Network Reduction Techniques and Network Theorems

Network Reduction Techniques and Network TheoremsChapter 4, Sections 9-16, in the text by Nilsson covers several useful networkreduction techniques and network theorems.

The purposes of these techniques and theorems are:• To provide alternate analysis methods• To provide methods for simplifying circuits• To provide methods for representing circuits in the simplest possible form• To gain insight into circuit behavior

Topics to be covered• Source Transformations• Superposition• Thevenin’s and Norton’s Theorems• Maximum Power Transfer Theorem

Reading Assignment: Sections 4.9 - 4.16, in Electric Circuits, 9th Ed. by Nilsson

1Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I

Source TransformationsBefore covering the actual technical of source transformations, somebackground on types of sources is necessary.

There are two broad categories of voltage and current sources:1) Ideal sources2) Real sources (or practical sources)

So far in the course we have only considered ideal voltage and current sources. We will now consider real sources.

Example: • The voltage supplied by an ideal 12V source will maintain 12V for any

current required by the circuit.• The voltage supplied by a real 12V source (such as a car battery) will drop

as the current required increases.


Real Voltage Source

A real voltage source is modeled using an ideal voltage source, VS and a series

resistance, RS.

+ _ VS

RS V +

_

I

Real Voltage Source

Plot the characteristics of a real voltage source.Also show the characteristics of an ideal voltage source.

I

V


Develop an expression for I as a function of V and put it in the form y = mx + b.

Example: A car battery has a voltage of 13V and a current of 0A when nothingis being powered by the battery, but the voltage drops to 9V and the current is300A while starting the car.A) Draw the characteristics for the battery.B) Determine the resistance of the battery, RS.C) Draw a model of the battery.


Example: (continued)D) If two headlight were left on (Sylvania 9006ST Halogen Headlamps, rated

for 55W at 12.8V), use the model to determine the current. Show the point on the graph.

E) If someone was “jump-starting” another car with this battery and accidentally touched the jumper cables together, determine the current through the cables (before they melted)! Show this point on the graph.


Real Current Source

A real current source is modeled using an ideal current source, IP and a parallel

resistance, RP. Develop an expression for I as a function of V and put it in the form y = mx + b.

IP RP V +

_

I

Real Current Source

Plot the characteristics of a real current source.Also show the characteristics of an ideal current source.

I

V


Source Transformations (or source conversion)There are two types of source transformations:• Transform a real voltage source into a real current source• Transform a real current source into a real voltage source

In order for the two types of sources to be equivalent, they should provide the same voltage and current to any load. This can be accomplished by equating their characteristics. In particular, the x-intercepts, y-intercepts, and slopes should be equal (actually equating any 2 of these 3 items fixes the remaining item).

+ _ VS

RS V

+

_

I

Real Voltage Source

Load IP RP V

+

_

Real Current Source

I

Load


Equating the y-intercepts yields: VS/RS = IP

Equating the x-intercepts yields: VS = IPRP

Equating the slopes yields: RS = RP

I

V

IP

IP RP

Slope = 1/RP

I

V

VS /RS

VS

Slope = 1/RS

Real voltage source characteristics Real current source characteristics


Source Transformations - Conversion Formulas:

sp p s

s

Converting a real voltage sourceto a real current source:

VI and R RR

s p p s p

Converting a real current sourceto a real voltage source:V I R and R R

5 A 4 + _ 20 V

4 Convert V- source to I- source

Convert I- source to V- source

I

V +

_

I

V +

_

Simple Example:


Example: Determine VX in the circuit below using source transformations.

+ _ 27 V

3

VX +

_

2 6 5


Example: Determine VX in the circuit below using source transformations.

+ _ 80 V

20

VX + _

10 60 25 +

_ 60 V

100


Example: Determine VX in the circuit below.

+ _ 60 V VX

+

_

10

4 A 40 12


Important notes on source transformations:1) Transformed sources are equivalent in that they provide the same terminal

voltage and terminal current (V and I) to any connected load.2) Transformed sources are not equivalent internally. For example, the current

through or the voltage across RS and RP is not the same. To assume that they are the same is a common error. Example: In the circuit below, V1 V2.

3) All sources are not transformable. Note that a voltage source MUST have a SERIES resistor to be transformable. Note that a current MUST have a PARALLEL to be transformable.

4) Dependent sources can be transformed also.

+ _ VS

RS

Load IP RP V2 +

_

I

Load

+ V1 -


I

Source Transformations - not always possibleThe last page stated that all sources are not transformable so source transformations cannot be used on all circuits.

Example: Draw a circuit that cannot be analyzed using source transformations.


Example: Determine VX in the circuit below. Show a correct and an incorrect approach.

+ _ 24 V

4

VX + _

6 3 A


Example: Determine VX in the circuit below.

+ _ 20 V

4

VX + _

6

+ _

2VX

18 V


SuperpositionThe superposition theorem essentially states that independent sources act separately. In particular, the current or voltage in any part of the circuit can be calculated by determining the current or voltage due to each independent source (with all other independent sources killed) and then by adding the results algebraically.

Independent sources are killed by:• Shorting voltage sources (which is equivalent to setting the voltage to zero)• Opening current sources (which is equivalent to setting the current to zero)

Notes:• Never kill a dependent source• Superposition applies to voltage and current, but not to power


Example: Determine the current IX using superposition.

+ _ 18 V

3 IX 6 2 A


Example: Determine the voltage VX using superposition.

+ _ 10 V 5

VX

+

_ 6

2 A 6

+ _

30 V 12


Example: Determine the voltage VX using superposition.

+ _ 18 V

3

VX + _

6 2 A 2VX


Example: A) Determine the voltage VX using superposition.

+ _ 80 V 18

VX

+

_

18

3 A 36

18


Example: (continued)B) Show that superposition does not apply to power (i.e., show that PT P1 +

P2 for the top right 18 ohm resistor.)


Thevenin’s & Norton’s TheoremsThevenin’s and Norton’s theorems are two related theorems that allow us to represent any circuit by a simple equivalent circuit. The equivalent circuit is easier to understand than the original circuit and gives us insight into circuit behavior. Engineer’s often use equivalent systems to help them understand the original system. In Statics, for example, multiple forces are often replaced by an equivalent resultant force, as illustrated in the example below.

F1

F4

F3

F2

FR

Original System Equivalent System


Thevenin’s & Norton’s TheoremsAny one-port network N may be represented by either of the following types of equivalent circuits:Thevenin Equivalent Circuit (TEC) – consisting of a voltage source and a series impedanceNorton Equivalent Circuit (NEC) – consisting of a current source and a parallel impedance

independent sources,

dependent sources, and resistors

RTH

V

I

+

_ VTH

TEC

V

I

+

_ Load

Network N

V

I

+

_ IN

NEC

+ _ RN Load Load

TH OC

N SC

OCTH N EQ seen by the load with sources killed

SC

V V Thevenin voltage or open-circuit voltageI I Norton current or short-circuit current

VR R Thevenin or Norton resistance RI

where


Illustration of VOC and ISC:

TH OCV V Thevenin voltage or open-circuit voltage

Load Network N

VOC

+

_ Network N

Remove the load and the voltage across the open terminals is VOC

Replace the load by a short circuit (wire) and the current through the short is ISC

Load Network N

ISC

Network N

N SCI I Norton current or short-circuit current


There are 3 ways to find the TEC or NEC for a given circuit:Examples using each of the three methods will be provided on the following pages.

sources.dependent with circuitsfor choicebest heprobably t is and

method generalmost theis This .IV R calculate Also . I and V Find )3

current Terminal voltageTerminal

IV R

:findingby and minalsoutput ter the to)(any value sourcecurrent or voltage externalan addingby found becan R sources,dependent hascircuit theIf • s.R' parallel & series combiningby done beoften can thiscircuit, simple aFor •

R R find Also . Ior V Find 2)

able transformare sources allnot that Recall • useful bemay reduction partial a though sources,dependent with possibleNot •

ations transformsource using NECor TEC a of form theintocircuit theReduce 1)

sc

ocThscoc

T

TTh

Th

killed sourcest independen with load by theSeen eqThscoc


Example: Find the TEC and the NEC seen by R in the circuit shown below. Discuss which method to use.

+ _ 18 V

3

6

2

R


Example: Find the TEC seen by R in the circuit shown below. Discuss which method to use.

+ _ 100 V 20

30

60

60

R


Example: Find VL for R = 10, 20, 40, and 80 ohms in the circuit shown below. Hint: Use the TEC instead of the original circuit.

+ _ 100 V 20

30

60

60

R VL

+

_


Example: Find the TEC seen by R in the circuit shown below. Discuss which method to use.

+ _ 12 V

2

12 2VX R

4

VX + _


Finding Thevenin resistance by adding external sources As indicated in Method 2 for finding a TEC or NEC, RTH can also be found as follows:

TH

thwith independent sources killed andany value source added to the terminals where R is to be measured

terminal voltageR terminal current

Discussion: If the dead circuit really acts like a resistor (RTH), then we can add any voltage source across the circuit (resistor), solve for the current, and use Ohm’s Law to find RTH. Similarly, we can add any value current source and solve for the voltage. This technique is illustrated below.

Dead Circuit(independent

sources killed)

+

_2A = IT

Add any value of current source and solve for VT

Dead Circuit(independent

sources killed) + _ 10V = VT

IT

Tth

T

V terminal voltageIn either case above, calculate R = I terminal current

Add any value of voltage source and solve for IT

VT

Case 1: Add an external voltage source Case 2: Add an external current source


Example: Find the RTH seen by R in the circuit shown below using Method 2 (i.e., add an external source). Compare the results to the value found previously using Method 3.

+ _ 12 V

2

12 2VX R

4

VX + _


Maximum Power Transfer TheoremSuppose that a general network N has a resistive load as shown below.

independent sources,

dependent sources, and resistors

Network N

RL

Now we might consider two questions:• For what value of RL is maximum power delivered to RL?• What is the maximum power that can be delivered to RL?

To answer these questions (see next page),1) Replace N by a Thevenin Equivalent Circuit2) Determine a general expression for power to RL

3) LL

L

dPSolve 0 to find where P is max imumdR


Maximum Power Transfer TheoremShow that:

2th

maxth

VP 4R

L thR R for maximum power


The relationship between PL and RL can be illustrated by the graph shown below.

RTH RL

PL 2th

th

V 4R


Example: A) For what value of R in the circuit below is maximum power delivered to R?B) Determine the maximum power that could be delivered to R.

+ _ 150 V

300

600 R


Example: C) Determine the power delivered to R when R = 20, 50, 100, 200, 400, 800,

and 2000 ohms.

D) Plot PL versus R.

R I V P2050

1002004008002000


+ _ 100 V 20

30

60

60

R

Example: Find R such that maximum power is delivered to R. Also find Pmax.

(Recall that this circuit was used in an earlier example.)


Applications of the Maximum Power Transfer TheoremThe maximum power transfer theorem is commonly used by engineers. Sometimes the requirement that RL = RTH is referred to as “impedance matching.” Max Power Transfer Theorem application - HF transmitter and antenna


Max Power Transfer Theorem application - stereo amplifier and speaker


PSPICE Demonstration:Reference:PSPICE Assignment #2Read Chapters 1 - 3 in Schematic Capture Using Cadence PSPICE by HerniterHandout: PSPICE Example - Maximum Power Transfer (Varying a Component Value)Handout: PSPICE Example - Op Amp Circuit using a Library Model ( uA741) Handout: PSPICE Example - Op Amp Example using a General Op Amp Model DC Sweep: Illustrate how to vary the following quantities:• A voltage source• A current source• A resistor

– Insert the part PARAM when varying a resistor value. – Use a potentiometer symbol (part R_VAR) for the resistor– Be sure to change the property SET from 0.5 to 1

PROBE: Illustrate with PSPICE’s graphing window how to:• Add traces• Add text, arrows, boxes• Add one or two cursors (control with left or right mouse and with + or Shift +) • Mark points on a graph• Find maxima and minima


Network Reduction Techniques and Network Theorems

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