Characteristics of Sinusoidal
Phasors
Phasor Relationships for R, L and C
Impedance
Parallel and Series Resonance
Examples for Sinusoidal Circuits
Analysis
Single Phase AC
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
Characteristics of Sinusoids
Outline:1. Time Period: T 2. Frequency: f (Hertz)3. Angular Frequency: (rad/sec)4. Phase angle: Φ5. Amplitude: Vm Im
Characteristics of Sinusoids :
tVv mt sin i
R
+
_
i
R
-
+
v ,i
tt1 t20
Both the polarity and magnitude of voltage are changing.
Radian frequency(Angular frequency): = 2f = 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 = Imsint , v =Vmsint
v ,i
t 20
Vm , Im
Characteristics of Sinusoids :
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.
RIRdtiT
T 2
0
21
Effective Value of a Periodic Waveform T
eff dtiT
I0
21
221
22cos1sin1 2
0
2
0
22 mm
Tm
T
meffITI
Tdtt
TItdtI
TI
21
0
2 mT
effVdtv
TV
Characteristics of Sinusoids :
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 :
)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)
221
v, i
t
v
i
21Out of phase
t
v, iv
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 :
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 + & -
2sin cos cos3 2
cos sin2
Characteristics of Sinusoids :
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 :
Phase difference
v, i
t
vi
-/3 /3• ••
3sin tVm
3sin tIm
Characteristics of Sinusoids :
Outline:1. Complex Numbers 2. Rotating Vector3. 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
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 tComplex form:
cosmv t V t
Phasor form:
j tmv t V e
Angular frequency ω is known in the circuit.
|| mVV
|| mVV
Phasors
Rotating Vector
tIti m sin)(
i
Im
t1
i
t
Im
t
x
y
max
cos sin
sin
j tm m m
j tm m
I e I t jI t
i t I t I I e
A complex coordinates number:
Real value:
i(t1)
Imag
Phasors
Rotating Vector
Vm
x
y
0
)sin( tVv m
Phasors
Complex Numbers
jbaA — Rectangular Coordinates
sincos jAA
jeAA — Polar Coordinates
jeAAjbaA
conversion : 22 baA
abarctg
jbaeA j cosAa sinAb
|A|
a
b
Real axis
Imaginary axis
jjje j 090sin90cos90
Phasors
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
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
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
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
)sin()cos()( tAjtAeAAe tjtj
)cos(||}Re{ tAAe tj
Complex Exponentials
jeAA
A real-valued sinusoid is the real part of a complex exponential. Complex exponentials make solving for AC steady state an algebraic problem.
Phasors
Phasor Relationships for R, L and C
Outline:I-V Relationship for R, L and C,
Power conversion
Phasor Relationships for R, L and C
v~i relationship for a resistor
_
v
i
R
+
U
I
tItR
VRvi m
m sinsin
tVv m sin
Relationship between RMS:RVI
Wave and Phasor diagrams : v 、 i
t
v
i
I
V
RVI
Resistor
Suppose
Time domain Frequency domainResistor
With a resistor θ﹦ϕ, v(t) and i(t) are in phase .
)cos()()cos()(
wtItiwtVtv
m
m
IRV
RIVeRIeV
eRIeV
mm
jm
jm
wtjm
wtjm
)()(
Phasor Relationships for R, L and C
PowerResistor
_
v
i
R
+
U
I
P 0
tItVvip mm sinsin tVI mm 2sin
tVI mm 2cos12
tIVIV 2cos
v, i
t
v
i P=IV Tpdt
TP
0
1 T
VIdttVIT 0
2cos11
RVRIIVP
22
• Average Power
• Transient Power
Note: I and V are RMS values.
Phasor Relationships for R, L and C
Resistor
, R=10, Find i and Ptv 314sin311
VVV m 2202
3112
ARVI 22
10220
ti 314sin222 WIVP 484022220
Phasor Relationships for R, L and C
v~i relationshipInductor
dtdiLvv AB
tLIdt
tIdLdtdiLv m
m cossin
90sin tLI m
90sin tVm
tvdt
Li 1
tvdt
Lvdt
L 0
0 11 tvdt
Li
001
tIi m sin Suppose
Phasor Relationships for R, L and C
v~i relationshipInductor
90sin tLIm dtdiLv 90sin tVm
LIV mm Relationship between RMS: LIV
LVI
fLLX L 2
For DC , f = 0 , XL = 0.
fX L
v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º
Phasor Relationships for R, L and C
v ~ i relationshipInductor
v, i
t
vi
eL
V
I
LXIjV
Wave and Phasor diagrams :
Phasor Relationships for R, L and C
PowerInductor
vip tItV mm sin90sin ttIV mm sincos
tIV mm 2sin
2 tVI 2sin
P
t
v, i
t
vi
++
--22max 2
1 LILIW m
2
00 21 LiLidividtW
it Energy stored:
T T
tdtVIT
pdtT
P0 0
02sin11 Average Power
Reactive PowerL
L XVXIIVQ
22 ( Var)
Phasor Relationships for R, L and C
Inductor
L = 10mH , v = 100sint , Find iL when f = 50Hz and 50kHz.
14.310105022 3fLX L
Atti
AXVI
L
L
90sin25.22
5.2214.3
2/10050
31401010105022 33fLX L
mAtti
mAXVI
L
Lk
90sin25.22
5.2214.3
2/10050
Phasor Relationships for R, L and C
v ~ i relationshipCapacitor
_
v
i
+
U
I
C
dtdvC
dtdqi
tVv m sinSuppose:
90sincos tCVtCVi mm 90sin tIm
t ttidt
cvidt
cidt
cidt
cv
0
000
1111
i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º
Relationship between RMS:CX
V
C
VCVI
1
fCC
X C 211
For DC , f = 0 , XC f
X C1
mm CVI
Phasor Relationships for R, L and C
_
v
i
+
U
I
C tj
m
tjm eCVjdtedVC
dttdvCti
)()(
v(t) = Vm ejt
Represent v(t) and i(t) as phasors:CjX
VVCωjI ==
• 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 IwCj
Phasor Relationships for R, L and C
v ~ i relationshipCapacitor
v, i
t
vi
I
V
CXIjV
Wave and Phasor diagrams :
Phasor Relationships for R, L and C
PowerCapacitor
Average Power: P = 0
Reactive PowerC
C XVXIIVQ
22 ( Var)
90sinsin tItVvip mm tVItIV mm 2sin2sin2
P
t
v, i
t
vi
++
--
Energy stored:
t vv
CvCvdvdtdtdvCvvidtW
0 0
2
0 21
22max 2
1 CVCVW m
Phasor Relationships for R, L and C
Capacitor
Suppose C=20F , AC source v=100sint , Find XC and I for f = 50Hz, 50kHz 。
1592
11Hz50fCC
Xf c
A44.02
c
m
c XV
XVI
159.02
11KHz50fCC
Xf c
A4402
c
m
c XV
XVI
Phasor Relationships for R, L and C
Review (v – i Relationship)
Time domain Frequency domain
iRv IRV
ICj
V 1
ILjV dtdiLvL
dtdvCiC C
X C 1
LX L ,
,
, 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
Summary: R : RX R 0
L : ffLLX L 22 iv
C: ffccX C
12
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
Impedance (Z)
Outline:Complex currents and voltages.ImpedancePhasor Diagrams
• 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
vmj
m VeVV v
imj
m IeII i
ZeZeIV
IVZ jj
m
m iv )(
‘Z’ is called impedancemeasured in ohms ()
Impedance (Z)
Complex Impedance
ZeZeIV
IVZ 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)
Complex Impedance
ZR = R = 0; or ZR = R 0
Resistor——The impedance is R
cj
c jXCje
CZ
21
)2
( iv
or 901
CZC
Capacitor——The impedance is 1/jωC
Lj
L jXLjLeZ 2
)2
( iv
or 90 LZL
Inductor——The impedance is jωL
Impedance (Z)
Complex ImpedanceImpedance in series/parallel can be combined as resistors.
_
U
U
Z1 +
Z2 Zn
I
n
kkn ZZZZZ
121 ...
_
In
Zn
+
U
I
Z2 Z1
n
k kn ZZZZZ 121
11...111
21
12
21
21 ZZ
ZIIZZ
ZII
Current divider:
n
kk
ii
Z
ZVV
1
Voltage divider:
Impedance (Z)
Complex Impedance
_
+
V
I
1I Z1
Z2 Z
2121
2
2121
21
21
1
2
21
11
ZZZZZZZVI
ZZZZZZZZV
ZZZ
VI
ZZZII
Impedance (Z)
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 = 377Find 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)
Complex Impedance
20kW+
-1mF10V 0 VC
+
-
w = 377Find VC
20kW 0
+
-2.65kW -9010V 0 VC
+
-
Now use the voltage divider to find VC:
46.82 V31.154.717.20
9065.20 10VCV
)0209065.2
9065.2(010
kk
kVVC
Impedance (Z)
Impedance allows us to use the same solution techniquesfor 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)
Kirchhoff’s Laws
KCL and KVL hold as well in phasor domain.
KVL : 01
n
kkv vk- Transient voltage of the #k branch
01
n
kkV
KCL: 01
n
kki
01
n
kkI
ik- Transient current of the #k branch
Impedance (Z)
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/jL
• Capacitor:– The admittance is jC
Impedance (Z)
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)
Parallel and Series Resonance
Outline:RLC Circuit,
Series Resonance
Parallel Resonance
v
vR
vL
vC
CLR vvvv
CLR VVVV Phasor
I
V
LV
CV
RVIZ
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 :
22 XRZ 22 )1(c
LR
IZVVVV CLR 22 )(
Z X = XL-XC
R
V
RVCLX VVV R
XXV
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 :
CLR VVVV CL XIjXIjRI
ZIjXRIXXjRI CL )()]((
)( CL XXjRIVZ
ZjXRZ22 )( CL XXRZ
RXX CL 1tan
iv
v
vR
vL
vC
Series RLC Circuit
Parallel and Series Resonance :
Series Resonance (2nd Order RLC Circuit )
CLR VVVV CL XIjXIjRI R
XXarctgV
VVarctg CL
R
CL
CLCL VVLC
XXWhen 1,
VVR 0and —— Series Resonance
Resonance condition
I
LV
CV
VVR
LCfor
LC
211
00
f0 f
X
CfX C 2
1
fLX L 2
Resonant frequency
Parallel and Series Resonance :
Series Resonance
RV
ZVIRXXRZ CL
00
220 )(•
Zmin ; when V = constant, I = Imax= I0
RXX CL RIXIXI CL 000 VVV CL
• Quality factor Q,
RX
RX
VV
VVQ CLCL
CLCL VVLC
XX )1(
Resonance condition:
When,
Parallel and Series Resonance :
Parallel RLC Circuit
V
I
LI CI
)(
1/11
222222 LRLCj
LRR
CjLjRLjR
LjR
CjLjRCjLjR
Y
Parallel Resonance
Parallel Resonance frequencyL
CRLC
2
0 11
LXR In generally )2
1( 0LC
f
LC1
0
0)( 222
LR
LCWhen
2220 LRRY
,
In phase withV I
VL
RC
CLR
RVL
LCR
RVLR
RVVYII
22222
0200 1Zmax Imin:
Parallel and Series Resonance :
Parallel RLC Circuit
V
I
LI CI
VLCj
LVj
LjRVIL
00
1
VLCjVCjIC 0
0|||||| 0 III CL Z .
RCRLQ
0
0 1
0IjQI L
0IjQIC
•Quality factor Q,
0000 YY
YY
II
IIQ CLLC
Parallel and Series Resonance :
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 :