1. Complex Power – Background Concepts
ECE Circuit Analysis
Lecture Set #23
Complex Power – Background Concepts
Dr. Dave Shattuck
Associate Professor, ECE Dept.
W326-D3

2. Overview of this Lecture Set Complex Power – Background Concepts
In this set of lecture notes, we will cover the following topics:
Review of Sinusoidal Sources and Phasors
Review of RMS
Power with Sinusoids
You can click on the blue text to jump to the subject that you want to learn about now.

3. Textbook Coverage
This material is introduced in different ways in different textbooks. Approximately this same material is covered in your textbook in the following sections:
Electric Circuits 6th Ed. by Nilsson and Riedel: Sections 10.1 through 10.3
You should also read these sections in your text. This material is intended to complement your textbook coverage, not replace it.

4. Power in the Sinusoidal Steady State (Complex Power)
We studied Phasor Transforms. Using these transforms, we can find things we want to know, more quickly and more easily.
Now we are going to do a similar thing with power absorbed or delivered in circuits with sinusoidal sources. Again, we will only consider what is happening in the steady state. We will find that:
The use of phasors and transforms can be used for power calculations, and
Some very useful new concepts will help us deliver more power to where we want it, with fewer losses at the same time.
The power we use is often sinusoidal. That is, the wall plugs, and some other sources, are voltages and currents that vary as a sine wave. Thus, this subject is very useful to us.

5. AC Circuit Analysis Using Transforms
Let’s remember first and foremost that the end goal is to find the solution to real problems. We will use the transform domain, and discuss quantities which are complex, but obtaining the real solution is the goal.

6. Power with Sinusoidal Voltages and Currents
It is important to remember that nothing has really changed with respect to the power expressions that we are looking for. Power is still obtained by multiplying voltage and current.
The fact that the voltage and current are sine waves or cosine waves does not change this formula.

7. Some Review – Sinusoids
The figure below is taken from Figure 6.2 in Circuits by A. Bruce Carlson. A general sinusoid has the equation given below. Note that in this equation there are three parameters, the amplitude (Xm), the frequency (w), and the phase (f). The time, t, is the independent variable. The sine function is just as good as the cosine function, but in electrical engineering the cosine function is used more often.

8. Definition of RMS – Review
We are now going to review an important term, the rms value of a voltage or current. This was covered before. The rms value, also called the effective value, has the most meaning in terms of power calculations.
It is so useful, that we will redefine our phasor transforms in terms of rms values, to make our formulas simpler and easier to use. Thus, it is worth the time to review rms concepts.

9. Derivation of RMS – 1
We want the effective value that could be used in power calculations, for average power, in the formula below.
We will do the derivation for a resistance, since we want the formula to work with the resistance power formulas. Let’s arbitrarily choose to work with the voltage. What we want to get is a value that will work in the formula,
With T as the period, the average value of the power is obtained by the formula,

10. Derivation of RMS – 2
Now, to get the formula, we simply set the two equations from the previous slide equal to each other,
Now, we need to simplify. The resistance is assumed to be a constant, and so it can be taken out of the integral. When we multiply both sides by R, we get

11. Derivation of RMS – 3
Finally, we can solve for the rms value of the voltage, by taking the square root of both sides,
This is the result that we have been working toward. We only need to interpret this result. We have taken the periodic voltage, v(t), and squared it. Then, by integrating it over a period and dividing by the period, we are taking the mean value of the squared function. Finally, we take the square root of the mean value of the squared function. We call this rms.

12. RMS Value of a Sinusoid Remember, this only holds for sinusoids!
The rms value for a general periodic function, x(t), is
Now, this was derived for any periodic function. The function must be periodic for the formula for the mean value to apply.
If we perform the calculus to get the rms value for a sinusoid, we find the rms value is equal to the zero-to-peak value (or amplitude) divided by the square root of 2, or
Remember, this only holds for sinusoids!

13. Power with Sinusoids – 1
If we have sinusoidal voltages and currents, we can get the power by multiplying the two. We could plot the power as shown below. This curve was obtained by taking a couple of arbitrary sinusoids for voltage and current. If you change the magnitudes and phases, this curve will change.

14. Power with Sinusoids – 2
Let’s approach this same issue using equations. Let’s assume that our voltage and current have the formulas,
Now, we can use the formula from trigonometry, which most of us learned but forgot,
Applying this here, we get

15. Power with Sinusoids – 3 In the last slide we found that
Now, we can use another formula from trigonometry,
Applying this here in the second term, with
we get

16. Power with Sinusoids – 4 So, we have that
Now, we can look at the plot that we had, and understand it somewhat better. Let’s note some special properties in the slides that follow.

17. Power with Sinusoids – Note 1
The power curve is not centered at zero, so the average is not zero.
First, note that even though v(t) and i(t) were zero-mean sinusoids, the product, p(t), does not generally have a zero mean.

18. Power with Sinusoids – Note 2
In fact, the mean, or average value, of the power is equal to the first term of this equation, since the average of the sinusoids is zero.

19. Power with Sinusoids – Note 3
We use a capital letter P to represent the average value of the power, p(t). The average power is very useful to know.

20. Power with Sinusoids – Note 4
This average power P is a function of the magnitudes of the voltage and current, but also of the difference in phase between voltage and current.

21. Power with Sinusoids – Note 5
The power varies with time, and is in fact sinusoidal with a frequency twice that of the voltage and current.

22. Shifting the Time Axis General case Shifted case
For notational reasons, electrical engineers take the general case, which we have been considering, and then shift the time axis, so that the phase of the current is zero. The new phase of the voltage is now reduced by the phase of the current, and now is qv-qi. We redefine this phase of the voltage as q, and get a new set of formulas, below.
General case
Shifted case

23. Shifting the Time Axis – Note 1
Some of you may be disturbed by the relationship between the two cases below. It may look like we have used q = qv + qi in some cases, and q = qv - qi in other cases. We have not. Remember that the t in the General case is different from the t' in the Shifted case.
General case
Shifted case

24. Shifting the Time Axis – Note 2
Since we shifted the time axis, we changed the t to a t' in the previous slide. However, the original choice of t was arbitrary, and there is no reason to keep the prime any longer. Therefore, for the rest of this material, we will use the notation below. Remember that q is the phase of the voltage with respect to the phase of the current. This way of expressing the phases will be useful to us. We will generally get q by subtracting the phase of the current from the phase of the voltage.

25. Power with Sinusoidal Sources
The formulas below are important, and are the beginning of the concepts that follow in the next two parts. We have found two things. First, the average power is a function of the product of the magnitudes of the voltage and current, and also a function of the difference between the phase of the voltage and current, q. Second, we found that the expression for the power as a function of time has a constant term, which is that average value, and terms at twice the frequency of the voltage and current.

26. So what is the point of all this?
This is a good question. First, our premise is that since electric power is usually distributed as sinusoids, the issue of sinusoidal power is important.
In addition, there are some significant problems that will arise when we connect loads to our power lines that act like inductors. This problems can be addressed using phasor analysis, and some additional concepts that we will lay out in the next set of lecture notes. These concepts involved quantities called real and reactive power.
Go back to Overview slide.