ββ β
β β
− −−
−
− +
+ + − + + − = = + − − ( ) ( )
( ) ( )
( ) ( )
0
0 00 0
00
cos sin tan( )
tan( )cos sin
j l j l j l j l
L
L L
LL
e e Z e e
Z l jZ l Z jZZ Z
Z jZZ l jZ l
β β β β
β β β
ββ β
− − + + −
+ + = =+ +
l
l
7.3 Lossless lineFig. 7.5 Transmission line impedance
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β,Z0
inZ
7.3 Lossless line7.3.7 Quarter-wave transformer
n4 2
λ λ= +l
( )
2 2n n n
4 2 4 2 2
tan tan n2
λ λ π λ π λ πβ = β + β = + = + π
λ λ
π ⇒ β = + π = ∞
l
l
2Z jZ tan( ) jZ tan( ) Z+ β β
0 L inZ Z Z=
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How to do impedance matching for a complex load using quarter-wave transformer?
2
L 0 0 0in 0 0
0 L L L
Z jZ tan( ) jZ tan( ) ZZ Z Z
Z jZ tan( ) jZ tan( ) Z
+ β β⇒ = = =
+ β β
l l
l l
7.3 Lossless line7.3.8 Special cases of lossless terminated lines
Terminated in a short circuit
Terminated in open circuit
sc L 0in 0 0
0 L
Z jZ tan( )Z Z jZ tan( )
Z jZ tan( )
+ β= = β
+ β
ll
l
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Terminated in open circuit
Terminated with matched load
oc L 0in 0 0
0 L
Z jZ tan( )Z Z jZ cot( )
Z jZ tan( )
+ β= = − β
+ β
ll
l
0ZZ
ZZ
00
00 =+
−=Γ
7.3 Lossless line Another important observation is that
if we measure open and
short circuit input impedances
of a lossless transmission line and
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multiply those two values and
take the square root
what we have is the characteristic impedance of the line (one of the methods for finding the characteristic impedance of a given line in laboratory)
7.3 Lossless line7.3.9 Reflection and transmission at the transmission line junction
Γ τ
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Fig. 7.7 Junction of two transmission line with different characteristic impedance
7.3 Lossless line For z<0,
characteristic impedance Z0;
z>0, characteristic impedance Z1 and
the junction of the two transmission lines is at z=0
At the junction,
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At the junction, looking from z<0 towards the right, it sees an infinite transmission line of characteristic impedance Z1 and
hence it is equivalent to ZL=Z1 for the transmission line z<0
Assuming ζ is the transmission coefficient and IL is insertion loss in dB
7.3 Lossless line
01
01
ZZ
ZZ
+
−=Γ
z 0< ( ) ( )zβjzβj0
eeVzV Γ+= −+
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z 0> j z
0V(z) V e+ − β= τ
z 0=01
1
01
01 211
ZZ
Z
ZZ
ZZ
+=
+
−+=Γ+=τ
IL 20log= − τ
7.4 Lossy lines One type of metal loss is I2R loss
In transmission lines, the resistance of the conductors is never equal to zero
except for superconductors
Whenever current flows through one of these conductors,
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Whenever current flows through one of these conductors, some energy is dissipated in the form of heat
7.4 Lossy lines Another type of loss is due to skin effect
Current in the center of the wire becomes smaller and
most of the electron flows on the wire surface
When the frequency applied is in the GHz range, the electron movement in the center is so small
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the electron movement in the center is so small
that the center of the wire could be removed without any noticeable effect on the current
7.4 Lossy lines Note that the effective cross-sectional area
decreases as the frequency increases
Since resistance is inversely proportional to the cross-sectional area (R=ρl/A),
the resistance will increase as the frequency is increased
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the resistance will increase as the frequency is increased
Also, since power loss increases as resistance increases,
power losses increase with an increase in frequency
because of the skin effect
7.4 Lossy lines Dielectric losses result from
the heating effect on the dielectric material between the conductors
Power from the source is used in heating the dielectric
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The heat produced is dissipated into the surrounding medium When there is no potential difference between two conductors, the atoms in the dielectric material between them are normal and
the orbits of the electrons are circular
7.4 Lossy lines When there is a potential difference between two conductors, the orbits of the electrons change
The excessive negative charge on one conductor repels electrons on the dielectric toward the positive conductor
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repels electrons on the dielectric toward the positive conductor and
thus distorts the orbits of the electrons
A change in the path of electrons requires more energy, introducing a power loss
7.4 Lossy lines Induction losses occur
when the electromagnetic field about a conductor cuts through any nearby metallic object and
a current is induced in that object
As a result,
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As a result, power is dissipated in the object and
is lost
7.4 Lossy lines Radiation losses occur
because some magnetic lines of force about a conductor do not return to the conductor when the cycle alternates
These lines of force are projected into space as radiation, and
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projected into space as radiation, and
these results in power losses
That is, power is supplied by the source,
but is not available to the load
7.4 Lossy lines7.4.1 Ideal lossy line characteristics
( )( )j R j L G j Cγ α β ω ω= + = + + R G( j L)( j C) ( 1)( 1)
j L j C
= ω ω + +
ω ω
−
+−=
RGj
Cω
G
Lω
R1LCωj
2
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−
+−=LCω
jCωLω
1LCωj2
CωG,LωR <<<< 2RG LC<< ω1
12
R Gj LC j
L Cγ ω
ω ω
≅ − +
0
0
1 1
2 2
C L RR G GZ
L C Zα
∴ ≅ + = +
LCβ ≅ ω 0
R j L LZ
G j C C
ω
ω
+= ≅
+
Low loss case
7.4 Lossy lines7.4.2 Terminated lossy lines
γ,Z0
( ) ( )lleeVlV
γγ −++ Γ+=− 0
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( ) ( )0
0
Z
eeVlI
ll γγ −++ Γ−=−
( ) ( ) ( ) lβj2lα2lβj2lα2lγ2
lγ
lγ
0
0 eeee0e0e
e
V
Vl −−−−−
+
−
+
−
Γ=Γ=Γ==Γ
Fig. 7.8 (a) A lossy transmission line terminated with load impedance ZL
7.4 Lossy lines
l lL 0l l
0 L 0
in 0 0l l
0 l lL 0
L 0
Z Ze e
V e e Z ZV( )Z Z Z
I( ) V e e Z Ze e
Z Z
γ −γ
+ γ −γ
+ γ −γ
γ −γ
−+ + Γ +− = = =
− − Γ − − +
l
l
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( ) ( )
( ) ( )
( ) ( )( )
L 0
l l l ll lL 0L 0 L 0
0 0l l l lL 0 L 0 0 L
Z Z
Z e e Z e eZ Z e Z Z eZ Z
Z Z e Z Z e Z e e Z
γ −γ γ −γγ −γ
γ −γ γ −γ
+
+ + − + + − = = + − − + + ( )
( ) ( )
( ) ( )
l l
L 0 L 00 0
0 L0 L
e e
Z cosh l Z sinh l Z Z tanhZ Z
Z Z tanhZ cosh l Z sinh l
γ −γ −
γ + γ + γ = =+ γγ + γ
l
l
7.4 Lossy lines
in
1P Re V( )I ( )
2
∗= − −l l( ) ( )
Γ−Γ+=
−++−++
*
0
00Re
2
1
Z
eeVeeV
llll
γγγγ
2
20 * * * * *
0
1Re
2
l l l l l l l lV
e e e eZ
γ γ γ γ γ γ γ γ
+
+ − − + − −= − Γ + Γ − Γ
2
V+
V2
+
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What happens to Ploss when α increases?
2
20 2 * 2 2 2
0
1Re
2
l j l j l lV
e e e eZ
α β β α
+
− −= − Γ + Γ − Γ llee
Z
Vαα 222
0
2
0
2
1 −
+
Γ−=
( )2
0
2
01
2
1Γ−=
+
Z
VPL
( ) ( )[ ]112
1 222
0
2
0+−Γ+−=−= −
+
ll
Linloss eeZ
VPPP
αα
7.4 Lossy lines7.4.3 Introduction to electromagnetic resonators:
λ
inZ
inZ
λ
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2
λ
0,Z jγ α β= + 0
LZ =
inZ
0,Z jγ α β= + L
Z = ∞
inZ
4
Fig. 7.8 (b) Series RLC resonant circuit (c) Tank or shunt RLCresonant circuit (d) O.C. terminated transmission line of length λ/4and (e) S.C. terminated transmission line of length λ/2
7.4 Lossy lines Microwave/electromagnetic resonators are used in many applications: filters,
oscillators,
frequency meters,
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum57
frequency meters,
tuned amplifiers, etc.
Its operations are very similar to the series and
parallel RLC resonant circuits
7.4 Lossy lines
We will review the series and
parallel RLC ciruits and
discuss the implementation of the microwave resonators using distributive elements such as microstrip line,
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum58
microstrip line,
rectangular and
circular waveguides, etc.
Series RLC resonant circuits
Consider the series RLC resonator
The input impedance Zin is given by1
inZ R j Lj C
ww
= + +
7.4 Lossy lines The average complex power delivered to the resonator is
The average power dissipated by the resistor is
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The average power dissipated by the resistor is
21
2lossP I R=
7.4 Lossy lines The time-averaged energy stored in the inductor is (recall the energy stored in the inductor)
Similarly, the time-averaged energy stored in the capacitor is
21
4mW L I=
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum60
2 22
2 2 2
1 1 1
4 4 4e c
C I IW C V
C Cw w= = =
7.4 Lossy lines The input impedance can then be expressed as follows:
At resonance,
( )2 2
2 m elossinin
P j W WPZ
R R
w+ -= =
( )2in m elossP P j W Ww\ = + -
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At resonance, the average stored magnetic and electric energies are equal,
therefore, we have
m eW W=21
2
lossin
PZ R
I
= =
7.4 Lossy lines Hence, the resonance frequency is defined as
The quality factor is defined as the product of the angular frequency and
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum62
the angular frequency and
the ratio of the average energy stored to
energy loss per second
7.4 Lossy lines Q is a measure of loss of a resonant circuit,
lower loss implies higher Q and
high Q implies narrower bandwidth
As R increases, power loss increases and
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum63
power loss increases and
quality factor decreases
Let us see what the approximate Zin near resonance
The input impedance can be rewritten in the following form:
7.4 Lossy lines
The above form is useful for
Near by the resonance
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The above form is useful for finding equivalent circuit
near the resonance,
for example, we can find out
the resistance at resonance and so as L
7.4 Lossy lines Half power fractional bandwidth
When the real power delivered to the circuit is half that of the resonance, occurs when
2inZ R=
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum65
7.4 Lossy linesShunt RLC Resonant Circuits
Now let us turn our attention to the parallel RLC resonator
The input impedance is equal to
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum66
7.4 Lossy lines The average complex power delivered to the resonator is
The average power dissipated by the resistor is
*1
2inP V I=
*
*
1
2 in
VV
Z=
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum67
21
2loss
VP
R=
7.4 Lossy lines The time-averaged energy stored in the inductor is (recall the energy stored in the inductor)
Similarly, the time-averaged energy stored in the capacitor is
2 22
2 2 2
1 1 1
4 2 2m L
V VW L I L
L Lw w= = =
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Similarly, the time-averaged energy stored in the capacitor is
21
4cW C V=
7.4 Lossy lines The input impedance can then be expressed as follows:
At resonance, the average stored magnetic and electric energies are equal,
( )2
2
22
1
2
loss m cinin
P j W WPZ
I I
w+ -= =
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the average stored magnetic and electric energies are equal, therefore,
we have (same results as in series RLC )
( )2in loss m cP P j W Ww\ = + -
0
1
LCw =
7.4 Lossy lines The quality factor, however, is different
Contrary to series RLC,
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Contrary to series RLC, the Q of the parallel RLC increases
as R increases
7.4 Lossy lines Similar to series RLC,
we can derive an approximate expression for parallel RLC near resonance
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7.4 Lossy lines As in the series case,
the half-power bandwidth is given by2
2
2in
RZ =
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum72
7.4 Lossy lines We discuss the use of transmission lines to realize the RLC resonator
For a resonator, we are interested in Q and therefore, we need to consider lossy transmission lines
Short-circuited λ/2 line
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Short-circuited λ/2 line
Note that
tanh(A+B)
=(tanhA + tanh B)/(1+ tanhA tanh B)
7.4 Lossy lines
Consider the transmission line equation
tan( )[ ] / ( )
[ ] /x
e e j
e e
jx jx
jx jx====
−−−−
++++
−−−−
−−−−
2
2
tanh( )[ ] /
[ ] /x
e e
e e
x x
x x====
−−−−
++++
−−−−
−−−−
2
2
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For a short-circuited line
7.4 Lossy lines Our goal here is to
compare the above equation
with input impedance of Series or
shunt RLC resonant circuit near resonance
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum75
so that we can find out the corresponding R, L and C
For a length l=λ/2 of the transmission line, assuming a TEM line so that
7.4 Lossy lines
For low-loss transmission lines, αl is small, hence
ββββ ωωωω µεµεµεµε ωωωω==== ==== / vp l vp o==== ====λλλλ ππππ ωωωω/ /2
ββββωωωω ωωωω
ππππωωωω
ll
v
l
v
l
vp
o
p p o
==== ==== ++++ ==== ++++∆ω∆ω∆ω∆ω ∆ωπ∆ωπ∆ωπ∆ωπ
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum76
For low-loss transmission lines, αl is small, hence
tan tan( ) tan( )ββββ ππππωωωω ωωωω ωωωω
lo o o
==== ++++ ==== ≈≈≈≈∆ωπ∆ωπ∆ωπ∆ωπ ∆ωπ∆ωπ∆ωπ∆ωπ ∆ωπ∆ωπ∆ωπ∆ωπ
7.4 Lossy lines Note that the loss is usually very small and
therefore, the input impedance can be rewritten as:
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum77
7.4 Lossy lines This equation can be compared favorably
with the input impedance of a series RLC resonant circuit near the resonance
It behaves like a series RLC resonator with
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum78
7.4 Lossy lines As α increases,
Q decreases
which is according to our expectation
Open-Circuited λ/4 Line
For a lossy line of length l
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For a lossy line of length l
with propagation constant γ and
characteristic impedance Z0,
we can find the input impedance for a load of ZL as follows:
7.4 Lossy lines For o.c.,
For
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l vp o==== ====λλλλ ππππ ωωωω/ / ( )4 2
ββββωωωω ωωωω ππππ
ωωωωl
l
v
l
v
l
vp
o
p p o
==== ==== ++++ ==== ++++2 2 2 2 2
∆ω∆ω∆ω∆ω ∆ωπ∆ωπ∆ωπ∆ωπ
7.4 Lossy lines
Knowing that tan d = d when d is small
The input impedance can be written as,
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum81
The input impedance can be written as,
7.4 Lossy lines
This equation can be compared favorably with the input impedance of a series RLC resonant circuit near the resonance
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum82
the resonance
It behaves like a series RLC resonator with
7.4 Lossy lines As α increases,
Q decreases
which is according to our expectation
We can extend this analysis for a s.c. λ/4 lines,
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum83
s.c. λ/4 lines,
o.c. λ/2 lines
and so on
7.5 Smith chart7.5.1 Impedance Smith chart
Smith chart is basically a graphical representation of
transmission line impedance transformation formula:
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L 0
in 0
0 L
Z jZ tan( )Z Z
Z jZ tan( )
+ β=
+ β
l
l
7.5 Smith chart If we represent this in x-y coordinates with
x as real part and y as imaginary part of and
then it becomes a semi-infinite plane, not practical
We know that the
LZinZ
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum85
We know that the modulus of reflection coefficient (|Γ|) is always less than or equal to 1
And there is one to one correspondence between Γ andinZ
7.5 Smith chart
we will draw normalized constant resistance and
in 0
in 0
Z ( ) Z( )
Z ( ) Z
−Γ =
+
ll
l
inin
0
Z 1 ( )Z
Z 1 ( )
+ Γ= =
− Γ
l
l
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constant reactance contours
in the reflection coefficient plane which is a circle of
A movement of d distance along the transmission line is equivalent to change in the reflection plane
1Γ ≤
2 j de− β
7.5 Smith chart Distance in movement in terms of wavelength is given in the circumference of the circle
It could be either towards load (WTL) or
source (WTG)
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source (WTG)
At first glance, Smith chart looks intimidating with so many contours of
constant resistance and
reactance
7.5 Smith chart Smith chart as a polar plot of Γ
(o.c. open circuit and
s.c. short circuit)
It can be simply
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum89
Γ
θ
j 0 0e , 180 180θΓ = Γ − ≤ θ ≤
It can be simply interpreted as a polar plot of
7.5 Smith chart The real utility of Smith chart lies
in the fact that we can read the corresponding normalized impedance value of Γ
from the constant reactance and resistance contours
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum90
in r iin in in
0 r i
Z 1 j1Z R jX
Z 1 1 j
+ Γ + Γ+ Γ= = + = =
− Γ − Γ − Γ
7.5 Smith chart constant resistance circles
constant reactance circles
( )2 2
2inr i
inin
R 1
R 1 R 1
Γ − + Γ =
+ +
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( )22
2 111
=
−Γ+−Γ
inin
irXX
7.5 Smith chart Constant resistance circles
(WTG Wavelength towards generator and
WTL Wavelength towards load)
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2Rin =
1=inR
5.0=inR
+1-1
WTG
WTL
7.5 Smith chart Constant reactance circles of an impedance smith chart
2=inX
1=inX
5.0=X
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2−=inX
1−=inX
5.0=inX
5.0−=inX
7.5 Smith chart In many applications,
transmission line and impedances are connected in parallel (shunt),
then, the admittance analysis is more convenient than the impedance analysis
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impedance analysis
L 0 L 0 0 L L L
L 0 0 L L L
L 0
1 1
Z Z Y Y Y Y 1 Y Y 1
1 1Z Z Y Y 1 Y Y 1
Y Y
−− − − −
Γ = = = = = −+ + + ++
7.5 Smith chart Rules for conversion of impedance (say ZL at N) to admittance (say YL at N )
Γ
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum95
7.5 Smith chart The admittance smith chart is therefore obtained by
rotating the impedance Smith chart by π and
replacing r by g and x by b
Since it is just a matter of rotation, there is no need to have separate Smith charts for impedance
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum96
there is no need to have separate Smith charts for impedance and admittance
Although r and x can be interchanged with g and b respectively and
a point (r,x) and (g,b) will have the same spatial location on the Smith chart for r=g and x=b,
7.5 Smith chart But, the physical interpretation corresponding to the two will not be identical
Upper half of the impedance Smith chart with +jx represent inductive loads
whereas +jb represents
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum97
whereas +jb represents capacitive load on the admittance Smith chart
Point B on impedance Smith chart represents s.c.
whereas point B on admittance Smith chart represent o.c.
7.5 Smith chart Interchange on location of o.c./s.c. and
location of VSWR on an impedance Smith chart
Inductive/Capacitive
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum98
AB CD
Capacitive/Inductive
7.5 Smith chart Point A on impedance Smith chart which represents o.c.
whereas point A on admittance Smith chart which represents s.c.
Note that the distance between o.c. and s.c. is λ/4
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7.6 Summary Transmission line analysis
Introduction
Distributed element concept
Lossless line
Transit time effect
Telegrapher’s equations
λ/4 transformer
Line impedance
Ideal
Lossy lineSmith chart
Ideal
Impedance
Admittancev(z, t) i(z, t)
Ri(z, t) Lz t
δ δ= − −
δ δi(z, t) v(z, t)
Gv(z, t) Cz t
δ δ= − −
δ δ
j j LCγ = α + β = ω
0
LZ
C=
0 L inZ Z Z=
0
0
1 1
2 2
C L RR G GZ
L C Zα
∴ ≅ + = +
LCβ ≅ ω
0
R j L LZ
G j C C
ω
ω
+= ≅
+
5/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum100
Line junctionWave equation
Fig. 7.1 Transmission line in a nutshell
Line impedance
Terminated
Terminated
0V(z)γdz
V(z)d 2
2
2
=−
0I(z)γdz
I(z)d 2
2
2
=−
( )( )j R j L G j Cγ α β ω ω= + = + +
0 00
0 0
V VZ
I I
+ −
+ −= = −
j z j z
0 0V(z) V e V e+ − β − β= +
j z j z0 0
0 0
V VI(z) e e
Z Z
+ −
− β β= −L 0
in 0
0 L
Z jZ tan( )Z Z
Z jZ tan( )
+ β=
+ β
l
l
01
1
01
01 211
ZZ
Z
ZZ
ZZ
+=
+
−+=Γ+=τ
( ) ( )zβjzβj0 eeVzV Γ+= −+
( ) ( )zβjzβj
0
0 eeZ
VzI Γ−= −
+
Γ−
Γ+==
1
1
min
max
V
VVSWR
02 2L C Z
( ) ( )lleeVlV
γγ −++ Γ+=− 0
( ) ( )0
0
Z
eeVlI
ll γγ −++ Γ−=−
( )( )lZZ
lZZZZ
L
Lin
γ
γ
tanh
tanh
0
00
+
+=