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Lecture 26 2
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.
Lecture 26 5
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
Lecture 26 6
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.
Lecture 26 7
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.
Lecture 26 8
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-express currents in terms of node voltages.
4. Solve the resulting system of linear equations.
Lecture 26 9
Reference Node
The reference node is called the ground node.
+
-
V 500
500
1k
500
500I1 I2
Lecture 26 10
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-express currents in terms of node voltages.
4. Solve the resulting system of linear equations.
Lecture 26 11
Node Voltages
V1, V2, and V3 are unknowns for which we solve using KCL.
500
500
1k
500
500I1 I2
1 2 3
V1 V2 V3
Lecture 26 12
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-express currents in terms of node voltages.
4. Solve the resulting system of linear equations.
Lecture 26 17
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-express currents in terms of node voltages.
4. Solve the resulting system of linear equations.
Lecture 26 18
System of Equations
• Node 1:
• Node 2:
12
1 500500
1
500
1I
VV
0500500
1
k1
1
500
1
5003
21
VV
V
Lecture 26 20
Equations
• These equations can be written by inspection-the left side:
– 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.
Lecture 26 21
Equations
• The right side of the equation:
– The right side of the equation is the sum of currents from sources entering the node.
Lecture 26 22
Matrix Notation
• The three equations can be combined into a single matrix/vector equation.
2
1
3
2
1
0
500
1
500
1
500
10
500
1
500
1
k1
1
500
1
500
1
0500
1
500
1
500
1
I
I
V
V
V
Lecture 26 23
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
Lecture 26 24
Solving the Equation with MATLAB
I1 = 3mA, I2 = 4mA
>> A = [1/500+1/500 -1/500 0;
-1/500 1/500+1/1000+1/500 -1/500;
0 -1/500 1/500+1/500];
>> i = [3e-3; 0; 4e-3];