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ELECTRIC CIRCUITS THEORY 1 These lecture slides have been compiled by Mohammed LECTURE 7 SalahUdDin Ayubi. Frequency Response (Phasor Domain Analysis) 22 August 2005
Engineer M S Ayubi
1
Dynamic Circuit Analysis Two Approaches to solving circuits with energy storage elements and time-varying inputs: • Transient Analysis (time-domain) • Phasor Analysis (frequency-domain)
22 August 2005
Engineer M S Ayubi
2
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
3
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
4
Complex Numbers • Notation – Rectangular form – Polar form – Exponential form
• Operations – Addition, subtraction – Multiplication, division – Conjugate 22 August 2005
Engineer M S Ayubi
5
Rectangular Form z = a + jb z = complex value a, b = real values j = √-1 a = Re(z) b = Im(z) 22 August 2005
Engineer M S Ayubi
6
Polar Form z = |z| ∠ θ z = complex value |z| = magnitude = √ a2+b2 θ = phase angle = arctan(b/a) a = |z| cos θ b = |z| sin θ 22 August 2005
Engineer M S Ayubi
7
Exponential Form z = |z| ejθ Derived by combining rectangular & polar: z = |z| (cos θ + j sin θ ) and Euler’s Identity: ejθ = cos θ + j sin θ 22 August 2005
Engineer M S Ayubi
8
Complex Numbers • Notation – Rectangular form – Polar form – Exponential form
• Operations – Addition, subtraction – Multiplication, division – Conjugate 22 August 2005
Engineer M S Ayubi
9
Complex Addition x = a + jb y = c + jd x + y = (a + c) + j(b + d) x – y = (a - c) + j(b - d)
22 August 2005
Engineer M S Ayubi
10
Complex Multiplication x = |x| ∠ θ y = |y| ∠ φ xy = |x| |y| ∠ θ +φ x/y = |x| / |y| ∠ θ -φ
22 August 2005
Engineer M S Ayubi
11
Complex Conjugate x = a + jb x* = a - jb x x* = |x|2
22 August 2005
Engineer M S Ayubi
12
Example x = a + jb y = c + jd Compute x/y directly:
22 August 2005
Engineer M S Ayubi
13
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
14
Phasor Analysis Charles Steinmetz developed phasor analysis, introducing it at the International Electrical Congress in Chicago in 1893 “The method of calculation is considerably simplified. Whereas before we had to deal with periodic functions of an independent variable ‘time’, now we obtain a solution through the simple addition, subtraction, etc of constant numbers … Neither are we restricted to sine waves, since we can construct a general periodic function out of its sine wave components … With the aid of Ohm’s Law in its complex form any circuit or network of circuits can be analysed in the same way, and just as easily, as for direct current, provided only that all the variables are allowed to take on complex values.” -- Steinmetz, C. P., ‘Die Anwendung complexer Grössen in der Elektrotechnik’, Elektrotechnische Zeitschrift, 42, 597-99; 44, 631-5; 45, 641-3; 46, 653-4, 1893 22 August 2005
Engineer M S Ayubi
15
Phasor Analysis Charles Steinmetz developed phasor analysis, introducing it at the International Electrical Congress in Chicago in 1893 “The method of calculation is considerably simplified. Whereas before we had to deal with periodic functions of an independent variable ‘time’, now we obtain a solution through the simple addition, subtraction, etc of constant numbers … Neither are we restricted to sine waves, since we can construct a general periodic function out of its sine wave components … With the aid of Ohm’s Law in its complex form any circuit or network of circuits can be analysed in the same way, and just as easily, as for direct current, provided only that all the variables are allowed to take on complex values.” -- Steinmetz, C. P., ‘Die Anwendung complexer Grössen in der Elektrotechnik’, Elektrotechnische Zeitschrift, 42, 597-99; 44, 631-5; 45, 641-3; 46, 653-4, 1893 22 August 2005
Engineer M S Ayubi
16
Phasor Analysis • Original problem (time domain)
22 August 2005
Engineer M S Ayubi
17
Phasor Analysis • Original problem (time domain) V
Ref
22 August 2005
Engineer M S Ayubi
18
Phasor Analysis • New, easier problem (phasor domain) (No, Really!)
5e2jt 22 August 2005
Engineer M S Ayubi
19
Phasor Analysis • New, easier problem (phasor domain)
22 August 2005
Engineer M S Ayubi
20
Phasor Analysis Guess a solution for v(t). Try: v(t)=Vkej2t, where Vk is a constant
22 August 2005
Engineer M S Ayubi
21
Phasor Analysis
22 August 2005
Engineer M S Ayubi
22
Phasor Analysis For R=10Ω and C=2F,
0.31-j1.25 = 1.25∠ -88.6° V=Vkej2t =(1.25∠ -88.6°) ej2t 22 August 2005
Engineer M S Ayubi
23
Phasor Analysis Back to original problem: Original input was Im(5ej2t ), so solution is Im(V) = Im{(1.25∠ -88.6°) ej2t} =1.25 sin(2t - 88.6°) V
22 August 2005
Engineer M S Ayubi
24
Phasor Analysis Summary Time Domain Circuit
Phasor Domain Circuit
Difficult
Time Domain Solution
22 August 2005
Easier
Phasor Domain Solution
Engineer M S Ayubi
25
Phasor to Time Domain • Convert solution from phasor domain to time domain – If input was cos, output is cos (take real part) – If input was sin, output is sin (take imaginary part)
22 August 2005
Engineer M S Ayubi
26
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
27
Phasor Domain Sources • Convert time domain cos/sin source to phasor domain by keeping magnitude and phase Time Domain
6sin(2t+30°)
22 August 2005
Phasor Domain
6∠ 30°
Engineer M S Ayubi
28
Phasor Domain Impedance • Resistor – Time Domain:
vR(t) = RiR(t)
– Phasor Domain:
VRejω t = RIRejω t
VR = RIR
22 August 2005
Engineer M S Ayubi
29
Phasor Domain Impedance • Inductor – Time Domain:
– Phasor Domain:
22 August 2005
Engineer M S Ayubi
30
Phasor Domain Impedance • Capacitor – Time Domain:
– Phasor Domain:
22 August 2005
Engineer M S Ayubi
31
Impedance • Each phasor circuit element (R, L, C) relates phasor voltage and current by a complex constant • V=ZI Ohm’s Law in the Phasor Domain • Z = Impedance • Y = 1/Z = Admittance 22 August 2005
Engineer M S Ayubi
32
Impedance ZR = R ZL = jω L ZC = 1/(jω C)
22 August 2005
Engineer M S Ayubi
33
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
34
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
35
Phasor Analysis Example 1 • Find i(t)
22 August 2005
Engineer M S Ayubi
36
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
37
Phasor Analysis Example 1
22 August 2005
Engineer M S Ayubi
38
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
39
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
40
Phasor Analysis Example 1
22 August 2005
Engineer M S Ayubi
41
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
42
Phasor Analysis Example 1 i(t)=Im(Iej5t) i(t)=0.09sin(5t + 42.4º) A From phasor domain
22 August 2005
Copy from time domain
Engineer M S Ayubi
43
Phasor Analysis Example 2 • Find i(t)
22 August 2005
Engineer M S Ayubi
44
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
45
Phasor Analysis Example 2
22 August 2005
Engineer M S Ayubi
46
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
47
Phasor Analysis Example 2
22 August 2005
Engineer M S Ayubi
48
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
49
Phasor Analysis Example 2 i(t) = Re(Iej500t) = 2.88 cos (500t – 5.9º) mA
22 August 2005
Engineer M S Ayubi
50
Phasor Analysis Example 3 • Find |Z|(ω )
22 August 2005
Engineer M S Ayubi
51
Phasor Analysis Example 3
22 August 2005
Engineer M S Ayubi
52
|Z|(w) 5.0E+02
4.0E+02
|Z|
3.0E+02
2.0E+02
1.0E+02
0.0E+00 5.0E+01
1.5E+02 Frequency (rad/s)
22 August 2005
Engineer M S Ayubi
53
Phasor Analysis Example 4 • Find ω so that |Z| = 20 Ω
22 August 2005
Engineer M S Ayubi
54
Phasor Analysis Example 4
22 August 2005
Engineer M S Ayubi
55
Phasor Analysis Example 4 ω =14,434 rad/s ω =2π f f = 2.3KHz
22 August 2005
Engineer M S Ayubi
56
Engineer M S Ayubi
1
Dynamic Circuit Analysis Two Approaches to solving circuits with energy storage elements and time-varying inputs: • Transient Analysis (time-domain) • Phasor Analysis (frequency-domain)
22 August 2005
Engineer M S Ayubi
2
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
3
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
4
Complex Numbers • Notation – Rectangular form – Polar form – Exponential form
• Operations – Addition, subtraction – Multiplication, division – Conjugate 22 August 2005
Engineer M S Ayubi
5
Rectangular Form z = a + jb z = complex value a, b = real values j = √-1 a = Re(z) b = Im(z) 22 August 2005
Engineer M S Ayubi
6
Polar Form z = |z| ∠ θ z = complex value |z| = magnitude = √ a2+b2 θ = phase angle = arctan(b/a) a = |z| cos θ b = |z| sin θ 22 August 2005
Engineer M S Ayubi
7
Exponential Form z = |z| ejθ Derived by combining rectangular & polar: z = |z| (cos θ + j sin θ ) and Euler’s Identity: ejθ = cos θ + j sin θ 22 August 2005
Engineer M S Ayubi
8
Complex Numbers • Notation – Rectangular form – Polar form – Exponential form
• Operations – Addition, subtraction – Multiplication, division – Conjugate 22 August 2005
Engineer M S Ayubi
9
Complex Addition x = a + jb y = c + jd x + y = (a + c) + j(b + d) x – y = (a - c) + j(b - d)
22 August 2005
Engineer M S Ayubi
10
Complex Multiplication x = |x| ∠ θ y = |y| ∠ φ xy = |x| |y| ∠ θ +φ x/y = |x| / |y| ∠ θ -φ
22 August 2005
Engineer M S Ayubi
11
Complex Conjugate x = a + jb x* = a - jb x x* = |x|2
22 August 2005
Engineer M S Ayubi
12
Example x = a + jb y = c + jd Compute x/y directly:
22 August 2005
Engineer M S Ayubi
13
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
14
Phasor Analysis Charles Steinmetz developed phasor analysis, introducing it at the International Electrical Congress in Chicago in 1893 “The method of calculation is considerably simplified. Whereas before we had to deal with periodic functions of an independent variable ‘time’, now we obtain a solution through the simple addition, subtraction, etc of constant numbers … Neither are we restricted to sine waves, since we can construct a general periodic function out of its sine wave components … With the aid of Ohm’s Law in its complex form any circuit or network of circuits can be analysed in the same way, and just as easily, as for direct current, provided only that all the variables are allowed to take on complex values.” -- Steinmetz, C. P., ‘Die Anwendung complexer Grössen in der Elektrotechnik’, Elektrotechnische Zeitschrift, 42, 597-99; 44, 631-5; 45, 641-3; 46, 653-4, 1893 22 August 2005
Engineer M S Ayubi
15
Phasor Analysis Charles Steinmetz developed phasor analysis, introducing it at the International Electrical Congress in Chicago in 1893 “The method of calculation is considerably simplified. Whereas before we had to deal with periodic functions of an independent variable ‘time’, now we obtain a solution through the simple addition, subtraction, etc of constant numbers … Neither are we restricted to sine waves, since we can construct a general periodic function out of its sine wave components … With the aid of Ohm’s Law in its complex form any circuit or network of circuits can be analysed in the same way, and just as easily, as for direct current, provided only that all the variables are allowed to take on complex values.” -- Steinmetz, C. P., ‘Die Anwendung complexer Grössen in der Elektrotechnik’, Elektrotechnische Zeitschrift, 42, 597-99; 44, 631-5; 45, 641-3; 46, 653-4, 1893 22 August 2005
Engineer M S Ayubi
16
Phasor Analysis • Original problem (time domain)
22 August 2005
Engineer M S Ayubi
17
Phasor Analysis • Original problem (time domain) V
Ref
22 August 2005
Engineer M S Ayubi
18
Phasor Analysis • New, easier problem (phasor domain) (No, Really!)
5e2jt 22 August 2005
Engineer M S Ayubi
19
Phasor Analysis • New, easier problem (phasor domain)
22 August 2005
Engineer M S Ayubi
20
Phasor Analysis Guess a solution for v(t). Try: v(t)=Vkej2t, where Vk is a constant
22 August 2005
Engineer M S Ayubi
21
Phasor Analysis
22 August 2005
Engineer M S Ayubi
22
Phasor Analysis For R=10Ω and C=2F,
0.31-j1.25 = 1.25∠ -88.6° V=Vkej2t =(1.25∠ -88.6°) ej2t 22 August 2005
Engineer M S Ayubi
23
Phasor Analysis Back to original problem: Original input was Im(5ej2t ), so solution is Im(V) = Im{(1.25∠ -88.6°) ej2t} =1.25 sin(2t - 88.6°) V
22 August 2005
Engineer M S Ayubi
24
Phasor Analysis Summary Time Domain Circuit
Phasor Domain Circuit
Difficult
Time Domain Solution
22 August 2005
Easier
Phasor Domain Solution
Engineer M S Ayubi
25
Phasor to Time Domain • Convert solution from phasor domain to time domain – If input was cos, output is cos (take real part) – If input was sin, output is sin (take imaginary part)
22 August 2005
Engineer M S Ayubi
26
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
27
Phasor Domain Sources • Convert time domain cos/sin source to phasor domain by keeping magnitude and phase Time Domain
6sin(2t+30°)
22 August 2005
Phasor Domain
6∠ 30°
Engineer M S Ayubi
28
Phasor Domain Impedance • Resistor – Time Domain:
vR(t) = RiR(t)
– Phasor Domain:
VRejω t = RIRejω t
VR = RIR
22 August 2005
Engineer M S Ayubi
29
Phasor Domain Impedance • Inductor – Time Domain:
– Phasor Domain:
22 August 2005
Engineer M S Ayubi
30
Phasor Domain Impedance • Capacitor – Time Domain:
– Phasor Domain:
22 August 2005
Engineer M S Ayubi
31
Impedance • Each phasor circuit element (R, L, C) relates phasor voltage and current by a complex constant • V=ZI Ohm’s Law in the Phasor Domain • Z = Impedance • Y = 1/Z = Admittance 22 August 2005
Engineer M S Ayubi
32
Impedance ZR = R ZL = jω L ZC = 1/(jω C)
22 August 2005
Engineer M S Ayubi
33
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
34
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
35
Phasor Analysis Example 1 • Find i(t)
22 August 2005
Engineer M S Ayubi
36
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
37
Phasor Analysis Example 1
22 August 2005
Engineer M S Ayubi
38
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
39
Outline • Complex numbers review – Notation – Operations
• Phasor analysis – – – – 22 August 2005
Overview Impedance Analysis steps Examples Engineer M S Ayubi
40
Phasor Analysis Example 1
22 August 2005
Engineer M S Ayubi
41
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
42
Phasor Analysis Example 1 i(t)=Im(Iej5t) i(t)=0.09sin(5t + 42.4º) A From phasor domain
22 August 2005
Copy from time domain
Engineer M S Ayubi
43
Phasor Analysis Example 2 • Find i(t)
22 August 2005
Engineer M S Ayubi
44
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
45
Phasor Analysis Example 2
22 August 2005
Engineer M S Ayubi
46
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
47
Phasor Analysis Example 2
22 August 2005
Engineer M S Ayubi
48
Phasor Analysis Steps 1. Convert time-domain circuit to phasor domain •
Keep magnitude and phase of source, write impedance for R, L, C
1. Solve for desired quantities in phasor-domain •
Use same techniques as for time-domain, but with complex-valued circuit elements
1. Convert phasor-domain solution to time-domain •
Keep sin/cos and frequency of source, use magnitude and phase from phasor solution
22 August 2005
Engineer M S Ayubi
49
Phasor Analysis Example 2 i(t) = Re(Iej500t) = 2.88 cos (500t – 5.9º) mA
22 August 2005
Engineer M S Ayubi
50
Phasor Analysis Example 3 • Find |Z|(ω )
22 August 2005
Engineer M S Ayubi
51
Phasor Analysis Example 3
22 August 2005
Engineer M S Ayubi
52
|Z|(w) 5.0E+02
4.0E+02
|Z|
3.0E+02
2.0E+02
1.0E+02
0.0E+00 5.0E+01
1.5E+02 Frequency (rad/s)
22 August 2005
Engineer M S Ayubi
53
Phasor Analysis Example 4 • Find ω so that |Z| = 20 Ω
22 August 2005
Engineer M S Ayubi
54
Phasor Analysis Example 4
22 August 2005
Engineer M S Ayubi
55
Phasor Analysis Example 4 ω =14,434 rad/s ω =2π f f = 2.3KHz
22 August 2005
Engineer M S Ayubi
56
