Download Circuit Theorems-Electrical Circuit Analysis-Lecture Slides and more Slides Electrical Circuit Analysis in PDF only on Docsity!
Circuit TheoremsCircuit Theorems -- Chapter 4Chapter 4
4 14.1 MotivationMotivation
4.2 Linearity Property
4.3 Superposition
4.4 Source Transformation
4.5 Thevenin’s Theorem
1
4.5 Thevenin s Theorem
4.6 Norton’s Theorem
4.7 Maximum Power Transfer
4.1: Motivation4.1: Motivation
- Kirchoff’s laws analyze the circuits without tampering with its original configuration. A major disadvantage is that for large complex circuit, tedious computation is involved.
- Thevenin and Norton theorems are applicable to analyze the linear circuits.
- The concepts of
- If you are given the above circuit, are there any other alternative(s) to
2
- What are they? And how?
- Can you work it out by inspection?
- The concepts of superposition, source transformation, and maximum power transfer are also used to make the circuit analysis simpler.
determine.
docsity.com
4.2: Linearity Property4.2: Linearity Property
- It is the property of an element describing a linear relationship between cause and effect.
- A linear circuit is one whose output is linearly related (or directly proportional) to its input.
- Linearity is combination of both homogeneity (scaling) property and additivity property.
1. Homogeneity (scaling) property: If the input, also called
as excitation, is multiplied by a constant, then the output, also called as response, is multiplied by a constant.
v = i R → k v = k i R
2 Additivity property: The additivity property requires that the
3
2. Additivity property: The additivity property requires that the
response to a sum of inputs is the sum of response to each input applied separately.
v 1 = i 1 R and v 2 = i 2 R
→ v = (i 1 + i 2 ) R = v 1 + v 2
- In general a circuit is linear if it is both additive and homogeneous.
4.2: Linearity Property4.2: Linearity Property
- Note that since p=i 2 R=v 2 /R (making it quadric function rather than a linear one)rather than a linear one), the relationship between power and voltage (or current) is nonlinear.
- Therefore, the theorems covered in this chapter are not applicable to power.
2 2 2
2 1 1 p Ri
p Ri
=
=
4
1 2
12
2 2
2 1
2 3 1 2
2 2
2
( )
p p
Ri Ri Rii
P Ri i
p Ri
≠ +
= + +
= +
docsity.com
Class Activity (5)Class Activity (5)
Practice Problem 4.2: Assume that Vo = 1V and use linearity to calculate the actual value of Vo in the circuit.
7
4.3: Superposition Theorem4.3: Superposition Theorem
- It states that the voltage across (or current through) an element in a linear circuit is theg ) algebraic sum of the voltage across (or currents through) that element due to EACH independent source acting alone while all the others independent sources are turned off.
- The principle of superposition helps us to analyze a linear circuit with more than one d d b l l h
8
independent source by calculating the contribution of each independent source separately.
docsity.com
4.3: Superposition Theorem4.3: Superposition Theorem
Practice Problem 4.
Using the superposition theorem, find v o in the circuit of Fig.
9
4.3: Superposition Theorem4.3: Superposition Theorem
Steps to apply superposition principle
- Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis.
- Repeat step 1 for each of the other independent sources
10
independent sources.
- Find the total contribution by adding algebraically all the contributions due to the independent sources.
docsity.com
4.3: Superposition Theorem4.3: Superposition Theorem
Practice Problem 4.
Use superposition to find v x in
the circuit below.
13
Dependant source keep unchanged
4.3: Superposition Theorem4.3: Superposition Theorem
Practice Problem 4.
Use superposition to find v x in
the circuit below.
2A is discarded by open-circuit
20 Ω (^) v1 20 Ω^ v
10V is discarded by open-circuit Dependant sourcekeep unchanged
14
10 V + −^4 Ω
(a)
0.1v 1 2 A 4 Ω
(b)
0.1v 2
docsity.com
4.4: Source Transformation4.4: Source Transformation
- An equivalent circuit is one whose v-i
characteristics are identical with the
original circuit.
- It is the process of replacing a voltage
source v (^) S in series with a resistor R by a current source i (^) S in parallel with a resistor
R or vice versa
15
R , or vice versa.
4.4: Source Transformation4.4: Source Transformation
- Arrow of the current source is directed toward the positive terminal of the voltage source.
- The source transformation is not possible when R = 0 for voltage source and R = ∞ for current source.
- • The two circuits are equivalent provided they have same voltageThe two circuits are equivalent provided they have same voltage- current relation at terminal a-b.
- If the sources are turned off the equivalent resistance at terminal a-b in both circuits is R.
+ +
-
- Also when the terminal a-b are short circuited, the short circuit current flowing from a to b is i (^) sc=v (^) s/R in the circuit on the left h d id d i i i h i i
16
(a) Independent source transform
(b) Dependent source transform
+ +
**- -
v v (^) s = isR or is = s
hand side and i (^) sc=i (^) s in the circuit on the right hand side. Thus v (^) s/R=i (^) s in order for the two circuits to be equivalent. Hence source transformation requires
docsity.com
4.5:4.5: Thevenin’sThevenin’s TheoremTheorem
MOTOVATION:
- Often it happens that an element in a circuit is variable, usually called load, while all the other elements are fixed e.g. household outlet connected withh h ld tl t t d ith different appliance.
- In such case each time the variable element is changed, the whole circuit has to analyzed all over again.
- To avoid such problem, Thevenin theorem provides a technique by which the fixed part of the circuit can be replaced by an equivalent circuit
19
replaced by an equivalent circuit.
- Thevenin theorem is very important in circuit analysis. It helps to simplify the circuit. A large circuit may be replaced by a single independent voltage source and a single resistor.
4.5:4.5: Thevenin’sThevenin’s TheoremTheorem
- It states that a linear two-terminal circuit (Fig. a) can be replaced by an equivalent circuit (Fig. b) consisting of a voltage source VTHTH in series with a resistor RTH , where;
- VTH is the open-circuit voltage at the terminals.
- To get the value of RTH if there are two cases:
- CASE 1: If the network has only independent sources, we can turn off all the independent sources. RTH is the i t i t f th t k l ki
20
input resistance of the network looking between terminals a-b.
docsity.com
Class ActivityClass Activity
Practice Problem 4.8: Using Thevenin’s theorem, find the
equivalent circuit to the left of the terminals in the circuit
shown belowshown below. Hence find i. Hence find i
21
4.54.5 Thevenin’sThevenin’s TheoremTheorem
- CASE 2: If the network has dependent sources, we turn off all independent sources. As with the superposition, dependent sources are not to be turned off because they are controlled by circuit variables.
- We apply a voltage vo at the terminals a and b and determine the resulting current i (^) o. Then Rth=vo /i (^) o.
- Alternatively we may insert a current source i (^) o at terminals a-b as shown in the fig(b) and find terminal voltage v
22
the fig(b) and find terminal voltage vo. Again Rth=vo /i (^) o.
- Either of the two approaches will give the same results. For example we may use vo =1 v or i (^) o =1 A.
docsity.com
Class ActivityClass Activity
Example 4.10: Evaluate the solution by connecting a 9 Ω
resistor and a 10 V source across the output terminals of the
original circuit and then across the Thevenin equivalent
circuitcircuit
25
Class ActivityClass Activity
Practice Problem 4.10: Obtain the Thevenin equivalent
circuit.
26
docsity.com
4.6 Norton’s Theorem (1)4.6 Norton’s Theorem (1)
It states that a linear two-terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN ,
Where
- IN is the short circuit current through the terminals.
- RN is the input or equivalent resistance at the terminals when the independent sources are turned off.
27
th
th N
N th
R
V I
R R
=
=
- The Thevenin’s and Norton equivalent circuits are related by a source transformation. For this reason the source transformation is often called Thevenin-Norton transformation.
4.6 Norton’s Theorem (1)4.6 Norton’s Theorem (1)
th
N th V I
R R
=
=
- Since V (^) TH, IN and RTH are related according to eq (i), to determine the thevenin or Norton equivalent circuit requires that we find: th
N R
circuit requires that we find: I =
- The open circuit voltage voc across the terminals a and b.
- The short circuit current i (^) sc at terminals a and b.
- The equivalent or input resistance Rin at terminals a and b when all the independent sources are (^) V = v
28
independent sources are turned off.
- We can calculate any two of the three using the method that takes least effort and use them to get the third using ohm’s law…
N sc
oc th
N sc
th oc
R i
v R
I i
V v
= =
=
=
docsity.com
4.6 Norton’s Theorem (2)4.6 Norton’s Theorem (2)
Practice Problem 4.12:
Find the Norton
equivalent circuit of the
circuit shown below.
Calculate the Isc
31
docsity.com
