thÉvenin and norton equivalent circuits -...

Contemporary Electric Circuits, 2nd ed., ©Prentice-Hall, 2008 Class Notes Ch. 12 Strangeway, Petersen, Gassert, and Lokken

CHAPTER 12

THÉVENIN AND NORTON EQUIVALENT CIRCUITS

Chapter Outline

12.1 What Is an Equivalent Circuit? Why Are Equivalent Circuits Needed?

12.2 Thévenin Equivalent Circuits

12.2.1 DC Signals

12.2.2 AC Signals

12.3 Norton Equivalent Circuits

12.3.1 DC Signals

12.3.2 AC Signals

12.4 Models of Sources and Maximum Power Transfer

12.4.1 DC Signals

12.4.2 AC Signals

12.1 WHAT IS AN EQUIVALENT CIRCUIT? WHY ARE EQUIVALENT

CIRCUITS NEEDED? Consider the circuit in Figure 12.1.

Figure 12.1

What does the arrow through RL indicate? _________________________________________________________

How can the voltage across and the current through RL be determined as the resistance of RL is varied?

Is the repetition of this analysis as the RL value is changed efficient use of your time? ___________


Alternative to this labor-intensive approach: determine an equivalent circuit.

Which results are the same at the load for both the equivalent circuit and the original circuit? (see Figure 12.2).

Then why is an equivalent circuit utilized? _________________________________________________________

Thévenin equivalent circuit: a voltage source in series with a resistance (Figure 12.2a).

Figure 12.2

What is an equivalent circuit?

12.2 THÉVENIN EQUIVALENT CIRCUITS

12.2.1 DC Signals

The procedure to determine the Thévenin equivalent circuit is general.

Does the general procedure change based on the circuit under consideration? ___________

Do the specific steps to execute the procedure change based on the circuit under consideration? __________

What are the two unknown quantities to determine in a Thévenin equivalent circuit (Figure 12.2a)? ___________

Thus, what are the two sub-strategies in the strategy to determine the Thévenin equivalent circuit? (see Figure 12.3)

Figure 12.3


Summary of the procedure to determine the Thévenin equivalent circuit:

Determine the load in the original circuit. Label the terminals at each end of the load.

Remove the load from the original circuit.

Label the polarity of the open circuit (Thévenin) voltage between the terminals.

Determine the open-circuit voltage between the terminals. This is VTh.

Deactivate all sources in the original circuit (with RL still removed).

Determine the total equivalent resistance between the terminals. This is RTh.

Build the Thévenin equivalent circuit and reattach the load per Figure 12.3.

Analyze the simplified circuit for the load voltage and the load current.

Why is the load removed from the original circuit? __________________________________________________

Why is the voltage between the terminals called the open circuit voltage?

How are sources deactivated: Voltage sources are replaced with _______________ to guarantee _________ volts.

Current sources are replaced with ________________to guarantee __________ amperes.

Example 12.2.1 (Explain each step).

a. Determine the Thévenin equivalent

circuit for the circuit shown in Figure

12.4 if the load is the 33 k resistance.

b. Determine the voltage across the load.

Given: Original circuit in Figure 12.4

The load is the _________ k

resistance

Figure 12.4

__________: Thévenin equivalent circuit

The voltage across the load

__________: Determine the Thévenin equivalent circuit.

Determine the load voltage using the Thévenin equivalent circuit.


Solution: What are the first steps in the procedure for determining the Thévenin equivalent circuit (see Fig. 12.5)?

Figure 12.5

Why does VTh = VR2? _________________________________________________________________________

2

2

Th

1 2

Th

( ) 30(45 k )20.769 V

20 k 45 k

20.769 V

S

R

V RV V

R R

V

What step is illustrated in Figure 12.6? ___________________________________________________________

Figure 12.6

Th 3 1 2 4

(20 k )(45 k )( ) 10 k 65 k 88.846 k 88.8 k

20 k 45 kR R R R R

What step is illustrated in Figure 12.7 and the calculation to the right of the figure?

Th

Th

( ) 20.769(33 k )5.625 V 5.63 V

88.846 k 33 k

L

t

L

V RV

R R

Figure 12.7


Example 12.2.2 (Fill in the steps.)

a. Determine the Thévenin equivalent circuit for the circuit shown in Fig. 12.1 (repeated below) if the load is RL.

b. Determine the voltage across the load if RL = 80 .

Figure 12.1

Given:

Desired:

Strategy:

What part of the procedure is illustrated in Figure 12.8?

Figure 12.8

Sub-strategy for the circuit in Figure 12.8: _______________________________________________


Figure 12.9 Figure 12.10

Solution:

Sub-strategy for the circuit in Figure 12.9:

Execute the sub-strategy: (Intermediate answers: Th15.556 , 14.000 V, 10.000 Vx xR V V )


Execute the sub-strategy: (Intermediate answers: Th25.4545 , 33.600 V, 24.000 Vy yR V V )


Thus, VTh = ____________________________________________________________________________ V

What is the step in the strategy illustrated in Figure 12.11? _________________________________________

Figure 12.11


Execute the sub-strategy and build the Thévenin equivalent circuit (results are shown in Figure 12.12):

Figure 12.12

Sub-strategy for the terminal voltage Vt: __________________________________________________________

Execute the sub-strategy (result 27.2 VtV ):


12.2.2 AC Signals

Is the procedure for finding the Thévenin equivalent circuit with AC signals similar to that for DC? _________

Are there any differences? Explain.

What is the limitation with respect to frequency when the Thévenin equivalent circuit is determined? Why?

Example 12.2.3 (Fill in steps.)

a. Determine the Thévenin equivalent circuit as “seen” by RL for the circuit shown in Figure 12.13.

b. Determine the value of the capacitance or the inductance in the Thévenin impedance if the source frequency is

1 MHz.

Figure 12.13

Given:

Desired:

Strategy:

Solution: (use separate paper to solve this problem; the answers appear on the next page)


a.

Figure 12.16

b. 4.77435 33.335 nFCX C

The Thévenin equivalent circuit and value for C are valid only at ___________________________ .

12.3 NORTON EQUIVALENT CIRCUITS

12.3.1 DC Signals

The Norton equivalent circuit is another equivalent circuit form. Explain the main ideas in Figures 12.17 and 12.18.

Figure 12.17

Figure 12.18


Summary of the procedure to determine the Norton equivalent circuit:

Determine the load in the original circuit. Label the terminals at each end of the load.

Remove the load from the original circuit. Place a short between the terminals.

Label the direction of the short-circuit (Norton) current between the terminals.

Determine the short-circuit current (including direction) between the terminals. This is IN.

Deactivate all sources in the original circuit (with RL still removed and the short removed).

Determine the total equivalent resistance between the terminals. This is RN.

Build the Norton equivalent circuit and reattach the load per Figure 12.17.

Analyze the simplified circuit for the load voltage and the current.

Example 12.3.1 (Explain each step).

a. Determine the Norton equivalent circuit

for the circuit shown in Figure 12.4

(repeated to the right) if the load is the 33

k resistance (note: same circuit that was

analyzed in Example 12.2.1).

b. Determine the voltage across the load.

Given: Original circuit in Figure 12.4

The load is the 33 k resistance.

Figure 12.4

__________: Norton equivalent circuit

The voltage across the ________ k load

__________: Determine the Norton equivalent circuit.

Determine the load voltage using the Norton equivalent circuit.

Solution: What are the first steps in the procedure for determining the Norton equivalent circuit (see Fig. 12.19)?

Figure 12.19


a. _________________________: Let 2 3 4( ),CR R R R voltage-divider rule VC, Ohm’s law ISC

2 3 4

2 3 4

1

SC

3 4

N

( ) ( ) (45 k )(10 k + 65 k )28.125 k

( ) 45 k (10 k + 65 k )

( ) 30(28.125 k )17.533 V

20 k 28.125 k

17.5330.23377 mA

75 k

0.23377 mA

C

S C

C

C

C

R R RR

R R R

V RV

R R

VI

R R

I

RN is found exactly as RTh was in Example 12.2.1; repeat it for practice (answer: RN = 88.846 k

b. What steps are illustrated in Figure 12.20 and the calculation that follows?

Figure 12.20

N N

(88.846 k )(33 k )( ) (0.23377 mA) 5.6250 V 5.63 V

88.846 k 33 kt LV I R R



a. Determine the Norton equivalent circuit for the circuit shown in Figure 12.1 (repeated below) if the load is RL

(note: same circuit that was analyzed in Example 12.2.2).

b. Determine the voltage across the load if RL = 80 .

Figure 12.1

Given:

Desired:

Strategy:

What parts of the procedure are illustrated in Figure 12.21?

Figure 12.21


Solution:

a. Sub-strategy: ___________________________________________________________________

Steps to execute sub-strategy in Figure 12.22:

Figure 12.22

Execute the sub-strategy: (intermediate answers: SC13.333 , 24.000 V, 1.200 Ax xR V I )

Steps to execute sub-strategy in Figure 12.23:

Figure 12.23


Execute the sub-strategy: (intermediate answers: 10yR , 10 VyV , SC 0.500 AI )

Thus, IN = ________________________________________________________________________________ A

Norton resistance: found exactly the same way as the Thévenin resistance (answer RN = 20.000

Repeat it for practice and build the Norton equivalent circuit (Figure 12.24).

Figure 12.24

b. Sub-strategy for the terminal voltage Vt: ____________________________________________________

Execute the sub-strategy (result 27.2 VtV ):


If the Thévenin and Norton equivalent circuits are both equivalent to the same original circuit, then what is the

relationship between the Thévenin and Norton equivalent circuits?

How does one convert back and forth between Thévenin and Norton equivalent circuits? (see Figure 12.25)

Figure 12.25

N Th I R (12.1)

What are these equation recognized to be? __________________________________________________________

Example 12.3.3

Perform a source conversion of the results from Example 12.2.2 and check against the results of Example 12.3.2.

Given: Thévenin equivalent circuit in Figure 12.12

(repeated to the right)

Desired: Norton equivalent circuit

Strategy:

Solution: (answers: RTh = RN = 20.000 N 1.7000 AI ) Figure 12.12


12.3.2 AC Signals

Is the procedure for finding the Norton equivalent circuit with AC signals similar to that for DC? _________

Are there any differences? Explain.

What is the limitation with respect to frequency when the Norton equivalent circuit is determined? Why?


a. Determine the Norton equivalent circuit as “seen” by RL for the circuit shown in Figure 12.13 (repeated below).

b. Check the results against those of Example 12.2.3 using a source conversion.

Figure 12.13

Given:

Desired:

Strategy:

Solution: (use separate paper to solve this problem; answers:

N 0.62823 20.855 AI , N Th 11.408 4.77435Z Z j , Th 7.7692 1.855 VV )


12.4 MODELS OF SOURCES AND MAXIMUM POWER TRANSFER

12.4.1 DC Signals

Compare the model of a DC voltage source (Figure 3.39) and the Thévenin equivalent circuit (Figure 12.3).


Hence, the model of a DC voltage source is really the _______________________________________________ .

Compare the model of a DC current source (Figure 3.40b) and the Norton equivalent circuit (Figure 12.17).


Hence, the model of a DC current source is really the _______________________________________________ .

Why is efficient power transfer from the source to the load important in some circuits?

What is the appropriate question to ask concerning maximum power transfer from the source to the load?


Consider the Thévenin equivalent circuit example shown in Figure 12.28 with a variable load resistance. If the

power in the load is calculated and plotted as RL is varied, the graph in Figure 12.29 results (check at RL = 535 Ω).


What is the value of the load resistance that results in the maximum amount of power in the load? _____________

In general, what is the value of the load resistance that results in the maximum amount of power to the load in terms

of the Thévenin equivalent circuit?

(12.2)

Develop the amount of power that transfers from the source to the load (result: 2

Th

MAX4 L

VP

R ) (12.3)

Repeat the analysis for the Norton equivalent circuit (results: RL = RN , 2

N

MAX4

LI RP ) (12.4), (12.5)

Thus, the condition on the load resistance for maximum power transfer from the source to the load in a DC circuit

is

LR (12.6)



For the circuit in Example 12.2.1, determine (a) the load

resistance for maximum power transfer, and (b) the power

dissipated in the load.

Given: Thévenin equivalent circuit that was determined in

Example 12.2.1 (Figure 12.7, repeated on the right)

Desired:

Strategy: Figure 12.7

Solution: (answers: RL = 88.846 kMAX 1.21 mWP )

12.4.2 AC Signals

What may the load of an AC circuit have that is of no importance in DC circuits? ___________________________

Two types of loads will be examined. In the first type of load, the load impedance must be the complex conjugate

of the Thévenin (or Norton) equivalent circuit impedance to achieve maximum power transfer:

* *

Th NLZ Z Z (12.7)

What does the asterisk mean in the previous equation? ________________________________________________

Determine and simplify the total impedance of the Thévenin equivalent circuit with the load connected?

(12.8), (12.9)



For the circuit in Example 12.2.3, determine (a) the load impedance for maximum power transfer, and (b) the power

dissipated in the load.

Given: Thévenin equivalent circuit that was determined in Example 12.2.3 (Figure 12.16, repeated below)

Figure 12.16 Desired:

Strategy:

Solution: (answers: MAX11.408 4.77435 , 1.3228 W 1.32 WLZ j P )

Second type of load for maximum power transfer in AC circuits: purely resistive only (no reactance). Why could

this type of load be important?

The condition on the load impedance for maximum power transfer from the source to the load when the load is a

pure resistance is:

Th N| | | |LR Z Z (12.10)

Interpret the previous equation in words:



For the circuit in Example 12.2.3, determine (a) the load impedance for maximum power transfer if the load is a pure

resistance, and (b) the power dissipated in the load.

Given: Thévenin equivalent circuit that was determined in Example 12.2.3 (Figure 12.16, repeated below)

Figure 12.16 Desired:

Strategy:

Solution: (answers: 12.367 12.4LR , MAX 1.2694 W 1.27 WP )

Was the maximum power in the load impedance that was a pure resistance equal to the maximum power in the load

impedance that was the complex conjugate of the Thévenin impedance? Why or why not?

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. Explain what an equivalent circuit is and why equivalent circuits are needed.

2. Determine the Thévenin equivalent circuit of a given circuit.

3. Determine the Norton equivalent circuit of a given circuit.

4. Convert between Thévenin and Norton equivalent circuits.

5. Solve for the voltages and currents for a given load using the equivalent circuit.

6. Determine the load resistance or impedance required for maximum power transfer.

thÉvenin and norton equivalent circuits -...

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