1. Additional Circuit Analysis Techniques
Chapter 3
Additional Circuit Analysis Techniques

2. 3.1 The Principle of Superposition
If a circuit composed of linear elements contains
N independent sources, any element voltage
or current in that circuit is composed of the sum of
N contributions, each of which is due to one of the
sources acting individually when all others
are set equal to zero (Deactivated)

3. Deactivated sources
Figure 3.1 Illustration of (a) replacing a deactivated voltage source with a short circuit, and (b) replacing a deactivated current source with an open circuit.

4. Example: determine v and i

5. As a check Direct Method
KCL at upper node gives
KVL around nonsimple loop
Solving for I gives

6. Ex 3.1: Determine V and I using Superposition

8. Alternatively, using direct method

9. 3.2 The Thevenin Equivalent Circuit
Figure 3.5 (a) separating a circuit into two parts, (b) writing the u–i relation at the terminals of the linear part, and (c) representing the linear part with the Thevenin equivalent circuit.

10. Figure 3.6 Illustration of the computation of (a) the open-circuit voltage VOC and (b) the Thevenin resistance RTH.

11. Ex. 3.3 Determine current i by reducing the circuit attached to the 5 resistor to a Thevenin equivalent

12. 3.3 The Norton Equivalent Circuit

13. Ex 3. 5 Determine i considered in Ex 3
Ex 3.5 Determine i considered in Ex 3.3 using a Norton equivalent circuit
Using Superposition

14. OR

15. The requested i

16. 3.4 Maximum Power Transfer
All practical sources have internal resistance represented by RS.
When the source is attached to a load RL. some voltage will drop
across RS.
What would be the optimum RL such that a maximum power will be
delivered from the source to the load?

17. The load current is
The delivered power to the load is
To maximize power, differentiate with respect to load and set result to zero
Solve for
Thus for maximum power transfer to a load, we should choose the load resistance
to be equal to the internal resistance of the source. The delivered power is

18. Example 3.8
Determine the value of load resistance RL to achieve maximum power transfer to that load for the following circuit
Thevenin Equivalent
The circuit attached to the load can be reduced to a Thevenin equivalent circuit.
From the circuit we determine
The maximum power is =

19. 3.5 The Node-Voltage Method+ -
There are a total of four nodes
We arbitrarily choose one node a
as the reference node to which the
Node voltages are reference
Then we define the voltages of the
Other three nodes c, b, d with respect
to this reference node

20. KCL at node c

21. KCL at node c
KCL at node b

22. KCL at node c
KCL at node b
KCL at node d

23. Matrix Form

24. Ex. 3.9 Write the node-voltage equations and solve for I
Select a reference node
Label the other nodes
Write KCL on each non reference node
You can also solve the circuit and finding
the current I using superposition
(See the book)

25. Choose a reference node and define node voltages as usual
Circuit Containing Voltage Source
General rule:
Choose a reference node and define node voltages as usual
Write the constraints that are imposed on the node voltages
by voltage sources
Draw supernodes around all voltage sources that have neither
end connected to the reference node
Write KCL for all supernodes and all remaining nodes except
those that have a voltage source connected between that node
and the reference node.
Substitute the constraints imposed by the voltage sources into
these equations, and place them in standard form.

26. Ex: 3.12 Determine Current I by writing node-voltage equations
Select a reference node and label the other non reference nodes
From constraints
Drawing supernode around node a and c
KCL for the supernode gives:
Substitution from constraints
The current I is:

27. 3.6 The Mesh-Current Method
A mesh is a circuit loop that does not enclose any elements
The mesh currents are fictitious currents that are defined to
flow only around the mesh
Figure 3.21 Illustrations of the concept of a mesh.

28. The general rules for writing
A three-mesh circuit and
the mesh currents
The general rules for writing
mesh-current equations
Define the mesh currents
Write the total current through each element in terms of the mesh
currents flowing through them
3. Write KVL around each mesh
4. Put these equations in standard form, and solve them for the mesh
currents

29. For example In matrix form KVL around each mesh give:
Grouping and placing in a standard form
In matrix form
KVL around each mesh give:
Once these equations are solved,
the current through each element
Can be written in terms of the
Mesh currents

30. Ex 3.14: Write the mesh-current equations and determine current I
Direction of mesh current does not affect solution
KVL around each mesh
Solving for mesh currents
In matrix form
Therefore:

31. We will solve the circuit using the Node Voltage method (comparison)
Only one node is unknown
KCL at node b

32. 3.6.1 Circuit Containing Current Sources
Hence only one mesh current is unknown
Voltage across current source is unknown, so we apply
KVL around mesh 2 and mesh 3
Substitute constraints:
Mesh-current equations when a circuit contains a current source
Define the mesh currents in the usual fashion
Write the constraints that are imposed on the mesh currents by any
current sources.
3. Draw loops around all pairs of meshes that share a current source
4. Write KVL for all these loops and all other meshes except those
meshes that have a current source in an outside branch
5. Substitute the constraints imposed on the mesh currents by the
current sources into these equations and place them in standard form

33. Ex 3.16 Determine V by writing mesh equations
Constraints:
KVL around mesh 2 and 3
Substituting current constraints
Hence the voltage is