FEE1-L3

INTRODUCTION
So far covered: Ohm’s law and Kirchhoff’s laws
This lecture covers powerful techniques for circuit analysis
nodal analysis - based on a systematic application of KCL
mesh analysis - based on a systematic application of KVL
With these techniques we can analyze any linear circuit by obtaining a set of simultaneous equations that are then solved to obtain the required values of current or voltage
substitution method
elimination method
Cramer’s rule
matrix inversion

Nodal Analysis
In nodal analysis, we are interested in finding the node voltages by applying KCL
Select a node as the reference node, assign voltages V1, V2, …, Vn-1 to the remaining nodes, the voltages are referenced with respect to the reference node
Apply KCL to each of the nonreference nodes, use Ohm’s law to express the branch currents in terms of node voltages.
Solve the resulting simultaneous equations to obtain the unknown node voltages
The reference node is commonly called the ground since it is assumed to have zero potential
common ground
ground
chassis
ground

Example
Calculate the node voltages in the circuit shown
The fact that i2 is negative shows that the current flows in the direction opposite to the one assumed

Problems
Determine the voltages at the nodes

Problems
Determine the voltages at the nodes

Nodal Analysis with Voltage Sources
CASE 1: If a voltage source is connected between the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source
CASE 2: If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we apply both KCL and KVL to determine the node voltages
A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

Example
For the circuit shown find the node voltages.

Problems
Find v and i in the circuit

Problems
Find v and i in the circuit
V1 = 14 V
6 = V3 - V2
V3 = 6 + V2
I1 = I2 + I3 + I4
(14 – V2)/4 = V2/3 + V3/2 + V3/6
42 – 3*V2 = 4*V2 + 6*V3 + 2*V3
42 = 7*V2 + 8*V3
42 = 7*V2 + 8*(6 + V2)
42 = 7*V2 + 48 + 8*V2
15*V2 = -6
V2 = -6/15 = -0.4 V
V3 = 6 + (-0.4) = 5.6 V
I = V3/2 = 5.6 / 2 = 2.8 A

Mesh Analysis (Planar / Nonplanar)
mesh analysis applies KVL to find mesh currents
Mesh analysis is not quite general because it is only applicable to a circuit that is planar
planar circuit is one that can be drawn in a plane with no branches crossing one another
Nonplanar circuits can be handled using nodal analysis
Planar circuit
Nonplanar circuit

Mesh Analysis
A mesh (independent loop) is a loop which does not contain any other loops within it
Assign mesh currents I1, I2, …, In to the n meshes
Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents
Solve the resulting n simultaneous equations to get the mesh currents.

Example
For the circuit, find the branch currents using mesh analysis

Problem
Use mesh analysis to find the current I0 in the circuit

Problem
Use mesh analysis to find the current I0 in the circuit

Mesh Analysis with Current Sources
the presence of the current sources reduces the number of equations
CASE 1: When a current source exists only in one mesh we set mesh current equal to algebraic sum of current sources in that mesh
CASE 2: When a current source exists between two meshes we create a supermesh by excluding the current source and any elements connected in series with it, and apply KVL around supermesh

Example
For the circuit, find i1 to i4 using mesh analysis
i3 = i2 + 3*i4
i4 = 0.8*i3 - 1
i3 = i2 + 3*(0.8*i3 - 1)
i3 = i2 + 2.4*i3 – 3
i3 = (3 – i2)/1.4
I3 = (– i1 – 2)/1.4
I4 = 0.8*(– i1 – 2)/1.4 - 1
i1 + 3*(i1 + 5) + 6*(– i1 – 2)/1.4 – 4*(0.8*(– i1 – 2)/1.4 – 1) = 0
i1 + 3*i1 + 15 – 4.3*i1 – 8.6 + 4 + 2.3*i1 + 4.6 = 0
2*i1 +15 = 0
i1 = -7.5A i2 = -2.5A i3 = 3.93A i4 = 2.143A

Nodal and Mesh Analyses by Inspection
(G1+G2)*v1 – G2*v2 = I1 – I2
(G2+G3)*v2 – G2*v1 = I2
(R1+R3)*i1 – R3*i2 = v1
(R2+R3)*i2 – R3*i1 = -v2
Voltage of the selected node multiplied by sum of conductances connected to that node minus all neighboring nodes multiplied with conductance that makes that node neighbor to the selected one equals to algebraic sum of current sources at the selected node
Current of the selected mesh multiplied by sum of resistanses around that mesh minus all neighboring meshes multiplied with resistance that makes that mesh neighbor to the selected one equals to algebraic sum of voltage sources around the selected mesh

Nodal Versus Mesh Analysis
first factor is the nature of the particular network
Networks that contain many series-connected elements, voltage sources, or supermeshes are more suitable for mesh analysis
whereas networks with parallel-connected elements, current sources, or supernodes are more suitable for nodal analysis
circuit with fewer nodes than meshes is better analyzed using nodal analysis
circuit with fewer meshes than nodes is better analyzed using mesh analysis
second factor is the information required
If node voltages are required, it may be better to apply nodal analysis
If branch or mesh currents are required, it may be better to use mesh analysis