Download Thevenin Theorem

Additivity and Multiplicativity
Theorem: (Additivity)
Group 1
Consider a circuit with linear resistors and
Group 2
independent sources.
Solve the circuit when only sources in Group 1 are active
whereas sources in Group 2 are set to zero
i ,v
1
1
2
2
Solve the circuit when only sources in Group 2 are active
whereas sources in Group 1 are set to zero
i ,v
Then, the solution when all the sources are active is
Proof: wT  w1  w2 ,
Circuit equations when all sources are active:
A
0

M
0
I
N
0   iT   0 
 AT  vT    0 
0  eT  wT 
~
A x  bT
~
xT  A 1bT
 iT   A
v    0
 T 
eT  M
0
I
N
iT  i1  i2 ,
vT  v1  v2
0 
 AT 
0 
1
~ 1  0   0  
xT  A      
 w1   w2  
0
0
 
wT 
If only group 1 is active:
A
0

M
0
I
N
0 i   0 
 AT  v   0 
0  e w1 
~
A x  b1
~
x1  A 1b1
 i1   A
v    0
 1 
e1   M
0
I
N
1
0
0
 
 w1 
1
0
0
 
 w2 
0 
 AT 
0 
~ 1  0 
x1  A  
w1 
If only group 2 is active:
A
0

M
0
I
N
0 i   0 
 AT  v    0 
0  e w2 
 i1   A
v    0
 1 
e1   M
0
I
N
~ 1  0 
x2  A  
w2 
~ 1  0  ~ 1  0  ~ 1  0   0  
x1  x2  A    A    A        xT
w1 
w2 
 w1   w2  
~
A x  b2
~ 1
x1  A b2
0 
 AT 
0 
Theorem: (Multiplicativity)
Consider a circuit with linear resistors and independent
sources vs . Assume that solutions are i, v .
If independent sources are set to k  vs ,
solutions are k  i, k  v .
Proof: ...........
Thevenin (1883) and Norton (1926) Theorems
Aim: To obtain a simple equivalent circuit for a 1-port circuit that
consists of linear, time-invariant n-port resistors and independent
sources.
What does «equivalent» mean?.....................................................................
Thevenin Equivalent:
RTH
i
1-port
circuit
+
v
_

+
_
i
+
VTH
v
_
i
RTH
+
_
+
v
VTH
_
v(t )  RTH i(t )  vTH (t )
RTH
Thevenin resistance
Equivalent resistor between terminals
when sources are set to zero.
VTH
Open circuit voltage
The voltage of the port when the port is
left as open circuit.
Thevenin Theorem: A 1-port circuit has a Thevenin equivalent circuit
if the port voltage can be uniquely determined for a given port
current, in other words, if the 1-port is current-controlled.
Norton Equivalent:
i
1-port
circuit
+
v
_

i
+
iN
GN
v
_
i
+
iN
GN
v
_
i(t )  GN v(t )  iN (t )
GN
Norton conductance
Equivalent conductance between
terminals when sources are set to zero.
iN
Short circuit current
The current through the port when the
port is short-circuited.
Norton Theorem: A 1-port circuit has a Norton equivalent circuit if
the port current can be uniquely determined for a given port voltage,
in other words, if the 1-port is voltage-controlled.
v (t )
v (t )
v (t )
vTH
iN
vTH
i (t )
iN
i (t )
i (t )
How to obtain Thevenin equivalent circuit?
i
+
1-port
circuit
v
v*
_
+
i
+
1-port
circuit
v
_
_
i*
•
Connect a current source to the port.
•
Solve the circuit and obtain a relation between
i* and v*.
•
Use i=i* and v=-v* to obtain a relation
between i and v.
•
Set the values of independent sources to
zero.
•
Calculate the equivalent resistance Rth = v / i.
•
Assume that i=0 and calculate Vth taking into
account all independent sources .
How to obtain Norton equivalent circuit?
i
1-port
circuit
+
+
v
v*
_
_
i
+
1-port
circuit
v
_
+
-
i*
•
Connect a voltage source to the port.
•
Solve the circuit and obtain a relation between
i* and v*.
•
Use i=-i* and v=v* to obtain a relation
between i and v.
•
Set the values of independent sources to
zero.
•
Calculate the equivalent conductance GN = i/v.
•
Assume that v=0 and calculate IN taking into
account all independent sources .
Interchange between Thevenin and Norton
V = RTH I +VTH
• Thevenin Equivalent:
If 1-port is not current-controlled there is no Thevenin eq..
• Norton Equivalent:
I = GNV + I N
If 1-port is not voltage-controlled there is no Norton eq..
From Thevenin to Norton:
•
RTH ¹ 0
1
VTH
I=
VRTH
RTH
GN
RTH = 0,
No Norton equivalent!
IN
From Norton to Thevenin:
•
GN ¹ 0
1
IN
V=
IGN
GN
RTH VTH
GN = 0,
No Thevenin equivalent!