Basis for Thevenin and Norton Equivalent Circuits

Objective of Lecture Describe the differences between ideal and real voltage

and current sources

Demonstrate how a real voltage source and real current source are equivalent so one source can be replaced by the other in a circuit.

Chapter 4.4 Fundamentals of Electric Circuits

Voltage SourcesIdeal Real

An ideal voltage source has no internal resistance.

It can produce as much current as is needed to provide power to the rest of the circuit.

A real voltage sources is modeled as an ideal voltage source in series with a resistor.

There are limits to the current and output voltage from the source.

Limitations of Real Voltage Source

Real Voltage Source

LLL

S

SL

LL

RVI

VRR

RV

/

VL

IL

RL

Voltage Source Limitations (con’t)RL = 0W RL = ∞W

W0

/

V0

max

L

SSL

L

P

RVI

V

W0P

A 0min

L

L

SL

I

VV

Current SourcesIdeal Real

An ideal current source has no internal resistance.

It can produce as much voltage as is needed to provide power to the rest of the circuit.

A real current sources is modeled as an ideal current source in parallel with a resistor.

Limitations on the maximum voltage and current.

Limitations of Real Current Source Appear as the resistance of the load on the source

approaches Rs.

Real Current Source

LLL

S

SL

SL

RIV

IRR

RI

VL

IL

RL

Current Source Limitations (con’t)RL = 0W RL = ∞W

W0

V0min

L

L

SL

P

V

II

W0P

A0

L

max

SSL

L

RIV

I

Electronic Response For a real voltage source, what is the voltage across the

load resistor when Rs = RL?

For a real current source, what is the current through the load resistor when Rs = RL?

Equivalence An equivalent circuit is one in which the i-v

characteristics are identical to that of the original circuit.

The magnitude and sign of the voltage and current at a particular measurement point are the same in the two circuits.

Equivalent Circuits RL in both circuits must be identical.

IL and VL in the left circuit = IL and VL on the left

Real Current SourceReal Voltage Source

1

2VL

IL

RL VL

IL

RL

Example #1 Find an equivalent current source to replace Vs and Rs

in the circuit below.

VL

IL

RL

Example #1 (con’t) Find IL and VL.

mWP

mAVVmAVP

PPP

mAkVI

RVI

VVkk

kV

VRR

RV

Vs

Vs

RsLVs

L

LLL

L

S

SL

LL

36

)2)(1218()2(12

26/12

/

121836

6

W

WW

W

VL

IL

RL

Example #1 (con’t) There are an infinite number of equivalent circuits that

contain a current source.

If, in parallel with the current source, Rs = ∞W

Rs is an open circuit, which means that the current source is ideal.

VL

IL

RL

mWPP

mWmAVIVP

VkmAV

II

IsL

LLL

L

LS

24

24)2(12

12)6(2

W

Example #1 (con’t) If RS = 20kW

mWP

mAmAVmAVP

IVIVPPP

VRIVV

mAmAk

kkI

IR

RRI

Is

Is

RsRsLLRsLIs

LLIsL

S

L

S

SLS

0.32

)267.2(12)2(12

12

67.2220

206

W

WW

Example #1 (con’t) If RS = 6kW

mWP

mAmAVmAVP

IVIVPPP

VRIVV

mAmAk

kkI

IR

RRI

Is

Is

RsRsLLRsLIs

LLIsL

S

L

S

SLS

48

)24(12)2(12

12

426

66

W

WW

Example #1 (con’t) If RS = 3kW

mWP

mAmAVmAVP

IVIVPPP

VRIVV

mAmAk

kkI

IR

RRI

Is

Is

RsRsLLRsLIs

LLIsL

S

L

S

SLS

72

)26(12)2(12

12

623

36

W

WW

Example #1 (con’t) Current and power that the ideal current source needs to

generate in order to supply the same current and voltage to a load increases as RS decreases. Note: Rs can not be equal to 0W.

The power dissipated by RL is 50% of the power generated by the ideal current source when RS = RL.

Example #2 Find an equivalent voltage source to replace Is and Rs

in the circuit below.

Example #2 (con’t) Find IL and VL.

mWP

mAmAV

mAVP

PPP

VmAV

RIV

mAI

II

Vs

Vs

RsLVs

L

LLL

L

SL

07.1

)714.05(214.0

)714.0(214.0

214.0)300(714.0

714.0

50300

50

W

WW

W

Example #2 (con’t) There are an infinite number of equivalent circuits

that contain a voltage source.

If, in series with the voltage source, Rs = 0W

Rs is a short circuit, which means that the voltage source is ideal.

mWPP

mWP

mAVIVP

mAI

VRVI

VVV

VsL

L

LLL

L

LLL

LS

153.0

153.0

)714.0(214.0

714.0

300/214.0/

214.0

W

Example #2 (con’t) If RS = 50W

mWP

mAVV

AVP

IVIVPPP

mARVII

VVV

VR

RRV

Vs

Vs

RsRsLLRsLVs

LLVsL

S

L

L

SLS

179.0

)714.0)(214.025.0(

)714.0(214.0

714.0/

25.0214.0300

50300

W

WW

Example #2 (con’t) If RS = 300W

mWP

mAVV

AVP

IVIVPPP

mARVII

VVV

VR

RRV

Vs

Vs

RsRsLLRsLVs

LLVsL

S

L

L

SLS

306.0

)714.0)(214.0418.0(

)714.0(214.0

714.0/

418.0214.0300

300300

W

WW

Example #2 (con’t) If RS = 1kW

mWP

mAVV

AVP

IVIVPPP

mARVII

VVk

V

VR

RRV

Vs

Vs

RsRsLLRsLVs

LLVsL

S

L

L

SLS

662.0

)714.0)(214.0927.0(

)714.0(214.0

714.0/

927.0214.0300

1300

W

WW

Example #2 (con’t) Voltage and power that the ideal voltage source needs to

supply to the circuit increases as RS increases. Rs can not be equal to ∞W.

The power dissipated by RL is 50% of the power generated by the ideal voltage source when RS = RL.

Summary An equivalent circuit is a circuit where the voltage

across and the current flowing through a load RL are identical.

As the shunt resistor in a real current source decreases in magnitude, the current produced by the ideal current source must increase.

As the series resistor in a real voltage source increases in magnitude, the voltage produced by the ideal voltage source must increase.

The power dissipated by RL is 50% of the power produced by the ideal source when RL = RS.