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Chapter 10 – Sinusoidal Steady-State Power Calculations
Study Guide
Objectives:
1. Understand the following ac power concepts, their relationships to one another, and how
to calculate them in a circuit:
 Instantaneous power;
 Average (real) power;
 Reactive power;
 Complex power; and
 Power factor.
2. Understand the condition for maximum real power delivered to a load in an ac circuit and
be able to calculate the load impedance required to deliver maximum real power to the
load.
3. Be able to calculate all forms of ac power in ac circuits with linear transformers and in ac
circuits with ideal transformers.
Mastering the Objectives:
1. Read the Introduction and Section 10.1.
a) Define instantaneous power.
b) What is the relationship between the frequency of the voltage and current signals
in Fig. 10.1 and the frequency of the instantaneous power?
c) Look at the plots in Fig. 10.2. The voltage and current plots are symmetric about
the time axis but the instantaneous power plot is not – why? (Hint – look at Eq.
10.8).
2. Read Section 10.2
a) Define average power. What are the units for average power? Is the average
power in a resistor positive or negative – why? What is the average power in a
capacitor or an inductor – why?
b) Define reactive power. What are the units for reactive power? What is the
reactive power in a resistor – why? Is the reactive power in an inductor positive
or negative – why? Is the reactive power in a capacitor positive or negative –
why?
c) Define power factor. What is the power factor angle? Rewrite the equations for
average power (Eq. 10.10) and reactive power (Eq. 10.11) in terms of the power
factor. What is the power factor of
i.
A purely resistive load?
ii. A purely inductive load?
iii. A purely capacitive load?
iv.
An RL load?
v.
An RC load?
d) Solve Assessment Problem 10.1
3. Read Section 10.3.
a) Rewrite the equations for average power (Eq. 10.10) and reactive power (Eq.
10.11) in terms of rms voltage, rms current, and power factor. What
simplification is achieved in these equations when using the rms values? We will
generally use rms values to take advantage of this simplification.
b) Start a table that will contain all of the formulas in this chapter. You should
consult this table as you solve power problems to help you choose the formula
that best suits the problem at hand. Add the formulas for instantaneous power,
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average power, reactive power, and power factor that you have encountered so
far.
c) Solve Assessment Problem 10.3.
4. Read Section 10.4.
a) Given the formula for complex power (Eq. 10.23), why is average power also
called real power? What units do we use for complex power to distinguish it from
average power and reactive power?
b) The power triangle is an important tool for solving power problems. Using right
triangle geometry, find the following relationships:
i.
Express |S| in terms of P and Q.
ii. Express  in terms of P and Q.
iii. Express Q in terms of |S| and .
iv.
Express P in terms of |S| and .
v.
Express the power factor in terms of P and Q.
c) Answer the following questions about power triangles:
i.
If a load consists only of resistors and inductors, what does the power
triangle look like?
ii. If a load consists only of resistors and capacitors, what does the power
triangle look like?
iii. If a load consists only of resistors, what does the power triangle look like?
d) Solve Chapter Problem 10.14.
5. Read Section 10.5.
a) This section contains lots of alternative formulas for complex power, average
power, and reactive power. Add these formulas to your table as you encounter
them and think about what quantities must be known to use a particular formula.
Note that several of these formulas use phasors, and be sure to distinguish phasor
voltages and currents from voltage and current magnitudes in your table – this can
be a source of confusion when using the formulas. Also note that Eq. 10.36 holds
only for a purely resistive element, and Eq. 10.37 holds only for a purely reactive
element.
b) Read Example 10.6 carefully. It presents a concept known as power-factor
correction. Note that because of the impedance of the line connecting the power
source to the load in Fig. 10.14, real power generated by the source is lost in the
line. One way to reduce this power loss is to reduce the resistive impedance of
the line, but in real applications there will always be some non-zero line
resistance. Another way to reduce the power loss is to reduce the magnitude of
the line current. This is particularly effective, since the real power lost in the line
is proportional to the square of the current magnitude. (See Eq. 10.33). This
example shows that the magnitude of the line current is minimum when the load
is purely resistive and its power factor is 1. To change, or correct, the power
factor, we place a component in parallel with the load that exactly cancels the
reactive power of the load – if the load is inductive, we place a capacitor in
parallel, and if the load is capacitive, we place an inductor in parallel. You can
see this cancellation in the power triangles of Fig. 10.16. Solve Assessment
Problem 10.6 and then calculate the value of a third load to be added in parallel
that would correct the power factor of the combined loads to 1.
c) Read Example 10.7 carefully. It demonstrates a very useful feature of complex
power calculations. Recall that in Chapter 4 we could check the results of our
circuit analysis by calculating the power associated with each component; we sum
the powers and expect the sum to equal zero and the power to balance. Up to this
point we had no analogous method to check the results of our analysis of
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frequency-domain circuits. Complex power gives us a method to compute power
for every component in our frequency-domain circuit, add the complex powers
together, and expect the sum to be zero, so that the complex power balances.
Perform this power balance by solving Assessment Problem 10.4 and Chapter
Problem 10.16.
6. Read Section 10.6.
a) The condition for maximum power transfer to a load is similar to the condition for
minimizing the power lost in the line connecting the source and the load in
Example 10.6. When the condition for maximum power transfer is satisfied for
the circuit in Fig. 10.19, what is the power factor of the impedance seen by the
voltage source? What is then true about the impedance of the impedance seen by
the voltage source?
b) List the three restrictions on impedance discussed on p. 475. Then review
Examples 10.9 – 10.11 that illustrate maximum power transfer for each of these
restricted impedance conditions. For each of the impedance restrictions on your
list, note the condition that must be satisfied for maximum power to be transferred
to the load.
c) Use the value for the load resistance calculated in Example 10.11 for Fig. 10.24
and verify that the current in the load resistor is 3 A by checking that the real
power balances for this circuit.
d) Solve Chapter Problems 10.33, 10.43, and 10.44.
Assessing Your Mastery:
Review the Objectives for this unit. Once you are satisfied that you have achieved these
Objectives, take the Chapter 10 Test.