16.360 Lecture 1
Units and dimensions
• Six fundamental International System of Units
Dimensions Unit symbol
Length meter m
Mass kilogram kg
Time second s
Electric Current Ampere A
Temperature Kelvin K
Amount of substance
mole mol
• any other dimension can be derived from the fundamental dimensions, e.g.:
2/ skgmdt
dvmmaF
3/ AskgmIdt
F
q
FE
16.360 Lecture 2
Electric field
• Electric forces on point charges, Columb’s law
,4 2
120
2112
^
12 R
qqRF
,4 2
210
2121
^
21 R
qqRF
,4 12
0121 Eq
R
qqF
,4 2
0
^
R
qRE
16.360 Lecture 2
Magnetic field by constant current
r
I B = 2r
I
= r 0,
r: relative magnetic permeability
r =1 for most materials
=2r
I
H = B
16.360 Lecture 3
Traveling wave
y(x,t) = Acos(2t/T-2x/),
(x,t) = 2t/T-2x/, y(x,t) = Acos(x,t),
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5
t = 0T
t = 0.2 T
t = 0.4T
t = 0.6T
16.360 Lecture 3
Traveling wave
y(x,t) = Acos(2t/T+2x/),
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.4 0.8 1.2 1.6 2 2.4
t = 0T
t = 0.2 T
t = 0.4T
t = 0.6T
Velocity = 0.6/0.6T = /T
Vp = dx/dt
= - /T
Phase velocity:
16.360 Lecture 3
• Phasor
VR(t)
Vs(t) VC(t)
i (t)
Vs(t) = V0Sin(t+0),
VR(t) = i(t)R,
VC(t) = i(t)dt/C,
Vs(t) = VR(t) +VC(t),
V0Sin(t+0) = i(t)dt/C + i(t)R, Integral equation,
Using phasor to solve integral and differential equations
16.360 Lecture 3
• Phasor
Z(t) = Re( Z ejt
)
Z is time independent function of Z(t), i.e. phasor
Vs(t) = V0Sin(t+0)
)j(0 - /2)= Re(V0 e
jte
jte= Re(V ),
V = V0 e j(0 - /2) ,
16.360 Lecture 3
• Phasor
i(t) = Re( I ejt
)
), = Re(I jte
i(t)dt = Re( I e jt )dt
j1
V0Sin(t+0) = i(t)dt/C + i(t)R,
time domain equation:
phasor domain equation:
)(tf f
)(tfdt
dfj
dttf )( fj
Time Phasor
VR(t)
Vs(t) VC(t)
i (t)
V + I R , = IjC
1
16.360 Lecture 3
• Phasor domain
Back to time domain:
V + I R , = IjC
1
I = V
R + 1/(jC)
= R + 1/(jC)
V0 e j(0 - /2)
,
i(t) = Re( I ejt
) = Re ( jt
) R + 1/(jC)
V0 e j(0 - /2)
e
VR(t)
Vs(t) VC(t)
i (t)
V0Sin(t+0) = i(t)dt/C + i(t)R,
16.360 Lecture 3
• An Example :
VL(t)
Vs(t) = V0Sin(t+0),
VR(t) = i(t)R,
VL(t) = Ldi(t)/dt,
Vs(t) = VR(t) +VL(t),
V0Sin(t+0) = Ldi(t)/dt + i(t)R, differential equation,
Using phasor to solve the differential equation.
VR(t)
Vs(t)
i (t)
16.360 Lecture 3
• Phasor
i(t) = Re( I ejt
)
), = Re(Ijt
e
di(t)/dt = Re(d I e jt )/dt
j
V0Sin(t+0) = Ldi(t)/dt + i(t)R,
time domain equation:
phasor domain equation:
jte Re(V ) Re( I e
jt), )L + = Re(I
jtej
16.360 Lecture 3
• Phasor domain
Back to time domain:
V + I R, = I jL
I = V
R + (jL)
= R + jL)
V0 e j(0 - /2)
,
i(t) = Re( I ejt
) = Re ( jt
) R + (jL)
V0 e j(0 - /2)
e
16.360 Lecture 3
• Steps of transferring integral or differential equations to linear equations using phasor.
1. Express time-dependent variables as phsaor.2. Rewrite integral or differential equations in phasor domain.3. Solve phasor domain equations4. Change phasors variable to their time domain value
16.360 Lecture 3
• Waves in phasor domain
Recall waves, traveling wave in time domain
)22
cos(),( 0
tT
xAtxy
In phasor domain
02
)(
xjAexy + x direction
- x direction02
)(
xjAexy
16.360 Lecture 3
• A question
Answer: a traveling wave in phasor domain
What’s this?
xjAexy
2
)(
Complex amplitude