AC Fundamentals

1.  Characteristics of Sinusoidal
 Phasors
 Phasor Relationships for R, L and C
 Impedance
 Parallel and Series Resonance
 Examples for Sinusoidal Circuits Analysis
Single Phase AC

2. 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

3. Characteristics of Sinusoids
Outline:
1. Time Period: T
2. Frequency: f (Hertz)
3. Angular Frequency:  (rad/sec)
4. Phase angle: Φ
5. Amplitude: Vm Im

4. Characteristics of Sinusoids :
  tVv mt sin iI1
I1
I1 I1 I1 I1
R1
R1
R
5
5
+
_
IS 
E
I1
U1
+
-
U
I

iI1 I1 I1 I1
R1
R1
R
5
5
-
+
IS 
E
I1
U1
+
-
U
I

v ,i
tt1 t20
Both the polarity and magnitude of voltage are changing.

5. Radian frequency(Angular frequency):  = 2f = 2/T (rad/s）
Time Period: T — Time necessary to go through one cycle. (s)
Frequency: f — Cycles per second. (Hz)
f = 1/T
Amplitude: Vm Im
i = Imsint， v =Vmsint
v ,i
t 20
Vm , Im
Characteristics of Sinusoids :

6. 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.
RIRdti
T
T
2
0
21

Effective Value of a Periodic Waveform 
T
eff dti
T
I
0
21
22
1
2
2cos1
sin
1 2
0
2
0
22 m
m
T
m
T
meff
IT
I
T
dt
t
T
I
tdtI
T
I 

 


2
1
0
2 m
T
eff
V
dtv
T
V  
Characteristics of Sinusoids :

7. Phase (angle)
   tIi m sin
  sin0 mIi 
Phase angle
-8
-6
-4
-2
0
2
4
6
8
0 0.01 0.02 0.03 0.04 0.05
<0
0
Characteristics of Sinusoids :

8. )sin( 1  tVv m
)sin( 2  tIi m
Phase difference
2121 )(   ttiv
021   — v(t) leads i(t) by (1 - 2), or i(t) lags v(t) by (1 - 2)
2
21

 
v, i
t
v
i
  21
Out of phase
t
v, i
v
i
v, i
t
v
i
021  
In phase
021   — v(t) lags i(t) by (2 - 1), or i(t) leads v(t) by (2 - 1)
Characteristics of Sinusoids :

9. Review
The sinusoidal waves 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

   

 
   
       
   
 
   
 
Characteristics of Sinusoids :

10. Phase difference
 
30314sin22201  tv
   
9030314sin222030314cos22202  ttv
 
120314sin2220  t

1501203021  
 
30314cos22202  tv
 
30314cos22202  tv  
18030314cos2220  t
  
210314360cos2220  t
 
90150314sin2220  t
 
60314sin2220  t

30603021  
Find ?
 
30314cos22202  tvIf
Characteristics of Sinusoids :

11. Phase difference
v, i
t
v
i
-/3 /3
• ••








3
sin

tVm







3
sin

tIm
Characteristics of Sinusoids :

12. Outline:
1. Complex Numbers
2. Rotating Vector
3. Phasors
A sinusoidal voltage/current at a given frequency, is characterized by only
two parameters : amplitude and phase
A phasor is a Complex Number which represents magnitude and phase of a sinusoid
Phasors

13. e.g. voltage response
A sinusoidal v/i
Complex transform
Phasor transform
By knowing angular
frequency ω rads/s.
Time domain
Frequency domain
  eR v t
Complex form:
   cosmv t V t  
Phasor form:
   j t
mv t V e
 

Angular frequency ω is
known in the circuit.
 || mVV
 || mVV
Phasors

14. Rotating Vector
   tIti m sin)(
i
Im
t1
i
t
Im

t
x
y
 
   
     
 max
cos sin
sin
j t
m m m
j t
m m
I e I t jI t
i t I t I I e
 
 
   
 


   
  
A complex coordinates number:
Real value:
i(t1)
Imag
Phasors

15. Rotating Vector
Vm
x
y
0

)sin(   tVv m
Phasors

16. Complex Numbers
jbaA  — Rectangular Coordinates
  sincos jAA 
j
eAA  — Polar Coordinates
j
eAAjbaA 
conversion： 22
baA 
a
b
arctg
jbaeA j

cosAa 
sinAb 

a
b
Real axis
Imaginary axis
jjje j

090sin90cos90 
Phasors

17. Complex Numbers
Arithmetic With Complex Numbers
Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d)
Real Axis
Imaginary Axis
AB
A + B
Phasors

18. Complex Numbers
Arithmetic With Complex Numbers
Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d)
Real
Axis
Imaginary
Axis
AB
A - B
Phasors

19. 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)
Phasors

20. Phasors
A phasor is a complex number that represents the magnitude and phase of
a sinusoid:
  tim cos  mI
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.
Phasors

21. )sin()cos()(

 
tAjtAeAAe tjtj
)cos(||}Re{ 
 tAAe tj
Complex Exponentials
j
eAA 
 A real-valued sinusoid is the real part of a complex exponential.
 Complex exponentials make solving for AC steady state an
algebraic problem.
Phasors

22. Phasor Relationships for R, L and C
Outline:
I-V Relationship for R, L and C,
Power conversion

23. Phasor Relationships for R, L and C
 v~i relationship for a resistor
_
v
i
R

 +
S 


tIt
R
V
R
v
i m
m
 sinsin 
tVv m sin
Relationship between RMS:
R
V
I 
Wave and Phasor diagrams：
v、i
t
v
i 
I 
V
R
V
I



Resistor
Suppose

24.  Time domain Frequency domainResistor
With a resistor θ﹦ϕ, v(t) and i(t) are in phase .
)cos()(
)cos()(




wtIti
wtVtv
m
m
IRV
RIV
eRIeV
eRIeV
mm
j
m
j
m
wtj
m
wtj
m



 


 )()(
Phasor Relationships for R, L and C

25.  PowerResistor
_
v
i
R
+

 P  0
tItVvip mm  sinsin  tVI mm 2
sin
 t
VI mm
2cos1
2
 tIVIV 2cos
v, i
t
v
i
P=IV 
T
pdt
T
P
0
1
  
T
VIdttVI
T 0
2cos1
1

R
V
RIIVP
2
2

• Average Power
• Transient Power
Note: I and V are RMS values.
Phasor Relationships for R, L and C

26. Resistor
, R=10，Find i and Ptv 314sin311
 V
V
V m
220
2
311
2

 A
R
V
I 22
10
220

ti 314sin222  WIVP 484022220 
Phasor Relationships for R, L and C

27.  v~i relationshipInductor
dt
di
Lvv AB 
  tLI
dt
tId
L
dt
di
Lv m
m


cos
sin

 
90sin  tLIm 
 
90sin  tVm 
 

t
vdt
L
i
1
 

t
vdt
L
vdt
L 0
0 11

t
vdt
L
i
0
0
1
tIi m sinSuppose
Phasor Relationships for R, L and C

28.  v~i relationshipInductor
 
90sin  tLIm 
dt
di
Lv   
90sin  tVm 
LIV mm 
Relationship between RMS: LIV 
L
V
I

 fLLXL  2  
For DC，f = 0，XL = 0.
fXL 
v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º
Phasor Relationships for R, L and C

29.  v ~ i relationshipInductor
v, i
t
v
i
eL
V
I
LXIjV 
Wave and Phasor diagrams：
Phasor Relationships for R, L and C

30.  PowerInductor
vip    tItV mm  sin90sin 
 ttIV mm  sincos 
t
IV mm
2sin
2
 tVI 2sin
P
t
v, i
t
v
i
++
--22
max
2
1
LILIW m 
2
00 2
1
LiLidividtW
it
 Energy stored:
  
T T
tdtVI
T
pdt
T
P
0 0
02sin
11
Average Power
Reactive Power
L
L
X
V
XIIVQ
2
2
 （Var）
Phasor Relationships for R, L and C

31. Inductor
L = 10mH，v = 100sint，Find iL when f = 50Hz and 50kHz.
  
14.310105022 3
fLX L
 
   Atti
A
X
V
I
L
L

90sin25.22
5.22
14.3
2/100
50



  
31401010105022 33
fLX L
 
   mAtti
mA
X
V
I
L
L
k

90sin25.22
5.22
14.3
2/100
50



Phasor Relationships for R, L and C

32.  v ~ i relationshipCapacitor
_
v
i
+


C
dt
dv
C
dt
dq
i 
tVv m sinSuppose:
 
90sincos  tCVtCVi mm   
90sin  tIm 
   

t tt
idt
c
vidt
c
idt
c
idt
c
v
0
0
0
0
1111
i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º
Relationship between RMS:
CX
V
C
V
CVI 


1
 
fCC
XC
 2
11

For DC，f = 0， XC  
f
XC
1

mm CVI 
Phasor Relationships for R, L and C

33. _
v
i
+


C
tj
m
tj
m
eCVj
dt
edV
C
dt
tdv
Cti 


)(
)(
v(t) = Vm ejt
Represent v(t) and i(t) as phasors:
CjX
V
VCωjI ==


• The derivative in the relationship between v(t) and i(t) becomes a
multiplication by in the relationship between and .
• 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.
 v ~ i relationshipCapacitor
V I
wC
j

Phasor Relationships for R, L and C

34.  v ~ i relationshipCapacitor
v, i
t
v
i
I
V
CXIjV  
Wave and Phasor diagrams：
Phasor Relationships for R, L and C

35. PowerCapacitor
Average Power: P = 0
Reactive Power
C
C
X
V
XIIVQ
2
2
 （Var）
 
90sinsin  tItVvip mm  tVIt
IV mm
 2sin2sin
2

P
t
v, i
t
v
i
++
--
Energy stored:
  
t vv
CvCvdvdt
dt
dv
CvvidtW
0 0
2
0 2
1
22
max
2
1
CVCVW m 
Phasor Relationships for R, L and C

36. Capacitor
Suppose C=20F，AC source v=100sint，Find XC and I for f = 50Hz,
50kHz。
 159
2
11
Hz50
fCC
Xf c

A44.0
2

c
m
c X
V
X
V
I
  159.0
2
11
KHz50
fCC
Xf c

A440
2

c
m
c X
V
X
V
I
Phasor Relationships for R, L and C

37. Review (v – i Relationship)
Time domain Frequency domain
iRv  IRV  
I
Cj
V  

1
ILjV   
dt
di
LvL 
dt
dv
CiC 
C
XC

1

LXL ,
,
, v and i are in phase.
, v leads i by 90°.
, v lags i by 90°.
R
C
L
Phasor Relationships for R, L and C

38. Summary:
 R： RX R  0
L： ffLLXL   2
2

  iv
C：
ffcc
XC
1
2
11

 2

  iv
 IXV 
 Frequency characteristics of an Ideal Inductor and Capacitor:
A capacitor is an open circuit to DC currents;
A Inductor is a short circuit to DC currents.
Phasor Relationships for R, L and C

39. Impedance (Z)
Outline:
Complex currents and voltages.
Impedance
Phasor Diagrams

40. • 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:
ZIV 
Complex voltage， Complex current， Complex Impedance
vm
j
m VeVV v


im
j
m IeII i



 
ZeZe
I
V
I
V
Z jj
m
m iv )(


‘Z’ is called impedance
measured in ohms ()
Impedance (Z)

41. Complex Impedance

 
ZeZe
I
V
I
V
Z jj
m
m iv )(


 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.
Impedance (Z)

42. Complex Impedance
ZR = R  ＝ 0; or ZR = R  0
Resistor——The impedance is R
c
j
c jX
C
j
e
C
Z 





21
)
2
(

  iv
or 
90
1

C
ZC

Capacitor——The impedance is 1/jωC
L
j
L jXLjLeZ  

2
)
2
(

  iv
or 
90 LZL 
Inductor——The impedance is jωL
Impedance (Z)

43. Complex Impedance
Impedance in series/parallel can be combined as resistors.
_
U
U
Z1
+

Z2 Zn


I



n
k
kn ZZZZZ
1
21 ...
_
US
In
I1
I1
I1 I1 I1 I1
 


R1
R1
Zn
5
5
5 5
+
+
_
US
IS 
U1
+
-
U

I

Z2Z1




n
k kn ZZZZZ 121
11
...
111
21
1
2
21
2
1
ZZ
Z
II
ZZ
Z
II



 
Current divider:

 n
k
k
i
i
Z
Z
VV
1

Voltage divider:
Impedance (Z)

44. Complex Impedance
_
+


V
 I


1IZ1
Z2 Z

 
2121
2
2121
2
1
2
1
1
2
2
1
11
ZZZZZZ
ZV
I
ZZZZZZ
ZZV
ZZ
Z
V
I
ZZ
Z
II





















Impedance (Z)

45. 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
20kW
+
-
1mF10V  0 VC
+
-
w = 377
Find VC
• How do we find VC?
• First compute impedances for resistor and capacitor:
ZR = 20kW = 20kW  0
ZC = 1/j (377 *1mF) = 2.65kW  -90
Impedance (Z)

46. Complex Impedance
20kW
+
-
1mF10V  0 VC
+
-
w = 377
Find VC
20kW  0
+
-
2.65kW  -
90
10V  0 VC
+
-
Now use the voltage divider to find VC:




46.82V31.1
54.717.20
9065.2
010VCV
)
0209065.2
9065.2
(010 





kk
k
VVC
Impedance (Z)

47. 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.
Complex Impedance
Impedance (Z)

48. Kirchhoff’s Laws
KCL and KVL hold as well in phasor domain.
KVL： 0
1

n
k
kv vk- Transient voltage of the #k branch
0
1

n
k
kV
KCL: 0
1


n
k
ki
0
1


n
k
kI
ik- Transient current of the #k branch
Impedance (Z)

49. 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
Impedance (Z)

50. 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.
2mA  40
–
1mF VC
+
–
1kW VR
+
+
–
V
I = 2mA  40, VR = 2V  40
VC = 5.31V  -50, V = 5.67V  -29.37
Real Axis
Imaginary Axis
VR
VC
V
Impedance (Z)

51. Parallel and Series Resonance
Outline:
RLC Circuit,
Series Resonance
Parallel Resonance

52. v
vR
vL
vC
CLR vvvv 
CLR VVVV  Phasor

I
V
LV
CV
RV
IZ
XRI
XXRI
IXIXIR
VVVV
CL
CL
CLR





22
22
22
22
)(
)()(
)(
）CL XXX (
22
XRZ 
22
)
1
(
c
LR

 
(2nd Order RLC Circuit )Series RLC Circuit
Parallel and Series Resonance :

53. 22
XRZ 
22
)
1
(
c
LR

 IZVVVV CLR  22
)(
Z
X = XL-XC
R


V
RV
CLX VVV   R
XX
V
VV
CL
R
CL





1
1
tan
-
tan
Phase difference:
XL>XC   >0，v leads i by  — Inductance Circuit
XL<XC   <0，v lags i by  — Capacitance Circuit
XL=XC   =0，v and i in phase — Resistors Circuit
Series RLC Circuit
Parallel and Series Resonance :

54. CLR VVVV   CL XIjXIjRI  
ZIjXRIXXjRI CL
  )()]((
)( CL XXjR
I
V
Z 


 ZjXRZ
22
)( CL XXRZ 
R
XX CL 
 1
tan
iv  
v
vR
vL
vC
Series RLC Circuit
Parallel and Series Resonance :

55. Series Resonance (2nd Order RLC Circuit )
CLR VVVV   CL XIjXIjRI  
R
XX
arctg
V
VV
arctg CL
R
CL 



CLCL VVL
C
XXWhen  

1
,
VVR  0and —— Series Resonance
Resonance condition
I
LV
CV
VVR
 
LC
for
LC 

2
11
00 
f0 f
X
Cf
XC
2
1

fLXL 2
Resonant frequency
Parallel and Series Resonance :

56. Series Resonance
R
V
Z
V
IRXXRZ CL 
0
0
22
0 )(•
Zmin；when V = constant, I = Imax= I0
RXX CL  RIXIXI CL 000  VVV CL 
• Quality factor Q,
R
X
R
X
V
V
V
V
Q CLCL

CLCL VVL
C
XX  )
1
( 

Resonance condition:
When,
Parallel and Series Resonance :

57. Parallel RLC Circuit
V

I
 LI
 CI
  
)(
1
/
11
222222
LR
L
Cj
LR
R
Cj
LjRLjR
LjR
Cj
LjRCjLjR
Y
























Parallel Resonance
Parallel Resonance frequency
L
CR
LC
2
0 1
1

LXR In generally )
2
1
( 0
LC
f


LC
1
0 
0)( 222



LR
L
C


When 2220
LR
R
Y

,
In phase withV I
V
L
RC
C
L
R
R
V
L
LC
R
R
V
LR
R
VVYII 






222
22
0
200
1
Zmax Imin:
Parallel and Series Resonance :

58. Parallel RLC Circuit
V

I
 LI
 CI
V
L
C
j
L
V
j
LjR
VIL


 


00
1

V
L
C
jVCjIC
  0
0|||||| 0  III CL
 Z  .
RCR
L
Q
0
0 1


0IjQIL
  0IjQIC
 
•Quality factor Q,
0000 Y
Y
Y
Y
I
I
I
I
Q CLLC

Parallel and Series Resonance :

59. Parallel RLC Circuit
Review
For sinusoidal circuit， Series ： 21 vvv  21 VVV 
21 iii  21 III 
？
Two Simple Methods:
Phasor Diagrams and Complex Numbers
Parallel :
Parallel and Series Resonance :