1 Lecture #5 EGR 272 – Circuit Theory II Phasor Analysis The use of complex numbers to solve ac circuit problems – the so-called phasor method considered

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Lecture #5 EGR 272 – Circuit Theory II

Phasor Analysis

The use of complex numbers to solve ac circuit problems – the so-called phasor method considered in this chapter – was first done by German-Austrian mathematician and electrical engineer Charles Proteus Steinmetz in a paper presented in 1893. He is noted also for the laws of hysteresis and for his work in manufactured lighting.

Steinmetz was born in Breslau, Germany, the son of a government railway worker. He was deformed from birth and lost his mother when he was 1 year old, but this did not keep him from becoming a scientific genius. Just as his work on hysteresis later attracted the attention of the scientific community, his political activities while he was at the University at Breslau attracted the police. He was forced to flee the country just as he had finished the work for his doctorate, which he never received. He did electrical research in the United States, primarily with the General Electric Company. His paper on complex numbers revolutionized the analysis of ac circuits, although it was said at the time that no one but Steinmetz understood the method. In 1897 he also published the first book to reduce ac calculations to a science.

(Electric Circuit Analysis, 2nd Edition, by Johnson, Johnson, and Hilburn, p. 307)

“I have found the equation that will enable us to transmit electricity through alternating current over thousands of miles. I have reduced it to a simple problem in algebra.”

Charles Proteus Steinmetz

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Read: Chapter 9 and Appendix B in Electric Circuits, 6th Edition by Nilsson

Development of “phasor analysis”AC circuit analysis (or phasor analysis) is very similar to DC analysis, but the justification for the method of analysis is somewhat complicated.

Recall from a study of differential equations that the forced response to a circuit will be in the form of the input (+ derivatives of the input). So if the input is sinusoidal, the output must also be sinusoidal as shown below.

AC Circuit

Input: Vpcos(wt)

Output: Acos(wt) + Bsin(wt)

= Kcos(wt +

Note that the input and the output on differ by the magnitude and the phase.

Recall that polar numbers are represented by a magnitude and a phase angle.

(More on this shortly)

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Examples: Convert to the other notation (time waveform or phasor)

Phasor – a complex number in polar form representing either a sinusoidal voltage or a sinusoidal current.

Symbolically if a time waveform is designated at v(t), then the corresponding phasor is designated by .V

rad/s 10 w , 3010 V

Solving complex equations

Two equations can be generated from a single complex equation by equating the real and imaginary parts of the equation.

Example: A + jB = C + jD (one complex equation)

so A = C and B = D

(real parts of the equation) (imaginary parts of the equation)

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Example: Use KVL on the circuit below to show how the phasor equation can be developed.

Further justification for “phasor analysis”Suppose that we replaced a real sinusoidal forcing function v(t) = Vpcos(wt + ) with a complex forcing function V(s) whose real part equals v(t).

Define V(s) = Vpej(wt+) = complex forcing function.

Recall Euler's Identity: e±jA = cos(A) ± jsin(A)

So V(s) = Vpej(wt+) = Vp[cos(wt + ) + jsin(wt + )]

And if the real part of V(s) is designated Re[V(s)] then

Re[V(s)] = Vpcos(wt + ) = v(t)

Replacing sinusoidal forcing functions with complex forcing functions will result in a cancellation of exponential terms leaving only phasor terms.

v(t) = Vpcos(wt+)

+

_

+ v1(t) -

+

v2(t) -

Since v1(t) and v2(t) can only vary from v(t) in magnitude and phase (as seen earlier) then v1(t) and v2(t) could be expressed as:

v1(t) = V1cos(wt+) v2(t) = V2cos(wt+)

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Note that the relationship above is similar to Ohm’s Law.

Now we will define impedance for resistors, inductors, and capacitors.

Resistors:

If i(t) = Ipcos(wt + ), find v(t) and show that

Before we get into the details of phasor analysis, we need a method of representing components in AC circuits. We will introduce a new term called impedance.

Complex Impedance

Z = impedance or complex impedance (in )

V phasor voltageZ

phasor currentI Z V

+

_

I

R

V RIZ R 0 R

II

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Inductors:

If i(t) = Ipcos(wt + ), find v(t) and show that L Z jwl j2 fL wl 90

Capacitors:

If v(t) = Vpcos(wt + ), find i(t) and show that C

1 1 -j 1 Z 90

jwC j2 fC wC wC

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Example: Find I(t) in the circuit below using phasor analysis.

100cos(500t) V

+

_ 400

2H

i(t)

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KVL and KCL in AC circuits:

KVL and KCL are satisfied in AC circuits using phasor voltages and currents. They are not satisfied using the magnitudes of the voltages or the currents.

Example: For the previous example, show that:

L R

L R

V V V (KVL is satisfied using phasors)

V V V (KVL is NOT satisfied using magnitudes only)

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Phasor diagrams:

KVL and KCL are only satisfied by phasors as indicated above, but it can be difficult to see exactly why certain voltages combine or cancel in some circuits. A phasor diagram is a vector diagram that illustrates graphically exactly how the phasors combine. Phasor diagrams will help the student to get a better feel for what happens when AC voltages or AC currents combine.

Example: For the previous example, show that

using a phasor diagram.

L RV V V

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AC Circuit Analysis Procedure:

1) Draw the phasor circuit (showing voltage and current sources as phasors and using complex impedances for the components).

2) Analyze the circuit in the same way that you might analyze a DC circuit.

3) Convert the final phasor result back to the time domain.

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Example: Analyze the circuit below using phasor analysis. Specifically,

A) Find the total circuit impedanceB) Find the total current, i(t)

C) Use current division to find i1(t) and i2(t)D) Show that KCL is satisfied using current phasors, but not current magnitudes.

120cos(100t)

+

_ 200

4H

i(t)

300

20 uF

i (t) 2

i (t) 1

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Example: (continued)E) Show that KCL is satisfied using a phasor diagram.

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1 Lecture #5 EGR 272 – Circuit Theory II Phasor Analysis The use of complex numbers to solve ac circuit problems – the so-called phasor method considered