EGR 2201 Unit 13AC Power Analysis

Read Alexander & Sadiku, Chapter 11. Homework #13 and Lab #13 due next

week. Final Exam and Lab Exam next week.

Review: Power Recall the following key points about power

from the first week of this course. An element’s power is the rate at which

that element supplies or absorbs energy:

Power’s unit of measure is the watt (W). By convention, we assign a positive sign to

a power value if the element is absorbing energy, and we assign a negative sign if the element is supplying energy.

dt

dwp

Supplies energy

Absorb energy

Review: The Power Law

An element’s power is equal to the product of its voltage times its current:

To get the correct sign (+ or ) on the power value when we use this equation, we must obey the passive sign convention, which says that we regard the positive direction for current as current into an element’s positive terminal.

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Review: Dissipation versus Storage

Recall also that resistors always absorb energy. They never supply energy. So a resistor’s power is always positive. The energy a resistor absorbs is lost (or

“dissipated”) as heat. In contrast, inductors and capacitors are

energy-storage elements. At times they may absorb energy, but at other times they may supply this energy back to the circuit. So an inductor’s or capacitor’s power may

be positive at one time but negative at another time.

Review: Other Power Formulas for Resistors

By combining the power law (p = v i) with Ohm’s law (v = i R or i = v R), we can easily derive two other useful formulas for the power dissipated by a resistor:

p = i 2 R

p = v 2 R There are no similar formulas for

capacitors or inductors in DC circuits.

Average Value of a Sinusoid (1 of 2)

Consider a sinusoid that represents any quantity (voltage, current, power, …) versus time.

If the sinusoid is symmetrical about the horizontal axis, then its average value is 0. In the circuits we’ve studied, a

graph of voltage or current versus time would look like this. Therefore the average voltage or average current is 0.

Average Value of a Sinusoid (2 of 2)

But if the sinusoid is “shifted up,” then its average value (seeblue dashed line) is a positive number. As we’ll see, a graph of power versus time in an AC

circuit would typically look like this. Therefore average power is usually not 0.

Shifting a Sinusoid Up

Mathematically, we can shift a sinusoid up by adding a positive constant to the sinusoid.

Example in MATLAB:>> fplot('5*cos(200*t)', [0, 0.1])>> hold on>> fplot('3 + 5*cos(200*t)', [0, 0.1], 'r')

What is the blue sinusoid’s average value? What is the red sinusoid’s average value?

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-6

-4

-2

0

2

4

6

8

Power in AC Circuits

In AC circuits we distinguish several kinds of power:

Quantity Symbol SI Unit Symbol for the Unit

Instantaneous power p(t) watt W

Average power (also called real power) P watt W

Apparent power S volt-ampere VA

Complex power S volt-ampere VA

Reactive power Qvolt-

ampere reactive

VAR

Instantaneous Power

To find an element’s or network’s instantaneous power, use the same power formula as for DC circuits:

The t’s remind us that in AC circuits, voltage and current change with time. So instantaneous power also changes with time.

This equation holds whether the source is sinusoidal, triangle, square, etc. But we’ll focus on the sinusoidal case.

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Multiplying Sinusoids

In a network connected to a sinusoidal source, v(t) and i(t) are sinusoids with the same frequency. And p(t) = v(t) i(t), so p(t) is the product of two sinusoids.

Question: What do you get when you multiply two sinusoids of the same frequency? Let’s use MATLAB to get an idea.

>> fplot('5*cos(200*t)', [0, 0.1])>> hold on>> fplot('8*cos(200*t+70*pi/180)', [0, 0.1],'r')>>fplot('5*cos(200*t)*8*cos(200*t+70*pi/180)',[0, .1],'g')

A Typical Graph of Instantaneous Power

In typical AC circuits, a network absorbs energy during part of the cycle and supplies energy back to the source during part of the cycle.

Therefore its power is sometimes positive and sometimes negative.

Positive p(t): network is absorbing energy.

Negative p(t): network is supplying energy.

Instantaneous Power with Sinusoidal Source

Suppose a network’s voltage and current are

Then its instantaneous power is

This term does not depend on t, and thus is constant. We call it the average power P.

This term is a sinusoid whose frequency is

twice the frequency of v(t) and i(t).

Graph of Instantaneous Power

On the previous slide we had

Constant term Sinusoid whose amplitude = .

Average Power

The constant term in our previous equation is the average power. It is measured in watts.

So . For any given network, is a constant

between 0 and 1, so P is a constant between 0 and .

Average power, P

Average Power is Real, Not Complex

We have .

Note that everything on the right-hand side of this equation is real, not complex.

Therefore, average power P always has a real value, not a complex value. So, for example, it would never be correct to

write something likeP = 4+j7 W or P = 830 W

Power Factor

We have .

We call the power factor. The symbol for power factor is pf. Its

value is just a number, with no units. For any given network, pf is a constant

between 0 and 1, so P is a constant between 0 and .

Power factor

Special Case #1: A Purely Resistive Network

Recall that for a resistor or a resistive network, current and voltage are in phase with each other:

So the power factor is 1: And average power

simplifies to

Other Average-Power Formulas for Resistors

We’ve just seen that, for a resistor,

By combining this with Ohm’s law, we can easily derive two other useful formulas for the average power dissipated by a resistor:

and

Summary for Resistors Compare the following formulas for computing

a resistor’s power in a DC circuit and computing a resistor’s average power in a sinusoidal AC circuit:

DC AC

Special Case #2: A Purely Inductive Network

Recall that for an inductor or an inductive network, current lags voltage by 90:

So the power factor is 0: And average power

simplifies to

Special Case #3: A Purely Capacitive Network

Recall that for a capacitor or a capacitive network, current leads voltage by 90:

So the power factor is 0: And average power

simplifies to

The General Case

We’ve looked at three special cases: A purely resistive network: pf=1 and

. A purely inductive network: pf=0 and P=0. A purely capacitive network: pf=0 and P=0.

In the general case, pf is a number between 0 and 1, and the formula for P does not simplify as it did in the special cases. We’re left with:

The General Case

In a general circuit containing sources, resistors, capacitors, and inductors, only the sources and the resistors have non-zero average power.

The general formula applies to each element, but for the capacitors and inductors it simplifies to 0, and for the resistors it simplifies to

Review: Maximizing the Load Power

In many applications, we wish to maximize the power transferred from a source to a load.

Replacing the source with its Thevenin-equivalent circuit, we have the following situation:

Thevenin-equivalent of source

Variable loadresistance

Review: The Load’s Power Depends on the Load Resistance

For this circuit, the load resistor’s power is given by:

Question: For fixed values of VTh and RTh, what value of RL will result in maximum load power? The answer is not obvious, since RL appears

in both the numerator and the denominator.

Review: Maximum Power Transfer Theorem

For DC resistive circuits, the maximum power transfer theorem says that maximum power is transferred to a load when the load resistance equals the source’s Thevenin resistance (RL = RTh).

What About for AC Circuits?

For AC circuits we have a similar situation, except instead of a Thevenin-equivalent resistance and a load resistance we have Thevenin-equivalent and load impedances.

Maximum Average Power Transfer Theorem for AC Circuits

The maximum average power transfer theorem says that maximum power is transferred to a load when the load impedance equals the complex conjugateof the source’s Thevenin impedance:

Also, , where is the real part of .

Different Ways to Give AC Values

We’ve seen two ways to specify the size of an AC current or voltage: Peak-to-peak value Peak (or maximum) value

A third way that is often used is called the effective value (or rms value).

These distinctions apply only to AC, not to DC.

Can We Compare AC and DC?

AC currents and DC currents are very different, but we can still draw some comparisons between them.

For example: if an AC current flows through a resistor and a DC current flows through a resistor of the same size, each current will deliver power to its resistor.

The Idea Behind Effective Values

For a given AC current, can we say what size DC current would deliver the same power to a resistor as the average power delivered by our AC current?

Yes, we can, and we call the answer the effective value.

Effective Value

So an AC current’s effective value is the DC current that delivers the same power to a resistor as the AC current delivers.

An AC voltage’s effective value is defined in the same way.

Root-mean-square

We’ve defined what we mean by effective value, but how can we compute effective values?

Answer: to compute an AC current’s effective value, take the square root of the mean (average) of its square:

Effective values are also called rms (root-mean-square) values.

Root-mean-square for Sinusoids For a sinusoidal current, , taking the

root-mean-square is equivalent to dividing the current’s amplitude by :

Similarly for sinusoidal voltages. If , then

Outlet Voltage in the USA The voltage at wall outlets in the USA

is 120 V rms. This voltage is also a sinusoid, and it

has a frequency of 60 Hz.

DC Versus AC on Multimeter

Most digital multimeters can measure DC voltage, DC current, AC voltage, AC current.

DC Voltage DC Current

AC Voltage AC Current

DC or AC? Current

Voltage

Fluke 45 Tektronix CDM250

DC or AC?

When a multimeter is set to measure DC voltage or current, it actually displays the average value of the voltage or current.

When a multimeter is set to measure AC voltage or current, it actually displays the rms (or effective) value of the voltage or current.