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Basis for Thevenin and Norton Equivalent Circuits
Objective of Lecture Describe the differences between ideal and real voltage
and current sources Chapter 8.1 and 8.2 Principles of Electric Circuits
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 6.6 Electric Circuit Fundamentals Chapter 8.3 Principles of Electric Circuits Chapter 4.4 Fundamentals of Electric Circuits Chapter 2.6 Electrical Engineering: Principles and Applications
Voltage Sources Ideal 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
SSL
LL
RVI
VRR
RV
/=+
=VL
IL
RL
Voltage Source Limitations (con’t) RL = 0Ω RL = ∞Ω
W0 /
V0
max
==
=
L
SSL
L
PRVI
V
W0PA 0min
==
=
L
L
SL
IVV
Current Sources Ideal 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
SSL
SL
RIV
IRR
RI
=+
=VL
IL
RL
Current Source Limitations (con’t) RL = 0Ω RL = ∞Ω
W0 V0min
==
=
L
L
SL
PV
II
W0P
A0
L
max
==
=
SSL
L
RIVI
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 Source Real Voltage Source
1
2 VL
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.
mWPmAVVmAVP
PPP
mAkVIRVI
VVkk
kV
VRR
RV
Vs
Vs
RsLVs
L
LLL
L
SSL
LL
36)2)(1218()2(12
26/12/
121836
6
=−+=
+=
=Ω==
=Ω+Ω
Ω=
+=
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 = ∞Ω
Rs is an open circuit, which means that the current source is ideal.
VL
IL
RL
mWPPmWmAVIVP
VkmAVII
IsL
LLL
L
LS
2424)2(12
12)6(2
=====
=Ω==
Example #1 (con’t) If RS = 20kΩ
mWPmAmAVmAVP
IVIVPPPVRIVV
mAmAkkkI
IRRRI
Is
Is
RsRsLLRsLIs
LLIsL
S
LS
SLS
0.32)267.2(12)2(12
12
67.2220
206
=−+=
+=+====
=Ω
Ω+Ω=
+=
Example #1 (con’t) If RS = 6kΩ
mWPmAmAVmAVP
IVIVPPPVRIVV
mAmAkkkI
IRRRI
Is
Is
RsRsLLRsLIs
LLIsL
S
LS
SLS
48)24(12)2(12
12
426
66
=−+=
+=+====
=Ω
Ω+Ω=
+=
Example #1 (con’t) If RS = 3kΩ
mWPmAmAVmAVP
IVIVPPPVRIVV
mAmAkkkI
IRRRI
Is
Is
RsRsLLRsLIs
LLIsL
S
LS
SLS
72)26(12)2(12
12
623
36
=−+=
+=+====
=Ω
Ω+Ω=
+=
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 0Ω. 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.
mWPmAmAV
mAVPPPP
VmAVRIV
mAI
II
Vs
Vs
RsLVs
L
LLL
L
SL
07.1)714.05(214.0
)714.0(214.0
214.0)300(714.0
714.050300
50
=−+
=+=
=Ω==
=Ω+Ω
Ω=
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 = 0Ω
Rs is a short circuit, which means that the voltage source is ideal.
mWPPmWP
mAVIVPmAI
VRVIVVV
VsL
L
LLL
L
LLL
LS
153.0153.0
)714.0(214.0714.0
300/214.0/214.0
===
===
Ω====
Example #2 (con’t) If RS = 50Ω
mWPmAVV
AVPIVIVPPPmARVII
VVV
VRRRV
Vs
Vs
RsRsLLRsLVs
LLVsL
S
LL
SLS
179.0)714.0)(214.025.0(
)714.0(214.0
714.0/
25.0214.0300
50300
=−+
=+=+=
===
=Ω
Ω+Ω=
+=
Example #2 (con’t) If RS = 300Ω
mWPmAVV
AVPIVIVPPPmARVII
VVV
VRRRV
Vs
Vs
RsRsLLRsLVs
LLVsL
S
LL
SLS
306.0)714.0)(214.0418.0(
)714.0(214.0
714.0/
418.0214.0300
300300
=−+
=+=+=
===
=Ω
Ω+Ω=
+=
Example #2 (con’t) If RS = 1kΩ
mWPmAVV
AVPIVIVPPPmARVII
VVkV
VRRRV
Vs
Vs
RsRsLLRsLVs
LLVsL
S
LL
SLS
662.0)714.0)(214.0927.0(
)714.0(214.0
714.0/
927.0214.0300
1300
=−+
=+=+=
===
=Ω
Ω+Ω=
+=
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 ∞Ω. 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.