UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE
Department of Electrical and Computer Engineering
Experiment No. 8 -Network Analysis
Overview:
The ability to analyze complex circuits is a necessary skill for design engineers. The purpose of
this experiment is to mathematically analyze an electrical network, build and test the network,
and then compare measured and calculated data. Four useful tools for network analysis are
described in this laboratory. They are:
1- Kirchhoff’s laws
2- Mesh analysis, a method based on Kirchhoff’s voltage law
3- Nodal Analysis, a method based on Kirchhoff’s current law
4- Superposition, a frequently used method when more than one source is present
1 - Kirchhoff’s laws
Most of the circuit problems we encounter can be solved by repeatedly applying the rules for
adding resistors in series or parallel, until the problem has been reduced to one of a battery
connected to a single resistor.
To solve more complex circuit problems, such as those with more than one battery, it is
sometimes necessary to write equations based on Kirchhoff’s Laws, which are formal
mathematical statements of two physical facts that you already know:
Kirchhoff’s law #1 (KVL) stands for Kirchhoff’s voltage-law: it states that the sum of voltage
changes around any closed path in a circuit will be zero. The voltage change when moving
across a battery from the - terminal to + terminal is positive (a rise in potential), and the voltage
change when moving across a resistor in the direction of conventional current flow is negative (a
drop in potential). Mathematically we have
∆Vn) = 0
Σn (∆
The first law simply restates what you already know about electrical potential: every point in a
circuit has a unique value of potential, so traveling around the circuit by any path must bring
you back to the potential you started from. Using the analogy to elevation, if you hike from any
starting point in the mountains and wander around by any choice of paths but finish at your
original starting point, the sum of the elevation changes along your path will add up to zero (i.e.,
the sum of ups will equal the sum of downs).
Kirchhoff’s law #2 (KCL) stands for Kirchhoff’s current-law: it states that the sum of all
currents entering a node (i.e., any junction of wires) will be equal to zero. A current leaving a
node is equivalent to a negative current entering that node. Mathematically we have
Σn (ΙΙn) = 0
The second law simply restates the fact that electric charge is conserved: electrons or protons are
not being created or destroyed in the node (or if they are, anti-particles with the opposite charge
are being created or destroyed along with them). A node is assumed to have negligible
capacitance, so that charge cannot just build up there. Thus, in any given time interval, the
charge that enters a node is equal to the charge that leaves the node, and likewise, the current that
enters a node is equal to the current that leaves the node. For example, at a point where three
wires are connected as in the diagram below, charge conservation requires that i1 = i2 + i3.
Figure 1. Node Currents
2 - Mesh Current Analysis
How do you do it?
1. Label each mesh (window pane) of the circuit with a mesh current, going in the
clockwise direction.
2. Write KVL equations for each mesh. Each term is a voltage, but you will write the terms
using Ohm’s law to put them in terms of the mesh currents (V=IR).
3. Solve the N equations in N unknowns. This will give you all the mesh current values. If
you are asked to find a voltage just use Ohm’s law to find the voltage in one more step.
Example: For the analysis of this circuit, you must write a KVL equation for each mesh. Around
each mesh the sum of voltage drops are equated to the sum of voltage rises. Recall that the
voltage across a resistor is IR, and this change is negative when moving in the direction of
conventional current flow.
2
Figure 2. Example Circuit for mesh analysis.
What are some difficulties I might run into?
If there is a current source between two meshes, then you must add an extra variable (call it Vx,
the voltage across the source). Then you will need an extra equation as well. The extra equation
is simply that the difference between the two mesh currents is the value of the source.
A dependent current source can be dealt with in exactly the way as the dependant voltage source.
Simply write the voltages as you normally would for an independent source (but writing down
the dependency for the value). Then write an extra equation to describe the dependency in terms
of the mesh currents.
Example: In the circuit below, a mesh has a current source in an outside branch. For the analysis
of this circuit, you must write a KCL equation relating the mesh current to the current source.
Figure 3. Example Circuit for mesh analysis when a constant current source is present.
3
3 - Nodal Analysis
This is a very powerful tool for analyzing complex circuits and it is based on Kirchhoff’s voltage
law: the algebraic sum of voltage rises and drops around a complete loop is zero.
How do you do it?
1. Select a reference node. Label it “REF’. If there is a ground node in the circuit you may
choose that as the reference, but it is not required. It is often advantageous to select the
node with the most wires coming into it.
A “node” is any point where three or more circuit branches come together. Points where
only two circuit branches come together may also be labeled (as auxiliary nodes), but this
is not required. It is often helpful in certain “difficult” circuits. Be careful! Just because
there is a “dot” showing wires coming together doesn’t mean it is a distinct node. Two
“dots” connected by a short (a wire) are the same node!
2. Label the other nodes in the circuit (V1, V2, etc.).
3. Write KCL equations for each node (in terms of the node voltages). Each term in the
equation should be a current, but the term is written using ohm’s law (I=V/R) so that they
contain the node voltages you labeled in step 2. If you write all the currents leaving a
node, then the node you are focusing on will show up positive in that equation and all
other node voltages will show up negative. The sum of all currents leaving equals zero
(don’t forget to write the = 0).
4. Solve the N equations in N unknowns. This will give you all the node voltage values. If
you are asked to find a current, just use Ohm’s law to find the current in one more step.
Example: For each node in the circuit below, the sum of currents entering the node must be
equated to the sum of currents leaving the node.
Figure 4: Example circuit for nodal analysis
4
The following three equations describe the nodal analysis. From these equations V1, V2 and V3
can be determined:
V1 = −4v
− V2 V3 − V2
+
=2
8
4
V3 − V2 V3 − 4
+
=1
4
1
What are some difficulties I might run into?
If there is a voltage source connecting two nodes (with no resistance), then you must add an extra
variable (call it Ix, the current through the source). Then you will need an extra equation as well.
The extra equation is simply that the difference between the two node voltages is the value of the
source.
A dependent source is slightly different. Simply write the currents as you normally would for an
independent source (but writing down the dependency for the value). Then write an extra
equation to describe the dependency in terms of the node voltages.
A capacitor or inductor can be a headache when writing the time-dependent node voltage
equations. If the element is in series with another element in a branch, this is a good time to use
an auxiliary node.
4-Superposition
Superposition is a process for calculating currents and/or voltages for a component in a
circuit that has more than one source. Contributions to a particular current or voltage are
calculate separately for each source and then algebraically added to give the final value.
The method of superposition is valid only for linearly related quantities, and for example,
is not an acceptable method for the calculation of power, a quantity proportional to the
square of current or voltage.
The superposition techniques involve the following steps:
Step 1
Remove all sources except one. It makes no difference which source you elect since
eventually each existing source will be the stand-alone source before the analysis is
complete. Replace the removed sources with their internal resistance (i.e., a short for the
ideal voltage source and an open for the ideal current source). Note: for this lab you will
use only voltage sources with a resistance of zero and will be instructed to replace these
sources with a short. Calculate the current(s) and/or voltage(s) with the one remaining
5
source in the circuit for the resistor(s) in which you have an interest. Record the amount
and direction of current and/or the magnitude and polarity of voltage across each resistor
of interest.
Step 2
Remove the source used in Step 1 and replace another source that was previously
removed. Calculate the current(s) and/or voltage(s) of interest, recording directions,
polarities and magnitudes of the current and voltage of interest.
Step 3
Repeat Step 2 until all sources in the original circuit have been used.
Step 4
The actual current and/or voltage for any one resistor will be the algebraic sum of the
currents and/or voltages found above for that particular resistor.
Percent Error = (Measured value – Calculated value) x 100%
Calculated value
6
Pre lab — Network Analysis
1. For the circuit in Figure 5, use mesh analysis to calculate the mesh currents. Call your
currents IA, IB, and IC. Record these currents (magnitude and direction) in Table 2.
2. Having determined all mesh currents, calculate the current through (magnitude &
direction) and the voltage across (magnitude & polarity) for each resistor. Record these
values in the Tables 3 & 4.
3. Write and solve node voltage equations for the circuit of Figure 5, and then, solve for the
current through (magnitude and direction) and the voltage across (magnitude and
polarity) for each resistor. Record these values in Tables 3 & 4.
4. Perform a computer simulation of the circuit using PSpice, and use the results to find the
current through (magnitude and direction) and the voltage across (magnitude and
polarity) for each resistor. Record these values in Tables 3 & 4. Print the simulation, and
submit it with this report. All current curves should be seen on one plot and all voltage
curves on a separate plot.
5. Refer to Figure 5. Use the following steps to calculate the current through and voltage
across R2 (the resistor of interest) by superposition techniques.
Step 1
V1 is present, V2 is removed. Referring to Figure 5, draw a new circuit with V2 removed and
replaced by a short (this assumes that the voltage source has zero internal resistance). In this
new circuit solve for the magnitude and direction of the current through R2. Determine the
magnitude and polarity of the voltage across R2. On this new circuit record the current
(magnitude and direction) and the voltage (magnitude and polarity) for R2. Also record these
values below. They represent the contribution from source 1.
V2 removed:
I R2 = _____________ VR2 = _____________
Step 2
V2 is present, V1 is removed. This time, referring to Figure 5, draw another circuit with V2
remaining and V1 removed and replaced by a short. As before calculate the current and voltage
for R2. Record these values on your second circuit and below. They represent the contribution
from source 2.
V1 removed:
I R2 = _____________VR2 = _____________
Step 3 The actual current through and voltage across R2 WILL BE the ALGEBRAIC SUM of
the results obtained in steps 1 and 2. Do these summations below and record the resultant current
(magnitude and direction) and voltage (magnitude and polarity) on Figure 5 and in Tables 3 & 4.
It is not required to do superposition calculations for any resistor other than R2.
Algebraic addition: VR2 = _________ + _________ = _______
I R2 = _________ + _________ = _______
(INSTRUCTOR’S SIGNATURE_______________________________DATE_____________)
7
Lab Session — Network Analysis
1. Show your pre-lab work to the instructor at the beginning of the lab session
2. Select and measure resistors for your circuit (see Fig. 5). Record the measured and colorcode values in Table 1.
3. Build the circuit on a protoboard and turn on the power supplies.
4. Measure the voltage across each resistor in your circuit, and record the values in Table 3.
Show the proper polarity for each voltage on Figure 5. Now, using these measured
voltages, calculate the current through each resistor. Record the values in Table 4 and
show the proper directions on Figure 5.
5. Measure the “mesh” currents IA, IB, and IC. Record these measured currents (magnitude
and direction) in Table 2.
Figure 5: Circuit for testing.
8
Data Sheet – Network Analysis
Table 1 Resistors
Mesh Resistance
R1
R2
R3
R4
R5
Measured Resistance (ohms)
Color Code Resistance (ohms)
Error (%)
Calculated Current (amp)
Error (%)
Table 2 Mesh Currents
Mesh Current
IA
IB
IC
Measured Current (amp)
Table 3 Component Voltages
Resistor
Voltages
VR1
VR2
VR3
VR4
VR5
Measured
(volt)
Mesh Method
(volt)
PSpice (volt)
Nodal Method Superposition
(volts)
(volt)
Mesh Method
(amp)
PSpice (amp)
Nodal Method Superposition
(amps)
(amp)
Table 4 Component Currents
Resistor
Currents
IR1
IR2
IR3
IR4
IR5
Measured
(amp)
INSTRUCTOR'
S INITIALS
DATE:
9
Post Lab – Network Analysis
1. Your report should include:
a. All tables and calculations including data signed by instructor.
b. All PSpice plots
c. Discussion on the validity of Mesh Analysis, Nodal Analysis, and superposition.
10