Download Ch4 Sinusoidal Steady State Analysis

Circuits and Analog Electronics
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
4.2 Phasors
4.3 Phasor Relationships for R, L and C
4.4 Impedance
4.5 Parallel and Series Resonance
4.6 Examples for Sinusoidal Circuits Analysis
4.7 Magnetically Coupled Circuits
References: Hayt-Ch7; Gao-Ch3;
Ch4 Sinusoidal Steady State Analysis
• Any steady state voltage or current in a linear circuit with a
sinusoidal source is a sinusoid
– All steady state voltages and currents have the same frequency as
the source
• In order to find a steady state voltage or current, all we need to know
is its magnitude and its phase relative to the source (we already know
its frequency)
• We do not have to find this differential equation from the circuit, nor
do we have to solve it
• Instead, we use the concepts of phasors and complex impedances
• Phasors and complex impedances convert problems involving
differential equations into circuit analysis problems
 Focus on steady state;
Focus on sinusoids.
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Key Words:
Period: T ,
Frequency: f , Radian frequency 
Phase angle
Amplitude: Vm Im
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
vt   Vm sin  t
I1
I1
I1
I1
I1
I1
+
U1
-

U
R1
5
R1
5
i
+
v、i
R
_
IS 
E
I1
I
0
I1
I1
I1
I
I1
t
t2
R1
5
R1
5
i
-
+
U1
-

U
t1
IS 
+
R
E
I1
Both the polarity and magnitude of voltage are changing.
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Period: T — Time necessary to go through one cycle. (s)
Frequency: f — Cycles per second. (Hz)
f = 1/T
Radian frequency(Angular frequency):  = 2f = 2/T (rad/s）
Amplitude: Vm Im
i = Imsint， v =Vmsint
v、i
Vm、Im
0

2 t
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Effective Roof Mean Square (RMS) Value of a Periodic
Waveform — is equal to the value of the direct current which is
flowing through an R-ohm resistor. It delivers the same average
power to the resistor as the periodic current does.
1
T

T
0
i 2 Rdt  I 2 R
Effective Value of a Periodic Waveform I eff 
I eff 
1
T

T
0
I m2
T
I sin  tdt 
2
m
2
Veff 
1
T

T
0

T
0
1
T
1  cos 2 t
dt 
2
v 2dt 
Vm
2

T
0
i 2dt
1 2 T
I
Im 
 m
T
2
2
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase (angle)
i  I m sin t   
Phase angle
08
6
4
2
i0   I m sin 
0
-2 0
<0
-4
-6
-8
0.01
0.02
0.03
0.04
0.05
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase difference
i  I m sin( t  2 )
v  Vm sin( t  1 )
  v  i  t  1  (t  2 )  1  2
  1   2  0 — v(t) leads i(t) by (1 - 2), or i(t) lags v(t) by (1 - 2)
  1  2  0 — v(t) lags i(t) by (2 - 1), or i(t) leads v(t) by (2 - 1)
  1   2  0
  1   2  
v、i In phase.
v、i
v

  1   2  
2
Out of phase。
v、i
v
v
i
i
i
t
t
t
Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Review
The sinusoidal means whose phases are compared must:
① Be written as sine waves or cosine waves.
② With positive amplitudes.
③ Have the same frequency.
360°—— does not change anything.
90° —— change between sin & cos.
180°—— change between + & 2 



 sin   cos       cos    
3 
2




 cos   sin    
2

Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase difference
P4.1, v1  220 2 sin 314t  30 
Find
v2  220 2 cos314t  30 
  ?
v2  220 2 cos314t  30   220 2 sin 314t  30  90 

 220 2 sin 314t  120 

  1  2  30 120  150


v2  220 2 cos314t  30   220 2 cos314t  30  180 


 220 2 cos 360   314t  210 

2 sin 314t  60 
 220 2 sin 314t  150   90 
 220
  1  2  30  60  30



Ch4 Sinusoidal Steady State Analysis
4.1 Characteristics of Sinusoidal
Phase difference


Vm sin  t  
3

P4.2,
v、i
v
i
•
-/3
•
/3
•


I m sin  t  
3

t
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
A sinusoidal voltage/current at a given frequency , is
characterized by only two parameters :amplitude an phase
Key Words:
Complex Numbers
Rotating Vector
Phasors
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
E.g. voltage response
v  t   Vm cos t   
Re v  t 
Time domain
Complex form: v  t   Vm e
Angular frequency ω is
known in the circuit.
Phasor form:
j t  
Frequency domain
A sinusoidal v/i
Complex transform
Phasor transform
By knowing angular
frequency ω rads/s.
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Rotating Vector
i1  I m1 sint  1 
i 2  I m2 sint   2 
i  i1  i2  I m sint   
y
i
i
t
Im
Im

x
i(t1)
A complex coordinates number: I m e
j t  

t1
t
 I m cos t     jI m sin t   

Real value: i  t   I m sin t     I max I m e
j t  

Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Rotating Vector
y
v  Vm sin( t   )

Vm

0
x
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
A  a  jb — Rectangular Coordinates
imaginary axis
A  A cos  j sin  
b
A  A e j— Polar Coordinates

real axis
a
conversion：
A  a  jb  A  A e j
A e j  a  jb

e  j 90  cos 90  j sin 90  0  j   j
A  a2  b2
  arctg
a  A cos 
b  A sin 
b
a
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
Arithmetic With Complex Numbers
Addition: A = a + jb, B = c + jd,
A + B = (a + c) + j(b + d)
Imaginary Axis
A+B
B
A
Real Axis
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
Arithmetic With Complex Numbers
Subtraction : A = a + jb, B = c + jd,
A - B = (a - c) + j(b - d)
Imaginary
Axis
B
A
A-B
Real
Axis
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Numbers
Arithmetic With Complex Numbers
Multiplication : A = Am  A, B = Bm  B
A  B = (Am  Bm)  (A  B)
Division: A = Am  A , B = Bm  B
A / B = (Am / Bm)  (A  B)
P4.3,


sint     60 sint  30 
i1  I m1 sint  1   100sin t  45
i2  I m2

2
Find：i  i1  i2
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Phasors
A phasor is a complex number that represents the
magnitude and phase of a sinusoid:
im cost   
I  I m 
Phasor Diagrams
• A phasor diagram is just a graph of several phasors
on the complex plane (using real and imaginary axes).
• A phasor diagram helps to visualize the relationships
between currents and voltages.
Ch4 Sinusoidal Steady State Analysis
4.2 Phasors
Complex Exponentials
A  A e j
Ae jt  A e j ( t  )  A cos(t   )  j A sin( t   )
Re{ Ae jt } | A | cos(t   )
 A real-valued
sinusoid is the real part of a complex exponential.
Complex exponentials make solving for AC steady state an
algebraic problem.

Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Key Words:
I-V Relationship for R, L and C,
Power conversion
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor  v~i relationship for a resistor


+

S
i
Suppose
R
v

_

v  Vm sin t
v Vm
i 
sin t  I m sin t
R R
V
Relationship between RMS: I 
R
v、i
v
Wave and Phasor diagrams：

i
t


V
I
R
I

V
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor  Time domain
frequency domain
With a resistor θ﹦φ, v(t) and i(t) are in phase .
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor  Power
i

+
• Transient Power
R
2
p  vi  Vm sin t  I m sin t  I mVm sin t
v

_

I mVm
1  cos 2t   IV  IV cos 2t
2
p0
• Average Power
v、i
v
i
P=IV
t
1
P
T
1 T
0 pdt T 0VI 1  cos 2t dt  VI
T
V2
P  IV  I R 
R
2
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Resistor
P4.4 ，
v  311sin 314t ,
R=10，Find i and P。
Vm 311
V

 220V 
2
2
I
V 220

 22 A
R 10
i  22 2 sin 314t
P  IV  220  22  4840W 
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor

v~i relationship
v  v AB
di
L
dt
Suppose i  I m sin t
vL
di
d I m sin t 
L
 I mL cos  t
dt
dt
 I mL sin t  90


 Vm sin t  90 
1 t
1 0
1 t
1 t
i   vdt  L vdt  L 0 vdt i0  0 vdt
L
L 
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor

v~i relationship
di
v  L  I mL sin  t  90   Vm sin t  90 
dt
Vm  I mL
Relationship between RMS: V  IL
V
X L  L  2fL  
I
L
XL  f
For DC，f = 0，XL = 0.
v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor

v ~ i relationship
i(t) = Im ejt
di
 I m jLe jt  jLi(t )
dt
Represent v(t) and i(t) as phasors: V  jLI


I  V  V
jL jX L
• The derivative in the relationship between v(t) and i(t) becomes a
multiplication by j in the relationship between V and I.
• The time-domain differential equation has become the algebraic equation in the
frequency-domain.
• Phasors allow us to express current-voltage relationships for inductors and
capacitors in a way such as we express the current-voltage relationship for a
resistor.
v (t )  L
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor

v ~ i relationship
Wave and Phasor diagrams：
V  jIX L
v、i
v
i
V
eL
t
I
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor

Power
p  vi  Vm sin t  90 I m sin t  Vm I m cos t  sin t
Vm I m

sin 2t VI sin 2t
2
t
i
1
Energy stored:W   vidt   Lidi  Li 2
0
0
2
1
Wmax  LI m2  LI 2
2
Average Power P  1
T

T
0
1 T
pdt   VI sin 2tdt  0
T 0
V2
Reactive Power Q  IV  I X L 
XL
2
（Var）
P
+
+
-
-
t
v、i
v
i
t
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Inductor
P4.5，L = 10mH，v = 100sint，Find iL when f = 50Hz and 50kHz.
X L  2fL  2  50  10  10 3  3.14
I 50 
V
100 / 2

 22.5 A
XL
3.14


iL t   22.5 2 sin t  90 A
X L  2fL  2  50  10 3  10  10 3  3140
V
100 / 2
I 50k 

 22.5mA
XL
3.14


iL t   22.5 2 sin t  90 mA
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor

+
v
_


v ~ i relationship
dq
dv
i
C
dt
dt
i
Suppose: v  Vm sin t
C
I m CVm

i  CVm cos t  CVm sin t  90   I m sin t  90
1 t
1 0
1 t
1 t
v   idt   idt   idt  v0   idt
c 
c 
c 0
c 0
Relationship between RMS: I  CV  V  V
1
XC

C
1
1

XC 

C 2fC
1
XC 
For DC，f = 0， XC  
f
i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º

Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor

+
v
_


v ~ i relationship
v(t) = Vm ejt
i
C
dv(t )
dVme j t
i (t )  C
C
 jCVme j t
dt
dt

V
Represent v(t) and i(t) as phasors: I = jωCV =
jX C
• The derivative in the relationship between v(t) and i(t) becomes a
multiplication by j in the relationship between V and I.
• The time-domain differential equation has become the algebraic equation in the
frequency-domain.
• Phasors allow us to express current-voltage relationships for inductors and
capacitors much like we express the current-voltage relationship for a resistor.
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor

v ~ i relationship
Wave and Phasor diagrams：
V   jI X C
v、i
I
i
v
t
V
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
Power
p  vi  Vm sin t  I m sin t  90  
Energy stored:
Vm I m
sin 2t  VI sin 2t
2
P
v
dv
1
W   vidt   v  C   dt   Cvdv  Cv 2
0
0
0
dt
2
1
Wmax  CVm2  CV 2
2
t
v
+
+
-
-
t
v、i
Average Power: P=0
i
2
V
Reactive Power Q  IV  I X C 
（Var）
XC
2
v
t
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Capacitor
P4.7，Suppose C=20F，AC source v=100sint，Find
XC and I for f = 50Hz, 50kHz。
f  50Hz  X c 
I
V
V
 m  1.38A
Xc
2Xc
f  50KHz  X c 
I 
1
1

 159
C 2fC
V

Xc
1
1

 0.159
C 2fC
Vm
 1380A 
2Xc
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Review (v-I relationship)
Time domain
R
L
C
v  Ri
Frequency domain
V  R  I ,
v and i are in phase.
vL  L
di
dt
V  jL  I
vC  C
dv
dt
1 
1
V 
I , XC 
, v lags i by 90°.
jC
C
, X L  L , v leads i by 90°.
Ch4 Sinusoidal Steady State Analysis
4.3 Phasor Relationships for R, L and C
Summary



R：
XR  R
  0
L：
X L  L  2fL  f
   v  i 
C：
XC 
1
1
1


c 2fc
f

2
   v  i  

2
V  IX
Frequency characteristics of an Ideal Inductor and Capacitor:
A capacitor is an open circuit to DC currents;
A Inducter is a short circuit to DC currents.
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Key Words:
complex currents and voltages.
Impedance
Phasor Diagrams
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex voltage， Complex current， Complex Impedance
• AC steady-state analysis using phasors allows us to express the
relationship between current and voltage using a formula that looks
likes Ohm’s law:
Z is called impedance.
V  IZ
measured in ohms ()
V  Vm e jv  Vm v
I  I m e ji  I mi
V Vm j (v i )
Z 
e
 Z e j  Z 
I I m
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
V Vm j (v i )
Z 
e
 Z e j  Z 
I I m
• Complex impedance describes the relationship between the
voltage across an element (expressed as a phasor) and the
current through the element (expressed as a phasor)
• Impedance is a complex number and is not a phasor (why?).
• Impedance depends on frequency
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Resistor——The impedance is R
ZR = R
 = 0（ ＝ 0); or ZR = R  0
Capacitor——The impedance is 1/jC
1 j2  j
Zc 
e

  jx c
C
C
=-/2

(    v  i   )
2
or
ZC 
1
  90
C
Inductor——The impedance is jL
Z L  Le
j

2
=/2
 jL  jxL
(    v  i 

2
or Z L  L90
)
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
I1
US
I1
I1
I1
I1
Impedance
in series/parallel can be combined as resistors.

I
 
+
I1
I1
Z1
Z2
I1
U
U
_

+
U1
-
Zn
R1
5
R1
5
+
IS 
n
Z  Z 1  Z 2  ...  Z n   Z k
k 1
Voltage divider:
Zi


Vi  V n
 Zk
k 1
U
Z1
Zn
Z2
_

US
_
I
 
+
n
1
1
1
1
1


 ... 

Z Z1 Z 2
Z n k 1 Z k
Current divider:
I1  I
Z2
Z1  Z 2
I2  I
Z1
Z1  Z 2
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
P4.8,

+
Z1
I1
 


Z2
V
_


I
Z
I  I Z 2
1
Z  Z2
V
V Z  Z 2 
I 

1
1
ZZ1  Z 2 Z1  ZZ 2
 1 1
Z1    
 Z2 Z 
Z
V
2
I 
ZZ1  Z 2 Z1  ZZ 2
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Phasors and complex impedance allow us to use Ohm’s law with
complex numbers to compute current from voltage and voltage
from current
P4.9
10V  0
+
-
20k
1F
+
-
VC
 = 377
Find VC
• How do we find VC?
• First compute impedances for resistor and capacitor:
ZR = 20k = 20k  0
ZC = 1/j (377 *1F) = 2.65k  -90
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
P4.9
+
10V  0
-
20k  0
20k
+
1F
 = 377
VC
Find VC
-
Now use the voltage divider to find VC:
2.65k  90
VC  10V0 (
)
2.65k  90  20k0

10V  0
+
-
+
VC
-
2.65k  -90
2.65  90
20.17  7.54
 1.31V   82.46
VC  10V 0
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Complex Impedance
Impedance allows us to use the same solution techniques
for AC steady state as we use for DC steady state.
• All the analysis techniques we have learned for the
linear circuits are applicable to compute phasors
– KCL & KVL
– node analysis / loop analysis
– superposition
– Thevenin equivalents / Norton equivalents
– source exchange
• The only difference is that now complex numbers are
used.
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Kirchhoff’s Laws
KCL and KVL hold as well in phasor domain.
n
KCL:
 ik
0
k 1
n
 Ik
ik- Transient current of the #k branch
0
k 1
n
KVL：
v
k 1
k
n
V
k 1
k
0
0
vk- Transient voltage of the #k branch
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Admittance
• I = YV, Y is called admittance, the reciprocal of
impedance, measured in siemens (S)
• Resistor:
– The admittance is 1/R
• Inductor:
– The admittance is 1/jL
• Capacitor:
– The admittance is j  C
Ch4 Sinusoidal Steady State Analysis
4.4 Impedance
Phasor Diagrams
• A phasor diagram is just a graph of several phasors on the complex
plane (using real and imaginary axes).
• A phasor diagram helps to visualize the relationships between currents
and voltages.
I = 2mA  40, VR = 2V  40
VC = 5.31V  -50, V = 5.67V  -29.37
2mA  40
+
1F
V
1k
–
+
Imaginary Axis
VC
–
+
–
VR
VR
Real Axis
V
VC
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Key Words:
RLC Circuit,
Series Resonance
Parallel Resonance
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit (2nd Order RLC Circuit )
v  vR  vL  vC
vR
v
Phasor
vL
V  VR2  (VL  VC ) 2
vC
 ( IR ) 2  ( IX L  IX C ) 2
VL
 I R 2  ( X L  X C )2
V

I
VC
V  VR  VL  VC
VR
 I R2  X 2
 IZ
Z R X
2
2
( X  X L  X C）
1 2
 R  (L  )
c
2
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit (2nd Order RLC Circuit )
V  VR2  (VL  VC )2  IZ
Z
X = XL-XC


2
 R 2  (L 
Phase difference:
VX  VL  VC
= arctg
XL
VR
XL>XC   >0，v leads i by ——Inductance Circuit
XL<XC
XL=XC
1 2
)
c
VL - VC
= arctg
VR
R
V
Z R X
2
  <0，v lags i by ——Capacitance Circuit
  =0，v and i in phase——Resistors Circuit
XC
R
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit (2nd Order RLC Circuit )
vR
V  VR  VL  VC  IR  jIX L  jIX C
 I( R  j( X L  X C )]  I( R  jX )  IZ
v
vL
vC
V
Z   R  j( X L  X C )
I
Z  R  jX  Z 
Z  R2  ( X L  X C )2
  arctg
X L  XC
R
   v  i
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series RLC Circuit (2nd Order RLC Circuit )
P4.9, R. L. C Series Circuit，R = 30，L = 127mH，C = 40F，Source
v  220 2 sin( 314t  20o ) , Find 1) XL、XC、Z；2) I and i；3) VR and
vR； VL and vL； VC and vC； 4) Phasor Diagrams
vR
v
vL
vC
P4.10，Computing I by (complex numbers) Phasors
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
V  VR  VL  VC  IR  jIX L  jIX C
VL  VC
X L  XC
  arctg
 arctg
VR
R
1
When X L  X C ,
 L  VL  VC
C
Resonance condition
0 
1
1
or f 0 
LC
2 LC
Resonance frequency
VR  V and   0 ——Series Resonance
VL
X
X L  2 fL
VR  V
I
VC
XC 
f0
1
2 f C
f
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Resonance condition:
•
X L  XC (
1
 L)
C
Z 0  R 2  ( X L  X C )2  R  I 0 
 VL  VC
V V

Z0 R
Zmin；when V=constant, I=Imax=I0。
•When X L  X C  R ,  I 0 X L  I 0 X C  I 0 R
•Quality factor Q,
Q
VL VC X L X C



V
V
R
R
VL  VC  V
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Series Resonance (2nd Order RLC Circuit )
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit
I

V
IL

IC

1
1
1


 jC
R  jL  j / C
R  jL
R  jL

 jC
R  jL R  jL 
R
L
 2

j
(

C

)
2 2
2
2 2
R  L
R  L
L
R
)

0
,
When (C  2
Y0  2
R   2 L2
R   2 L2
Y 
V In phase with I
Parallel Resonance
Parallel Resonance frequency
In generally R  X L
Zmax Imin:
0 
0 
1
1
LC
CR 2
1
L
( f0 
1
)
2

LC
LC
R
R
R
RC
I  I 0  VY0  V 2

V
V


V
1 2
L
R  02 L2
L
2
2
R 
L
R 
LC
C
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit
IL  V
I

V
IL

IC

1
V
C 
j
j
V
R  j0 L
0 L
L
C
IC  j0CV  j V
L
| IL || IC || I0 | 0
•Quality factor Q,
I C I L YL YC
Q



I 0 I 0 Y0 Y0
IL   jQI0
IC  jQI0
Q
0 L
R

1
0 RC
Z  .
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit
R1  3, X L  4, R2  8, X C  6
P4.10,
Find i1、 i2、 i
i
 i2
v
i1 
v  220 2 sin 314t
Ch4 Sinusoidal Steady State Analysis
4.5 Parallel and Series Resonance
Parallel RLC Circuit
Review
For sinusoidal circuit， Series ：v  v1  v2
Parallel : i  i1  i2
V  V1  V2
I  I1  I 2
Two Simple Methods:
Phasor Diagrams and Complex Numbers
？
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Key Words:
Bypass Capacitor
RC Phase Difference
Low-Pass and High-Pass Filter
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Bypass Capacitor
P4.11, Let i  3  10 3 2 sin t  3  10 3 2 sin 2ft  3  10 3 2 sin 1000  t
f ＝ 500Hz，Determine VAB before the C is connected . And VAB after
parallel C = 30F
Before C is connected
i
VAB  IR  3  103  500  1.5(V )
After C is connected
v
XC 
1
1

 10()
2fc 1000  30  10 6
1
1
Z   
 R  jX C
1

  0.2  10 j  10  88.85

 VAB  I C Z  3 103 10  30(mV )
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
RC Phase Difference
P4.12,

+
vi
_

f = 300Hz, R = 100。 If vo - vi= /4，C =？
R=100


+
Vi  IR  jX C   vi vi
Vo  I jX C   vo vo
1 5.3110 4
XC 


vo
C
C
C
vo   90
Vo  jX C


_

 5.3110 6
Vi R  jX C

vi arctg

C
6

 5.3110

vo  vi    arctg

2
C
4
6
 5.3110

arctg

C
4
 5.31106
 0.0411
C
C  1.29  10 4 F
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter

+
R=200


+
RC---- High-Pass Filter
vi
_

vo
C

XC 
_

VR
R

1
VC
C
1
f
VR X C  RVC
P4.13, The voltage sources are vi=240+100sin2100t(V), R＝200，
C＝50F， Determine VAC and VDC in output voltage vo.
VDC = 240V
V AC
V
32
 XC  
100  16(V )
Z 200
XC 
1
1

 32
6
2fc 2  100  50  10
Z  R 2  X C2  2002  322  200
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter
260V
250V
240V
230V
220V
50ms
V(2)
55ms
60ms
65ms
70ms
75ms
Time
80ms
85ms
90ms
95ms
100ms
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter
400V
300V
200V
100V
50ms
V(2)
55ms
V(1)
60ms
65ms
70ms
75ms
Time
80ms
85ms
90ms
95ms
100ms
Ch4 Sinusoidal Steady State Analysis
SEL>>
4.6 Examples for Sinusoidal Circuits Analysis
400V
300V
200V
100V
50ms
V(2)
55ms
V(1)
60ms
65ms
70ms
75ms
80ms
85ms
90ms
95ms
100ms
Time
300V
300V
200V
200V
100V
100V
0V
0Hz
0.2KHz
V(1)
0.4KHz
0.6KHz
0.8KHz
0V
0Hz
1.0KHz
0.2KHz
1.2KHz
V(2)
0.4KHz
1.4KHz
0.6KHz
1.6KHz
0.8KHz 2.0KHz1.0KHz
1.8KHz
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Low-Pass and High-Pass Filter
1.0V
0.5V
0V
1.0Hz
V(2)
3.0Hz
10Hz
30Hz
100Hz
300Hz
Frequency
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
Ch4 Sinusoidal Steady State Analysis
SEL>>
4.6 Examples for Sinusoidal Circuits Analysis
1.0V
0.5V
0V
0s
V(2)
50ms
V(1)
100ms
150ms
200ms
250ms
300ms
350ms
400ms
450ms
500ms
550ms
Time
800mV
400mV
SEL>>
0V
V(2)
1.0V
0.5V
0V
0Hz
50Hz
V(1)
100Hz
150Hz
200Hz
250Hz
300Hz
350Hz
400Hz
450Hz
500Hz
600ms
Ch4 Sinusoidal Steady State Analysis
SEL>>
4.6 Examples for Sinusoidal Circuits Analysis
1.0V
0.5V
0V
0Hz
800mV
50Hz
100Hz
150Hz
200Hz
250Hz
300Hz
350Hz
400Hz
450Hz
300Hz
350Hz
400Hz
450Hz
500Hz
V(2)
Frequency
400mV
SEL>>
0V
V(2)
1.0V
0.5V
0V
0Hz
50Hz
V(1)
100Hz
150Hz
200Hz
250Hz
500Hz
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Complex Numbers Analysis
P4.14, Find I1 I2 I3 V in the circuit of the following fig.
2
i3

v1=120sint
 i1
 i2
v2
Ch4 Sinusoidal Steady State Analysis
4.6 Examples for Sinusoidal Circuits Analysis
Complex Numbers Analysis
P4.15, Let Vm ＝ 100V. Use Thevenin’s theorem to find ICD
v
v
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Key Words:
Self- inductance and Mutual inductance
Magnetically Coupled Circuits and v ~ i relationship
Dot convention
Ideal transformer
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Coupled Circuits and v~i relationship
Magnetic flux:
1 ＝ f(i1) (1 = N11)
The flux is proportional to the current in linear inductor: 1(t) ＝ L1i1(t)
L is a lumped element abstraction for the coil.
i1
 
+
v1
-
i1

v1( t )
d 1
di1

 L1
dt
dt
v1
Voltage be proportional to
the time rate of change of
the magnetic field.
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Coupled Circuits and v~i relationship
1(t )  L1i1(t )  M12i2(t )
M12  M 21  M
i1
 
+
v1
-
i2
 
+
v2
-
2(t )  M 21i1(t )  L2i2(t )
d 1
di
di
 L1 1  M 2
dt
dt
dt
d 2
di
di
v2 
 M 1  L2 2
dt
dt
dt
v1 
——Ideal Coupled Circuits’ v ~ i relationship
 L1、L2、M represent Ideal Coupled Inductor

d 1
di
di
v1 
 L1 1  M 2
dt
dt
dt
Self- inductance voltage
Mutual- inductance voltage
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Coupled Circuits and v ~ i relationship
+
v1
-
+
i2
 
+
v2
-
i1
 
i2
 +
v2
i1
 
v1
-
-
1(t )  L1i1(t )  Mi2(t )
1(t )  L1i1(t )  Mi2(t )
2(t )  Mi1(t )  L2i2(t )
2(t )  Mi1(t )  L2i2(t )
v1 
d 1
di
di
 L1 1  M 2
dt
dt
dt
d 2
di1
di2
v2 
dt
M
dt
 L2
dt
v1 
v2 
d 1
di
di
 L1 1  M 2
dt
dt
dt
d 2
di1
di2
dt
 M
dt
 L2
dt
i2
i1

Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
v1

•
•
Dot convention
i2
 
+
v2
-
+
v1
-
+
1
 i
v1
-
i1
v1 
d 1
di
di
 L1 1  M 2
dt
dt
dt
d 2
di1
di2
v2 
dt
M
dt
 L2
dt

v1
i
2 +
v2
-
1
 i
•
i2

•
v2
v1 
v2 
d 1
di
di
 L1 1  M 2
dt
dt
dt
d 2
di1
di2
dt
 M
dt
 L2
dt
A current entering the dotted terminal of one coil produces an open
circuit voltage with a positive voltage reference at the dotted terminal of
the second coil.
Inversely , current leaving of the dotted terminal of one coil produces a
negative voltage reference at the dotted terminal of the other end.
v2
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Question：The terminal is dotted，how can we get v ~ i equations to
coupled inductor？
i2
i1


•
•
i2
i1


v1
v2
v1
u
2 of the i and
Suppose direction
M
with Dot convention!
•
v2
di is• consistent
dt
Steps to determine the coupled circuit voltage
1. For self inductance voltage
+i / i-i / i+
2. For mutual inductance voltage
+ vs
- vs
+ vm
- vm
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
P4.16，For the circuit shown in following figures, determine v1and v2.
i1

v1
•
L1
M
i2
i1

•
v
L2 u2 2

v1
•
L1
M
i2
i1

L2
v
• u2 2

v1
M
L1
•
i2
－
•
L2
uv2 2
＋
dv1
di
M 2
dt
dt
di
dv
v2  L2 2  M 1
dt
dt
v1   L1
dv1
di
M 2
dt
dt
di
di
v2  L2 2  M 1
dt
dt
v1   L1
di1
di
M 2
dt
dt
di
di
v2   L2 2  M 1
dt
dt
v1   L1
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Coupled Circuits and v ~ i relationship
1(t )  L1i1(t )  M12i2(t )
M12  M 21  M
i1
 
+
v1
-
i2
 
+
v2
-
2(t )  M 21i1(t )  L2i2(t )
d 1
di
di
 L1 1  M 2
dt
dt
dt
d 2
di
di
v2 
 M 1  L2 2
dt
dt
dt
v1 
——Ideal Coupled Circuits’s v~i relationship
For sinusoidal circuit,
V1  jL1I1  jMI2
V2  jMI1  jL2 I2
Ch4 Sinusoidal Steady State Analysis
4.7 Magnetically Coupled Circuits
Ideal transformer
n=N1/N2
(k = 1, L = , M = )


•
v1
v1( t )  nv2 ( t )
1
i1( t )   i2 ( t )
n
i2
i1
N1
•
N2
u2
v2
ZL
v1i1  v2i2  0
For ideal transformer, p=0
Z1
V1
nV2
Impedance Transformation Z1  
 n2Z L
I1  1 I
2
n