Transcript Thevenin Equivalence
THEVENIN’S AND NORTON’S THEOREMS
These are some of the most powerful analysis results to be discussed.
They permit to hide information that is not relevant and concentrate in what is important to the analysis
THEVENIN’S EQUIVALENCE THEOREM
LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART A
i v O
_
a b
LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART B
R TH v TH
i v O
_ PART A
Thevenin Equivalent Circuit for PART A
v TH
Thevenin Equivalent Source
R TH
Thevenin Equivalent Resistance
a b
LINEAR CIRCUIT PART B
NORTON’S EQUIVALENCE THEOREM
LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART A
i v O
_
a b
LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART B
i N R N
PART A
Norton Equivalent Circuit for PART A
i N
Thevenin Equivalent Source
R N
Thevenin Equivalent Resistance
i v O
_
a b
LINEAR CIRCUIT PART B
ANOTHER VIEW OF THEVENIN’S AND NORTON’S THEOREMS
v OC
+ _ R TH i +
v O
_
i SC R TH
Norton
i a
v O b
Thevenin
i SC
v OC R TH
This equivalence can be viewed as a source transformation problem It shows how to convert a voltage source in series with a resistor into an equivalent current source in parallel with the resistor
v
A General Procedure to Determine the Thevenin Equivalent
v TH
Open Circuit vo v oltage at a b ltage if Part B is removed
i SC
Short Circuit Current current through a b if Part B is replaced by a short circuit source
R TH
v TH
Thevenin Equivalent Resistance
i SC
1. Determine the Thevenin equivalent Remove part B and compute the OPEN CIRCUIT voltage
V ab
One circuit problem LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART A
i v OC
_ 0
a b
V ab
_ Second circuit problem 2. Determine the SHORT CIRCUIT current Remove part B and compute the SHORT CIRCUIT current
I ab
LINEAR CIRCUIT May contain independent and dependent sources with their controlling variables PART A
v
i SC
0 _
a b I ab TH
v OC
,
R TH
v OC i SC
LEARNING EXAMPLE USE THEVENIN TO COMPUTE Vo You have the choice on the way to partition the circuit. Make “Part A” as simple as possible Since there are only independent sources, for the Thevenin resistance we set to zero all sources and determine the equivalent resistance “Part B” For the open circuit voltage we analyze the following circuit (“Part A”) ...
R TH
10 3
k
Loop Analysis I
2 6
V
2
mA
4
kI
1 2
k
(
I
1
I
2 ) 0
V OC I
1 4
k
*
I
1 2
k
*
I
2 6 2
I
2
mA
6 20 / 3 4
V
5 3
mA
32 / 3 [
V
]
The circuit becomes...
LEARNING EXAMPLE: COMPUTE Vo USING NORTON
I SC R N I SC
R TH
I N
3
k
12
V
3
k
2
mA
PART B
2
mA
COMPUTE Vo USING THEVENIN
V TH
PART B
4
k I R N
2
k I N V O
2
kI
2
k
R N R
N
6
k I N
V O
2 3 9 ( 2 ) 4 3 [
V
]
V TH
3
k
12 2
mA
0
R TH
3
k
4
k R TH
V TH
+ 2
k V O
V O
2 2 7 ( 6
V
) 4 3 [
V
]
LEARNING EXAMPLE Find the Thevenin Equivalent circuit at A - B
Only dependent sources. Hence V_th = 0 To compute the equivalent resistance we must apply an external probe We choose to apply a current probe
R TH
V P I P
@V_1
V P I P
@V_2 Controlling variable “Conventional” circuit with dependent sources - use node analysis R TH A B Thevenin equivalent
(
I P
) 5
V
3 1 (
V
2 (
V
2 2
V
1 1 2
V
2 2
V
1
V
) 1 ) 5
V
2 0 6 3
V
2
V
1 6 [
V
2 (
V
1 ] 6 * * / / 2 5
V
2
V
2 ) 30 21 0 10 7 (
V P
V
2 ) (
I P
1
mA
)
R TH
V
2 1
mA
( 10 / 7 )
k