Methods of Analysis

1
Chapter 8

• Methods of Analysis

2
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.

3
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

4
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.

5
Source Conversions

• We can also convert from a current source to a
  voltage source.
• E IRS
• Place the voltage source in series with the
  resistor.

6
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.

7
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.

8
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.

9
Branch Current Analysis

• From KVL 6 - 2I1 2I2 - 4 0
• 4 - 2I2 - 4I3 2 0
• From KCL I3 I1 I2
• Solve the simultaneous equations.

10
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.

11
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.

12
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.

13
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.

14
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.

15
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.

16
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.

17
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.

18
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 ?.

19
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 ??. ?