CHAPTER 10 Power in AC Circuits HAPTER OUTLINEecampus.matc.edu/lokkenr/electronics/662_112... · Need a method to express both electrical energy conversion and electrical energy storage

Contemporary Electric Circuits, 2nd ed., ©Prentice-Hall, 2008 Class Notes Ch. 10 Page 1 Strangeway, Petersen, Gassert, and Lokken

CHAPTER 10 Power in AC Circuits

CHAPTER OUTLINE

10.1 Complex Power in Circuits with AC Signals

10.2 How to Calculate Complex Power

10.3 Complex Power Calculations in Series–Parallel Circuits

10.4 Power Factor and pf Correction

10.1 COMPLEX POWER IN CIRCUITS WITH AC SIGNALS What concept is illustrated in the plots in Figure 4.12?

Explain the concept in the following equation and relate it to the plots:

ave RMS RMS eff effV2

p pV IP I V I (10.1)

Note: The RMS and eff subscripts have identical meaning (RMS will be

used in this chapter, as it was in Ch. 4).

What is true power? Real power? Figure 4.12 (reproduced)

The previous power development is valid for what type of component in a circuit? Why?


What is the phase relationship between the voltage and the current for inductors? For capacitors? Why for each?

The graphical multiplication of v(t) and i(t) when they are ±90 out of phase is shown in Figure 10.1.

Which is leading, the voltage or the current leading?________________

What is the average power in this case? Why?

Is this result the same as the p(t) result for the resistive load?_______

Why or why not?

What is the significance of positive and negative power?

First, explain the passive sign convention:

Figure 10.1

What is positive power for inductors (Fig. 10.2a)?

What is positive power for capacitors (Fig. 10.2a)? Figure 10.2


What is negative power for inductors (Fig. 10.2b)?

What is negative power for capacitors (Fig. 10.2b)?

Thus, what is the net average power for ideal inductors and capacitors? _________________________________

How is energy storage in inductors and capacitors for AC signals quantified?

Need a power-like quantity that corresponds to energy storage for inductors and capacitors

Need a method to express both electrical energy conversion and electrical energy storage with AC signals

Identify the following equations: 2 21 12 2C LE CV E LI (10.2)

Note: calculus is needed to derive the stored energy expressions for AC signals. Alternatively, an explanation:

Calculus average value of energy stored in a capacitor or an inductor expressed with RMS values:

(ave) (ave)

2 21 12 2RMS RMSC LE CV E LI (10.3)

Notation: The capital letter E shall be used to indicate DC energy or average energy in the AC case.

Explain how the following equations were obtained:

2 RMS RMS1 1 12 2 2RMS RMS RMS RMS RMS

1

2C C

V IE CV CV I X CV I

C

(10.4)

2 RMS RMS RMS RMS1 1 12 2 2RMS RMS RMS

2L

L

V V V IE LI L I L I

X L

(10.5)

What is striking between these two general results? _________________________________________________

What is the average energy stored in an inductor or a capacitor directly proportional to? ____________________

What is the average energy stored in an inductor or a capacitor inversely proportional to? ____________________


Does (VRMSIRMS)/(2) in this energy expression contain a power-like quantity? If so, what is it? If not, why not?

What does the power-like quantity represent for inductors and capacitors?________________________________

How can this power-like quantity be utilized for inductors and capacitors? Recall impedance: Z R jX

What is the phase relationship between the voltage and the current for the real part of Z ? Why?

What is the phase relationship between the voltage and the current for the imaginary part of Z ? Why?

What is the phase relationship between the AC voltage and current when the power is real? __________________

What happens to electrical energy for real power? __________________________________________________

What is the phase relationship between AC voltage and current that represents stored energy? ________________

By analogy to impedance, what type of number is the power that represents energy storage? __________________

Based on the previous discussion, explain the following expression for complex power :

S P jQ (10.6)

where P = ___________________________________ power in watts (W),

Q = ___________________________________ power in volt-amperes reactive (VARs), and

S = ___________________________________ power in volt-amperes (VA).

Note: do not confuse reactive power Q with electric charge Q — know from context which one is appropriate

Now relate VRMSIRMS to complex power:

RMS RMS RMS RMS(in-phase part of ) 0 (90 out-of -phase part of ) 90S V I V I

RMS RMS RMS RMS(in-phase part of ) (90 out-of -phase part of )V I j V I (10.7)

How does one separate the VRMSIRMS product into real and imaginary parts? Start with:

RMS RMS RMS RMS RMS RMS( )( ) ( ) ???V I V IS V I V I V I

What is wrong with this equation? _______________________________________________________________


Has the total phase angle (V + I) ever appeared up to now? __________________________________________

What is the phase shift between voltage and current that is physically significant for impedance? ______________

Hence, the phase angle should be the __________________________________ not the sum:

RMS RMS ( )V IS V I (10.8)

How is the negative (opposite sign) of an angle obtained with complex numbers? Explain Figure 10.3.

* *

RMS RMS RMS( )I II I I (10.9)

Figure 10.3

Thus, the complex conjugate is__________________________________________________________________

Explain each step that follows:

* *

RMS RMS RMS RMS RMS RMS( )( ) ( )V I V IS V I V I V I (10.10)

RMS RMS RMS RMS( )V IS V I V I (10.11)

What is ? ___________________________________________________________________________________

Refer to Figure 10.4:

RMS RMS RMS RMS cos sin S V I jV I P jQ (10.12)

RMS RMSRe ( ) cos P S V I (10.13)

Figure 10.4

where Re means ______________________________________________________________________________

RMS RMSIm ( ) sin Q S V I (10.14)

where Im means ______________________________________________________________________________


RMS RMS RMS RMS

RMS RMS RMS RMS

cos sin

( cos ) 0 ( sin ) 90

S V I jV I

V I V I

P jQ

(10.15)

S P jQ S (10.16)

where: S = ___________________________________________ in units of__________________________

P = ___________________________________________ in units of__________________________

= ___________________________________________ of complex power in rectangular form,

Q = ___________________________________________ in units of__________________________

= ___________________________________________ of complex power in rectangular form,

S = ___________________________________________ in units of__________________________

= ___________________________________________ of complex power in polar form, and

= ___________________________________________ in units of__________________________

= ___________________________________________ of complex power in polar form.

Note: The unit watt is reserved for power that represents energy conversion.

The key complex power expression is * ( )V IS VI VI S P jQ (10.17)

Are the phasor voltage and current effective (RMS) or peak values? ______________ Explain.

Why is the VI product called apparent power for AC signals?

What is = V – I ? ____________________________________________________________________ (10.18)

What is Q = QL – QC ? __________________________________________________________________ (10.19)

Why is the sign for QL positive?

Why is the sign for QC negative?


10.2 HOW TO CALCULATE COMPLEX POWER There are two ways to calculate the total complex power provided by a source to a circuit:

a.

Determine the total phasor voltage and the total phasor current supplied by the source to the circuit (single

source circuits only)

Then use Equation (10.17): * ( )V IS VI VI S P jQ

or

b.

Determine the real or reactive power of each component in the circuit

Add all the real powers to obtain the total power PTOTAL

Add all the reactive powers to obtain the total reactive power QTOTAL (positive Q for inductors, negative Q

for capacitors)

Then form the total complex power: S P jQ S

(Multiple source circuits will be covered in the next section).

Example 10.2.1 (Explain each step.)

Determine the total complex power provided by the source to the circuit shown in Figure 10.5.

__________________: RMS10 0 VSV

R = 10

XC = 15

_____________________: S

Strategy:

Figure 10.5

Solution:

* *

10 15

10 00.55470 56.310 A

10 15

(10 0 )(0.55470 56.310 ) (10 0 )(0.55470 56.310 )

5.5470 56.310 3.0769 4.6154

5.55 56.3 VA (3.08 4.62) VA 3.08 W 4.62 V

T C

S

T

Z R jX j

VI

jZ

S VI

S j

S j j

AR


Why are both real power and reactive power present?

Which complex power calculation method was demonstrated in this example?

Explain each step as it relates to determining complex power using the second method:

Note: All phasors are assumed to have RMS magnitudes throughout this discussion.

For a resistance, * ( ) 0 0 0R R R R R V I R R R R RS V I V I V I S P j P (10.20)

2

2 0R

R R R R R

VP V I I R Q

R (resistances only) (10.21)

Which of the numbers is complex in the previous equation?____________________________________________

Explain:

For a capacitor, * ( ) 90 90 0C C C C C V I C C C C CS V I V I V I S jQ jQ (10.22)

2

20 C

C C C C C C

C

VP Q V I I X

X (capacitors only) (10.23)

For an inductance, * ( ) 90 90 0L L L L L V I L L L L LS V I V I V I S jQ jQ (10.24)


2

2

0 L

L L L L L L

L

VP Q V I I X

X (inductors only) (10.25)

Explain how complex power is determined using the next equation:

* [ ]L CS VI P j Q Q (10.26)

where the summation sign is designated by the uppercase Greek letter sigma ().

P is _______________________________________________________________________________________

QL is ______________________________________________________________________________________

QC is ______________________________________________________________________________________

Example 10.2.2 (Fill in the steps.)

Determine the complex power in the circuit shown in Figure 10.5 repeated below.

Given:

Desired:

Strategy: Figure 10.5

Solution:

Answer: 3.08 W 4.62 VAR (3.08 4.62) VA 5.55 56.3 VACS P jQ j j


The complex power can be visualized using the power triangle.

A plot in the complex number plane

Defined by three quantities: - the origin

- the real power on the real axis

- the reactive power on the imaginary axis

Example 10.2.3 Sketch and label the power triangle for the circuit in Figure 10.5.

Given:

3.08 W 4.62 VAR

5.55 56.3 VA

S j

P = 3.08 W

QC = 4.62 VAR

S = 5.55 VA

= –56.3

Desired: power triangle

Strategy:

Plot P and QC in approximate proportion

Sketch the triangle

Label P, Q, S,

Solution:

Explain why the power triangle ―flips‖ between the inductive and capacitive cases (Figure 10.7).

Figure 10.7


Determine the equivalent aspect of the following two equations:

* ( )V IS VI VI S (10.27)

( )V

V I

I

VV VZ Z

I II

(10.28)

For example, check out the results in Examples 10.2.1 and 10.2.2:

3.08 W 4.62 VAR 5.55 56.3 VA

10 15 18.0 56.3

T

T C

S j S

Z R jX j Z

This fact is a useful check in complex power calculations.

Form a summary statement for each relation that follows:

* ( )V IS VI VI S P jQ (10.29)

2

2 0R

R R R R R

VP V I I R Q

R (10.30)

2

20 C

C C C C C C

C

VP Q V I I X

X (10.31)

2

20 L

L L L L L L

L

VP Q V I I X

X (10.32)

* [ ]L CS VI P j Q Q (10.33)


10.3 COMPLEX POWER CALCULATIONS IN SERIES–PARALLEL CIRCUITS


Determine the complex power provided by the source to the

circuit shown in Figure 10.8 by (a) determining and sum-

ming the individual powers of the components, and (b) *.TS VI

Given: circuit in Figure 10.8

Desired: TS

Strategy: a. , ,T xZ I V using series–parallel analysis

QC = I2XC 2

2

( )

L x L

x

T L C

Q V X

P V R

S P j Q Q

Figure 10.8

b. *

T SS V I

Solution: a.

RMS

( )( ) (213)( 132)112.20 58.213

213 132

65 112.20 58.213 66.451 27.198

181.7 02.7343 27.198 A

66.451 27.198

(2.7343 27.198 )(112.20

R L

x

R L

T C x

S

T

x x

Z Z jZ

jZ Z

Z Z Z j

VI

Z

V IZ

RMS

2 2

2 2

2 2

58.213 ) 306.79 31.015 V

(2.7343) (65) 485.97 VAR

306.79713.03 VAR

132

306.79441.88 W

213

( ) 441.88 (713.03 485.97)

441.88 W 227.06 VAR 442 W 227 VAR

C C

x

L

L

x

T L C

Q I X

VQ

X

VP

R

S P j Q Q j

j j

b. * *(181.7 0 )(2.7343 27.198 )

(181.7 0 )(2.7343 27.198 )

496.83 27.198 VA 441.90 227.08 442 W 227 VAR

TS VI

j j

Is the circuit inductive or capacitive from an impedance viewpoint? Why?

Is the circuit inductive or capacitive from a complex power viewpoint? Why?


Multiple source circuits: Consider a circuit with more than one source.

Can the powers in each component due to each source be summed? ________ Why or why not?

Can the voltages (or currents) for each component be summed? _________ If so, under what condition?

The complex power for each component is determined from that total phasor voltage (or current) for that

component.

The total complex power of the circuit is determined by summing the complex powers of the individual

components.

Example 10.3.2 Determine the total complex power provided by the source to the circuit in Example 9.4.1. The circuit schematic is

repeated below (Figure 10.9).

Figure 10.9

Given: The circuit from Figure 9.18 is repeated in Figure 10.9 with the current in each branch labeled.

Desired: TS

Strategy:

Solution: (Perform all steps on separate paper.) Sub-answers and answer to check as you proceed:

For the 10+30 source: 1.0398 21.027 A, 1.1028 23.973 A, 0.8220 87.413 Ab a cI I I

For the 150 source: 2.8386 65.082 A, 1.6076 66.991 A, 1.2331 117.407 Ac a bI I I

Superposition: 1.9342 32.233 A, 0.8265 60.818 A, 2.1434 54.880 Aa b cI I I

Answer: S = (3.4155 + 14.9645) + j(7.4823 + 2.0493 – 27.5650) = 18.3800 – j18.0334 = 18.4 W – j18.0 VAR


10.4 POWER FACTOR AND PF CORRECTION What is the power factor angle ? _______________________________________________________________

Definition of power factor (pf): How does the P/S

ratio relate to in the power triangle (see Figure 10.7,

repeated to the right)?

Figure 10.7

Consider Table 10.1 for the physical significance of pf. Explain the trend in pf versus power factor angle.

TABLE 10.1

power factor

angle

power factor

+90 0

+60 0.5000

+45 0.7071

+30 0.8660

+0 1.0000

–30 0.8660

–45 0.7071

–60 0.5000

–90 0

When the power factor angle is ±90, what is the pf ? _____ Is the complex power real, reactive, or a mixture? Why?

When the power factor angle is 0, what is the pf ? _____ Is the complex power real, reactive, or a mixture? Why?

When the power factor is in the ―middle region‖ between 1 and 0, is the complex power primarily real, reactive, or a

mixture? Why?


Given the power factor, can one tell whether the power factor angle is positive or negative? Why or why not?

Terminology applied to pf: leading or lagging. It is applied to the current relative to the voltage:

If the power factor is leading, then the current leads the voltage. Is the circuit capacitive or inductive? Why?

If the power factor is lagging, then the current lags the voltage. Is the circuit capacitive or inductive? Why?

Major application of power factor: power factor correction

Example: Treat the circuit in Examples 10.2.1 and 10.2.2 as a ―block‖ with complex power (Figure 10.12):

Figure 10.12

Is the Q zero? Why or why not? _______________________________________________________________

Explain the following calculation:

3.0769 3.0769

pf 0.55470 0.555 leading| 3.0769 4.6154 | 5.5470

P

S j

Is real, reactive, or both powers significantly present in the circuit? Explain.


What was added to the circuit in Fig. 10.12(b) as shown in Fig. 10.13? __________________________________

Figure 10.13

Explain the mathematical statements that follow:

3.0769 ( 4.6154 4.6154) (3.0769 0) VA 3.08 WS P j Q j j

What is the total complex power? _______________________________________________________________

What is the total real power? ___________________________________________________________________

What is the total reactive power? ________________________________________________________________

3.0769 3.0769

pf 1.00| 3.0769 0 | 3.0769

P

S j

* *

* 3.0769 0 0.30769 0 A 0.308 0 A

10 0

S jS VI I

V

Note: The original current in the circuit without pf correction (from Ex. 10.2.1 or 10.2.2) is 0.555 56.3 A .

What are the primary effects of power factor correction with regards to each of the following aspects?

(1) The power factor__________________________________________________________________________

(2) The phase between the total circuit voltage and current ____________________________________________

(3) The circuit current _________________________________________________________________________

What is the impact of (3) on wire sizes, power dissipation in wires, and costs?

How should the component for pf correction be connected into the circuit (series or parallel)? Why?


What type of component is added in parallel with the load in this example? _________________________ Why?

What is the value of the component to add in parallel in Figure 10.13? Identify what is known and what is unknown

in the following equations:

2

S

L

L

VQ

X (10.35)

LX L (10.36)

Assume frequency is known. In this example, it is 60 Hz. Explain the following calculations:

2 210

21.667 4.6154

21.66757.5 mH

2 60

S

L

L

L

VX

Q

XL


Determine (a) the reactance for power factor correction in the

circuit shown in Figure 10.8 (repeated to the right).

(b) Determine the component value if the frequency is 60 Hz.

(c) Determine the pf both before and after pf correction.

__________: 441.90 W 227.08TS j VAR (from

Example 10.3.1)

RMS181.7 0 VSV

f = 60 Hz

_________: a. X for pf correction Figure 10.8

b. value of L or C for pf correction

c. pf before pf correction

pf after pf correction

________: pf correction |Im ( ) |TQ S (inductive or capacitive, as appropriate)

2

1

pf

S

C

C

C

VX

Q

CX

P

S


Solution: (Explain each step.)

pf correction |Im ( ) | 227.08TQ S VAR (capacitive)

2 2

5

before

after

181.7145.39 145

227.08

1 11.8245 10 18.2 F

2 60(145.39)

441.90 441.90pf 0.88944 0.889

| 441.90 227.08 | 496.83

pf 1.000

S

C

C

C

VX

Q

CX

P

S j

What is the difference and the significance of

Figure 10.14 with respect to Figure 10.8?

Figure 10.14

Thus, what is power factor correction?

What is the general approach to power factor correction?


Example 10.4.2 (Explain or fill in each step, as appropriate.)

Determine the reactance for power factor correction in the circuit shown in Figure 10.15.

Given: RMS120.0 0 VSV

load 1:

load 2: Figure 10.15

Desired:

_________: Determine for each load from pf: = cos

–1 (pf).

Determine S for each load from V, I, , and leading/lagging status

1 2

pf correction

2

| Im ( )| (inductive or capacitive, as appropriate)

T

T

S

S VI

S S S

Q S

VX

Q

Solution: (Perform the calculations per the strategy. Check against the answers provided on the next page.)


1

2

1

2

45.00 (positive due to lagging pf)

60.00 (negative due to leading pf)

24.00 45.00 kVA

12.00 60.00 kVA

S

S

(22.97 6.578) kVATS j

pf correction 6.578 kVAR (capacitive)

2.19 C

Q

X

Learning Objectives

Discussion: Can you perform each learning objective for this chapter? (Examine each one.)

As a result of successfully completing this chapter, you should be able to:

1. Describe why complex power is needed to express power in AC circuits.

2. Describe complex power, apparent power, real power, reactive power, power factor angle, and power factor and

the differences between them.

3. Calculate complex power, apparent power, real power, reactive power, power factor angle, and power factor for

components, groups of components, and entire circuits using two approaches:

a. complex power equation in terms of phasor voltage and phasor current, and

b. summing real or reactive powers of individual components.

4. Describe what power factor correction is and why it is important.

5. Determine the parallel reactance and component value required for power factor correction.

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CHAPTER 10 Power in AC Circuits HAPTER OUTLINEecampus.matc.edu/lokkenr/electronics/662_112... · Need a method to express both electrical energy conversion and electrical energy storage