Chapter 8

Methods of Analysis

Constant Current Sources• Maintains the same current in the branch of

the circuit regardless of how components are connected external to the source.

• The direction of the current source arrow indicates the direction of current flow in the branch.

• The voltage across the current source depends on how the other components are connected.

Constant Current Sources

• In a series circuit, the current must be the same everywhere in the circuit.

• If there is a current source in a series circuit, that will be the value of the current for that circuit.

• For the circuit shown,

I = 2 mA

Source Conversions

• In circuit analysis it is sometimes convenient to convert between voltage sources and current sources.

• To convert from a voltage source to a current source, calculate the current from E/RS.

• RS does not change.

• Place the current source and the resistor in parallel.

Source Conversions• We can also convert from a current source to

a voltage source.

• E = I•RS• Place the voltage source in series with the

resistor.

Source Conversions

• A load connected to a voltage source or its equivalent current source should have the same voltage across it and current through it for either source.

• Although the sources are equivalent, currents and voltages within the sources may differ.

• The sources are only equivalent external to the terminals.

Current Sources in Parallel and Series

• Current sources in parallel simply add together.

• The magnitude and direction of the resultant source is determined by adding the currents in one direction then subtracting currents in the opposite direction.

• Current sources should never be placed in series. This would violate KCL.

Branch Current Analysis• For circuits which have more than one source, we have

to use different methods of analysis.

• Begin by arbitrarily assigning current directions in each branch.

• Label the polarities of the voltage drops across all resistors.

• Write KVL around all loops.

• Apply KCL at enough nodes so all branches have been included.

• Solve the resulting equations.

Branch Current Analysis• From KVL: 6 - 2I1 + 2I2 - 4 = 0

4 - 2I2 - 4I3 + 2 = 0

• From KCL: I3 = I1 + I2

• Solve the simultaneous equations.

Mesh Analysis• Arbitrarily assign a clockwise current to each

interior closed loop.• Indicate the voltage polarities across all

resistors.• Write the KVL equations.• Solve the resulting simultaneous equations.• Branch currents are determined by

algebraically combining the loop currents which are common to the branch.

Mesh Analysis• Assign loop currents and voltage polarities.

• Using KVL: 6 - 2I1 - 2I1 + 2I2 - 4 = 0

4 - 2I2 + 2I1 - 4I2 + 2 = 0

• Simplify and solve the equations.

Format Approach• Mutual resistors represent resistors which

are shared between two loops.

• R12 represents the resistor in loop 1 that is shared by loop 1 and loop 2.

• The coefficients along the principal diagonal will be positive.

• All other coefficients will be negative. • The terms will be symmetrical about the

principal diagonal.

Format Approach

• Convert current sources into equivalent voltage sources.

• Assign clockwise currents to each independent closed loop.

• Write the simultaneous linear equations in the format outline.

• Solve the resulting simultaneous equations.

Nodal Analysis• Assign a reference node within the circuit and

indicate this node as ground.• Convert all voltage sources to current

sources.

• Assign voltages V1, V2, etc. to the remaining nodes.

• Arbitrarily assign a current direction to each branch where there is no current source.

Nodal Analysis

• Apply KCL to all nodes except the reference node.

• Rewrite each of the currents in terms of voltage.

• Solve the resulting equations for the voltages.

Format Approach• Mutual conductance is the conductance that

is common to two nodes.

• The mutual conductance G23 is the conductance at Node 2, common to Node 3.

• The conductances at particular nodes are positive.

• Mutual conductances are negative.• If the equations are written correctly, the terms

will be symmetrical about the principal diagonal.

Format Approach

• Convert voltage sources into equivalent current sources.

• Label the reference node as ground.

• Label the remaining nodes as V1, V2, etc.

• Write the linear equation for each node.• Solve the resulting equations for the voltages.

Delta-Wye Conversion• Any resistor connected to a point of the Y is

obtained by finding the product of the resistors connected to the same point in the Delta and then dividing by the sum of all the Delta resistors.

• Given a Delta circuit with resistors of 30, 60, and 90 , the resulting Y circuit will have resistors of 10, 15, and 30 .

Wye-Delta Conversions• A Delta resistor is found by taking the sum of

all two-product combinations of Y resistor values and then dividing by the resistance of the Y which is located directly opposite the resistor being calculated.

• For a Y circuit having resistances of 2.4, 3.6, and 4.8 , the resulting Delta resistors will be 7.8, 10.4, and 15.6 .