Download Nodal Analysis

Objective of Lecture
 Provide step-by-step instructions for nodal analysis,
which is a method to calculate node voltages and
currents that flow through components in a circuit.
 Partly covered in Chapter 5.5 Electric Circuits
Fundamentals
 Chapter 9.4 Principles of Electric Circuits
 Chapter 3.2 and 3.3 Fundamentals of Electric Circuits
 Chapter 2.4 Electrical Engineering: Principles and
Applications
Nodal Analysis
 Technique to find currents at a node using Ohm’s Law,
Kirchhoff’s Current Law, and the potential differences
betweens nodes.
 First result from nodal analysis is the determination of
node voltages (voltage at nodes referenced to ground).

These voltages are not equal to the voltage dropped across the
resistors.
 Second result is the calculation of the currents
Steps in Nodal Analysis
Vin
Steps in Nodal Analysis
 Pick one node as a reference node
 Its voltage will be arbitrarily defined to be zero
Vin
Step 1
 Pick one node as a reference node
 Its voltage will be arbitrarily defined to be zero
Vin
Step 2
 Label the voltage at the other nodes
Vin
Step 2
 Label the voltage at the other nodes
Vin
Step 3
 Label the currents flowing through each of the
components in the circuit
Step 4
 Use Kirchhoff’s Current Law
I 7  I1  I 2  I 6
I 2  I3  I 4
I 4  I5
I3  I5  I6
Step 5
 Use Ohm’s Law to relate the voltages at each node to
the currents flowing in and out of them.
 Current flows from a higher potential to a lower
potential in a resistor

The difference in node voltage is the magnitude of
electromotive force that is causing a current I to flow.
I  Va  Vb  R
Step 5
We do not write an equation for
I 7 as it is equal to I1
I1  V1  V2  R1
I 2  V2  V3  R2
I 3  V3  V5  R3
I 4  V3  V4  R4
I 5  V4  V5  R5
I 6  V5  0V  R6
Step 6
 Solve for the node voltages
 In this problem we know that V1 = Vin
Step 6
 Substitute the equations obtained using Ohm’s Law
into the equations obtained using KCL.
Vin  V2 
R1  V2  V3  R2  V5 R6
V2  V3 
R2  V3  V5  R3  V3  V4  R4
V3  V4 
R 4  V4  V5  R5
V3  V5 
R3  V3  V4  R4  V5 R6
Step 7
 Once the node voltages are known, calculate the
currents.
From Previous Slides
I 7  I1  I 2  I 6
I 2  I3  I 4
I 4  I5
I3  I5  I6
V 1  Vin
I1  V1  V2  R1
I 2  V2  V3  R2
I 3  V3  V5  R3
I 4  V3  V4  R4
I 5  V4  V5  R5
I 6  V5  0V  R6
Substituting in Numbers
I 7  I1  I 2  I 6
I 2  I3  I 4
I 4  I5
I3  I5  I6
V 1  10V
I1  10V  V2  9k
I 2  V2  V3  2k
I 3  V3  V5  5k
I 4  V3  V4  3k
I 5  V4  V5  1k
I 6  V5  0V  7 k
Substituting the results from
Ohm’s Law into the KCL equations
10V  V2  9k  V2  V3  2k  V5
7 k
V2  V3  2k  V3  V5  5k  V3  V4  3k
V3  V4  3k  V4  V5  1k
V3  V5  5k  V4  V5  1k  V5  7k
Chugging through the Math
Node Voltages
(V)
V1
10
V2
5.55
V3
4.56
V4
3.74
V5
3.46
 Node voltages must have a magnitude less than the sum of the
voltage sources in the circuit
 One or more of the node voltages may have a negative sign
 This depends on which node you chose as your reference node.
Chugging through the Math
Voltage across
resistors
(V)
VR1 = (V1 – V2)
VR2 = (V2 – V3)
4.45
0.990
VR3 = (V3 – V5)
VR4 = (V3 – V4)
1.10
0.824
VR5 = (V4 – V5)
0.274
VR6 = (V5 – 0V)
3.46
 The magnitude of any
voltage across a resistor
must be less than the
sum of all of the voltage
sources in the circuit.
 In this case, no voltage
across a resistor can be
greater than 10V.
Chugging through More Math
Currents
(mA)
I1
I2
495
495
I3
I4
220
275
I5
I6
275
495
I7
495
Check
 None of the currents should be larger than the current
that flows through the equivalent resistor in series
with the 10V supply.
 Note that this check is only valid if there is one voltage
source in the circuit.
Req  9k  2k  5k 3k  1k   7k
Req  20.2k
I eq  10V Req  495mA  0.495mA
Summary
Steps in Nodal Analysis
1. Pick one node as a reference node
2. Label the voltage at the other nodes
3. Label the currents flowing through each of the
components in the circuit
4. Use Kirchhoff’s Current Law
5. Use Ohm’s Law to relate the voltages at each node to the
currents flowing in and out of them.
6. Solve for the node voltage
7. Once the node voltages are known, calculate the
currents.