Nodal Analysis (3.1)

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Nodal Analysis (3.1)

• Prof. Phillips
• Feb 5, 2003

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Example A Summing Circuit

• The output voltage V of this circuit is
  proportional to the sum of the two input currents
  I1 and I2.
• This circuit could be useful in audio
  applications or in instrumentation.
• The output of this circuit would probably be
  connected to an amplifier.

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Summing Circuit
500W
500W

I1
V
500W
1kW
I2
500W

• Solution V 167I1 167I2

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Can you analyze this circuit using the techniques
of Chapter 2?
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Not This One!

• There are no series or parallel resistors to
  combine.
• We do not have a single loop or a double node
  circuit.
• We need a more powerful analysis technique
• Nodal Analysis

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Why Nodal or Loop Analysis?

• The analysis techniques in Chapter 2 (voltage
  divider, equivalent resistance, etc.) provide an
  intuitive approach to analyzing circuits.
• They cannot analyze all circuits.
• They cannot be easily automated by a computer.

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Node and Loop Analysis

• Node analysis and loop analysis are both circuit
  analysis methods which are systematic and apply
  to most circuits.
• Analysis of circuits using node or loop analysis
  requires solutions of systems of linear
  equations.
• These equations can usually be written by
  inspection of the circuit.

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Steps of Nodal Analysis

• 1. Choose a reference node.
• 2. Assign node voltages to the other nodes.
• 3. Apply KCL to each node other than the
  reference node express currents in terms of node
  voltages.
• 4. Solve the resulting system of linear equations.

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Reference Node
500W
500W

I1
V
500W
1kW
I2
500W

• The reference node is called the ground node.

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Steps of Nodal Analysis

• 1. Choose a reference node.
• 2. Assign node voltages to the other nodes.
• 3. Apply KCL to each node other than the
  reference node express currents in terms of node
  voltages.
• 4. Solve the resulting system of linear equations.

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Node Voltages
500W
500W
V1
V2
V3
1
2
3
I1
500W
1kW
I2
500W

• V1, V2, and V3 are unknowns for which we solve
  using KCL.

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Steps of Nodal Analysis

• 1. Choose a reference node.
• 2. Assign node voltages to the other nodes.
• 3. Apply KCL to each node other than the
  reference node express currents in terms of node
  voltages.
• 4. Solve the resulting system of linear equations.

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Currents and Node Voltages
500W
V1
V2
V1
500W
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KCL at Node 1
500W
V1
V2
I1
500W
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KCL at Node 2
500W
500W
V2
V3
V1
1kW
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KCL at Node 3
500W
V2
V3
500W
I2
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Steps of Nodal Analysis

• 1. Choose a reference node.
• 2. Assign node voltages to the other nodes.
• 3. Apply KCL to each node other than the
  reference node express currents in terms of node
  voltages.
• 4. Solve the resulting system of linear equations.

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System of Equations

• Node 1
• Node 2

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System of Equations

• Node 3

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Equations

• These equations can be written by inspection.
• The left side of the equation
• The node voltage is multiplied by the sum of
  conductances of all resistors connected to the
  node.
• Other node voltages are multiplied by the
  conductance of the resistor(s) connecting to the
  node and subtracted.

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Equations

• The right side of the equation
• The right side of the equation is the sum of
  currents from sources entering the node.

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Matrix Notation

• The three equations can be combined into a single
  matrix/vector equation.

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Matrix Notation

• The equation can be written in matrix-vector form
  as
• Av i
• The solution to the equation can be written as
• v A-1 i

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Solving the Equation with MATLAB

• I1 3mA, I2 4mA
• gtgt A 1/5001/500 -1/500 0
• -1/500 1/5001/10001/500 -1/500
• 0 -1/500 1/5001/500
• gtgt i 3e-3 0 4e-3

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Solving the Equation

• gtgt v inv(A)i
• v
• 1.3333
• 1.1667
• 1.5833
• V1 1.33V, V21.17V, V31.58V

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Matrix Refresher

• Given the 2x2 matrix A
• The inverse of A is

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Class Examples