Impedance

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Impedance
Young Won Lim
05/26/2015
Copyright (c) 2015 Young W. Lim.
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Young Won Lim
05/26/2015
Sinusoidal voltage and current
d vC
iC = C⋅
dt
d iL
v L = L⋅
dt
v C = A cos(ω t )
i L = A sin (ω t )
iC = −ω C A sin(ω t )
= A cos (ω t−π /2)
= ω C A cos(ω t + π/ 2)
v L = ω L A cos(ω t)
vC
cos(ω t)
1
=
iC
ω C cos(ω t + π/ 2)
vL
cos(ω t)
=ωL
iL
cos(ω t −π / 2)
[iC , v C ]
[v L , i L ]
leads by 90
Impedance
leads by 90
3
Young Won Lim
05/26/2015
Leading and Lagging Current
d vC
iC = C⋅
dt
d iL
v L = L⋅
dt
v C = cos(2 π t )
i L = sin (2 π t )
iC = −π sin (2 π t )
= cos(2 π t −π/ 2)
= π cos (2 π t + π / 2)
v L = π cos(2 π t)
C=0.5
L=0.5
vL
iC
vC
Impedance
iL
4
Young Won Lim
05/26/2015
Leading and Lagging Current
d vC
iC = C⋅
dt
d iL
v L = L⋅
dt
IC
−j
ωC
VL
VC
jωL
constant i
vL
iC
vC
Impedance
IL
iL
5
Young Won Lim
05/26/2015
Phasor and Ohm's Law
sinusoidal
i, v source
sinusoidal
i, v source
sinusoidal
i, v
sinusoidal
i, v
IC
VL
VC
IL
Impedance
6
Young Won Lim
05/26/2015
Impedance : a complex scaling factor
VC = A
0
IC = ωC A
ZC =
IL = A
π/2
VC
IC
VL= ωLA
phasor
phasor
ZL =
a complex scaling factor
V C = I C⋅Z C = I C
ZC =
Impedance
−π /2
(
1
ωC
VL
IL
0
phasor
phasor
a complex scaling factor
−π / 2
)
V L = I L⋅Z L = I L ( ω L
−j
1
=
ωC
j ωC
+π/ 2 )
ZL = j ω L
7
Young Won Lim
05/26/2015
Ohm's Law
VC = A
IC = ωC A
ZC =
0
IL = A
π/2
VL= ωLA
VC
1
=
−π / 2
IC
ωC
−j
1
=
=
ωC
j ωC
ZL =
VL
= ωL
IL
0
+π/2
= jω L
vL
iC
vC
Impedance
−π /2
iL
8
Young Won Lim
05/26/2015
Phasor Example (1)
A cos(ω t + 0)
A 0
A +π/2
A cos(ω t + π /2)
A
π/2
A −π/2
A cos(ω t − π /2)
A
−π/2
A −π
A
A cos(ω t − π)
−π
Impedance
9
Young Won Lim
05/26/2015
Phasor Example (2)
A cos(ω t + 0)
A 0
2 A cos(ω t − π / 2)
2 A −π/2
Impedance
10
Young Won Lim
05/26/2015
Phasor
A θ
A cos(ω t + θ)
= A cos(ω t) cos(θ) − A sin(ω t )sin(θ)
= A cos(θ) cos(ω t) − A sin(θ)sin(ω t)
= X cos(ω t) − Y sin(ω t )
A cos(θ) = X
A sin(θ) = Y
A =
√ X 2+Y 2
tan θ =
θ>0
Impedance
( X , Y ) = ( A cos θ , A sin θ)
Y
X
leading
X + j Y = A cos θ+ j A sin θ
θ<0
lagging
11
Young Won Lim
05/26/2015
Linear Combination of cos(ωt) & sin(ωt)
√ X 2 +Y 2 cos(ω t −θ)
X cos(ω t )+Y sin(ω t )
X cos(ω t )+Y sin (ω t)
[
X cos(ω t )+Y sin (ω t)
= √ X 2 +Y 2 √ X X+Y cos(ω t )+ √ XY+ Y sin (ω t )
2
2
2
2
]
= √ X 2 +Y 2 cos(ω t −θ)
= √ A 2 + B 2 [ cos(θ)cos(ω t)+sin (θ)sin (ω t ) ]
cos(θ) =
= √ X 2 +Y 2 cos(θ−ω t )
sin (θ) =
= √ X +Y cos(ω t −θ)
2
2
X
√ X 2 +Y 2
Y
√ X 2+Y 2
X cos(ω t )−Y sin(ω t )
√ X 2 +Y 2 cos(ω t +θ)
X
Y
cos(ω t )− sin (ω t )
A
A
A cos(ω t +θ)
cos (θ) cos(ω t )−sin (θ)sin (ω t )
Impedance
12
Young Won Lim
05/26/2015
RC Circuit
e S (t)
v C (t)
v R (t )
v R (t )
I s (t)
I s (t )
R = 6Ω
v C (t )
XC = 8 Ω
Impedance
13
Young Won Lim
05/26/2015
Current Source
I s (t)
5
0.927 rad
Impedance
14
Young Won Lim
05/26/2015
Resistor
v R (t )
30
v R = i R⋅R
0.927 rad
[i R , v R ]
same phase
R = 6Ω
Is (t) → 5
Is (t)⋅6 Ω → 30
Impedance
15
Young Won Lim
05/26/2015
Capacitor
π /2 = 1.571
v s (t)
40
d vC
iC = C⋅
dt
−0.644 rad
[iC , v C ]
1.571−0.927 = 0.644
leads by 90
XC = 8 Ω
Is (t) → 5
Is (t)⋅8 Ω → 40
Impedance
16
Young Won Lim
05/26/2015
vR(t) + vC(t)
v s (t)
v R (t )
`
30
0.927 rad
Impedance
40
−0.644 rad
17
Young Won Lim
05/26/2015
Phasor Addition
A 1 cos(ω t +θ1) =
A 1 cos(ω t )cos(θ 1 ) − A 1 sin (ω t)sin (θ1 )
A 2 cos(ω t +θ2 ) =
A 2 cos(ω t )cos(θ 2 ) − A 2 sin (ω t )sin (θ2 )
A 1 cos(ω t +θ1 ) + A 2 cos(ω t +θ2 ) = X cos(ω t ) − Y sin (ω t ) = A cos(ω + θ)
X = A 1 cos(θ 1) + A 2 cos (θ2 )
A 1 cos(θ1 ) = 30 cos(0.927 ) = 18.0
Y = A 1 sin (θ1 ) + A 2 sin (θ2 )
A 1 sin (θ1 ) = 30 sin (0.927 ) = 24.0
A 2 cos(θ1 ) = 40 cos(−0.644 ) = 32.0
X = 50.0
A 2 sin (θ1 ) = 40 sin (−0.644) = −24.0
Y = 0.0
A =
√ 50.02 +0.02
θ = tan−1
= 50.0
0.0
= 0
50.81
Impedance
18
Young Won Lim
05/26/2015
Is(t) and Es(t)
R = 6Ω
Z T = 10 Ω
XC = 8 Ω
ZT = 10
I s ZT = (5
Impedance
0.927)⋅(10
−0.927)
= 50
19
0
−0.927
E s = I s ZT
Young Won Lim
05/26/2015
Phasor
v s (t)
v s (t )
v R (t )
v R (t )
30
30
40
0.927 rad
0.927 rad
−0.644 rad
40
−0.644 rad
1.571−0.927 = 0.644
VR
ES
VC
Impedance
20
Young Won Lim
05/26/2015
Phasor
v R (t )
30
0.927 rad
Is
ES
ZT
Impedance
21
Young Won Lim
05/26/2015
Sinusoidal Functions
●
everlasting sinusoid function
causal sinusoid function
●
state at t = 0– = state at t = 0+
zero state at t = 0–
continuous state between t = 0– & 0+
non-zero state at t = 0+
sin (ω t)
sin (ω t)u (t)
cos(ω t )
cos (ω t )u(t)
zero conditions
−
at time t = 0
Impedance
22
creates
non-zero conditions
+
at time t = 0
Young Won Lim
05/26/2015
Steady State Response
Can apply impedance method
Cannot apply impedance method
IC
VL
VC
IL
Impedance
23
Young Won Lim
05/26/2015
References
[1] http://en.wikipedia.org/
[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003
Young Won Lim
05/26/2015
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