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7/29/2019 Thevenin and Norton Theorems
1/12
Thevenin Theorem in Sinusoidal Steady Analysis
Aim: To obtain a simple equivalent circuit for a 1-port circuit thatconsists of linear, time-invariant resistors, capacitors, inductorsand independent sources.
Thevenin Equivalent:
+
_v
i
1-port
circuit
+
_V
I
+_
ZTH
VTH
ZTh Thevenin impedance
Equivalent impedance between terminalswhen sources are set to zero.
VTh Open circuit voltage
THThVIZV
The voltage of the port when the port isleft as open circuit.
Thevenin Theorem insinusoidal steadyanalysis: A 1-port circuitthat consists of linearresistor, capacitor,
inductors andindependent sources hasa Thevenin equivalentcircuit in sinusoidalsteady state if the port
voltage phasor can beuniquely determined for agiven port currentphasor, in other words, ifthe 1-port is current-controlled.
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Norton Theorem in Sinusoidal Steady Analysis
Aim: To obtain a simple equivalent circuit for a 1-port circuit thatconsists of linear, time-invariant resistors, capacitors, inductorsand independent sources.
Norton Equivalent:
+
_v
i
1-port
circuit
Norton Theorem insinusoidal steadyanalysis: A 1-port circuitthat consists of linearresistor, capacitor,
inductors andindependent sources hasa Norton equivalentcircuit in sinusoidalsteady state if the port
current phasor can beuniquely determined for agiven port voltage phasor,in other words, if the 1-port is voltage-controlled.
NNIVGI
+
_V
I
YNIN
GN Norton conductance
IN Short circuit current
Equivalent conductance betweenterminals when sources are set to zero.
The current through the port when the
port is short-circuited.
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How to obtain Thevenin equivalent circuit?
+
_
V
I
1-portcircuit
Connect a sinusoidal current source to the port.
I*V*
+
_ Solve the circuit using sinusoidal steady analysis
and obtain a relation between phasors I* and V*.
Use I=I* and V=-V* to obtain a relationbetween I and V.
+
_V
I
1-portcircuit
Set the values of independent sources tozero.
Calculate the equivalent impedance ZTh = V / I. Assume that I=0 (open-circuit the port) and
calculate Vth=V taking into account allindependent sources .
There exist two methods for this!
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How to obtain Norton equivalent circuit?
Connect a sinusoidal voltage source to the port. Solve the circuit using sinusoidal steady analysis
and obtain a relation between phasors I* and V*.
Use I=-I* and V=V* to obtain a relationbetween I and V.
+
_V
I
1-portcircuit
Set the values of independent sources tozero.
Calculate the equivalent admitance GN = I / V. Assume that V=0 (short-circuit the port) and
calculate IN=I taking into account allindependent sources .
There exist two methods for this!
+
_
V
I
1-portcircuit
I*
V*
+
_
+-
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Thevenin Equivalent: THTH VIZV
If 1-port is not current-controlled there is no Thevenin eq..
Norton Equivalent:NNIVYI
If 1-port is not voltage-controlled there is no Norton eq..
0THZTH
TH
THZ
VVZ
I 1 ,0THZ No Norton equivalent!
,0NY No Thevenin equivalent!
NY
NI
0NYN
N
NY
II
YV
1
THV
THZ
Interchange between Thevenin and Norton
From Thevenin to Norton:
From Norton to Thevenin:
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Example 1: Find the Thevenin equivalent circuit for the following circuit!
Example 2: Find the Norton equivalent circuit for the following circuit! Check your answercomparing it to Example 1.
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Circuit(Network) Functions in Sinusoidal SteadyAnalysis
+
_E1
IS
linear,time-
independentelements
0
0
)()(0
000
s
d
TI
I
VV
jwNjwM
IAA
)(jwT
How does Vdk affect Is ? Depending on w!
Ssk
d IwT
wTcofactorwV
k )(det
)()( ,
Assume that
there is only one
source.
0
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+
_E1
IS
)(
)()(
jwQ
jwP
I
wV
S
dk
)(jwP are polynomials
in (jw) with real coefficients.)(jwQ
m
m
n
n
S
d
jwbjwbjwb
jwajwajwa
I
wVk
)(...)()(
)(...)()()(2
21
2
21
This depends on thecircuit but not on thevalue of Is .
linear,time-
independentelements
Circuit(Network) Functions in Sinusoidal SteadyAnalysis
How does Vdk affect Is ? Depending on w!
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One can define many other circuit functions:
)(
)(
wI
wV
s
dk Impedance Function
)(
)(1
wI
wV
s
Input Impedance Function
)(
)(
1 jwV
jwVkd Voltage Transfer Function
)()(
jwIjwI
s
k Current Transfer Function
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Symmetries of Circuit Functions
Lemma: Let be a polynomial in complex variable s withreal coefficients.
1)
2) (z is called as a root of n(z).)
jwssn ),(
)()( snsn
0)(0)( znzn
Proof: 011
1 ...)( nsnsnsnsnk
k
k
k
Rnnnn
kk
011 ,,...,
1) ...)( 0111 nsnsnsnsnk
k
k
k
01
1
1... nsnsnsn
k
k
k
k
01
1
1... nsnsnsn
k
k
k
k
)(sn
2) 00)(0)( znzn
0)(0)()()( znznznznFrom (1)
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Circuit function: )(
)(
)( jwd
jwn
jwH
)(
)()(jwHj
ejwHjwH
Symmetry Property: The magnitude of any circuit function is an evenfunction of w and its phase is an odd function of w.
Proof:
)()()(
jwdjwnjwH
)()()(
jwdjwnjwH
)()(
jwdjwn
jwjw and from Lemma)(
)()(
jwd
jwnjwH
)( jwH
)()()()( jwHjwHjwHjwHw
Since the phase of is .z z )()( jwHjwH
Theorem: For a circuit in sinusoidal steady state, any circuit function is
well-defined and is the ratio of two polynomials in (jw) with real
coefficients if det(T(jw)) is nonzero.
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+_ Vs (t) 1-portcircuit
so VjwHV )(
Result:In order to find the behaviour of the circuit for the frequency w, one
should find and .sVjwH ,)( sVjwH
),(
sVj
ss eVV
oVj
oo eVV
sV
s
jwH
o eVejwHV
)()(
sVjwH
so
eVjwHV
)(
)(
so VjwHV )( so VjwHV )(
sso VjwHwtVjwHtv )(cos)(
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