+
9V
−
+
9V
−
+
9V
−
+
9V
−
+
9V
−
+
9V
−
www.expresspcb.com
4
5
http://en.wikipedia.org/wiki/Electronic_color_code
We mostly
use the
boxed sizes,
which
increase in
1.5 multiples
6
Color Code Clock
7
Nodal Method for Circuit Analysis, AKA “The Circuit Solution Method of Choice”
A node is an equipotential surface, such the junction of two or more branches (i.e., wires, circuit
elements such as resistors, voltage sources, and current sources, etc.)
The reference (i.e., “ground”) node is the one at which the user defines as have zero voltage.
“Major” nodes are those having three or more non-combinable circuit elements attached to them.
Supernodes are perfect voltage sources having no series resistance, connected directly between two
nodes other than the reference node.
Having solved N equations and N unknowns for voltages at all major nodes, the circuit is considered
“solved,” and all branch voltages and currents can be found afterward.
Procedure:
1. Pick a reference node, “i.e., “ground”, to which all other node voltages are referenced. The
reference node is usually at the bottom of a circuit. Reference node voltage is zero by
definition.
2. Identify major nodes and supernodes, and wrap each in an imaginary surface for KCL equations
so that every “puncture” of the surface is easily identified. Supernodes connected to major
nodes are merged with those major nodes to become one super node.
3. If it helps, branch elements in series can be re-arranged in any order for the purpose of writing
KCL equations at major nodes.
4. Current sources – treat them as injectors (plus or minus) at the two nodes to which they are
connected.
5. Write Kirchhoff current law equations (KCL) for each major node, except the reference (because
its voltage is already zero).
6. Place the KCL equations in standard matrix form, and solve for major node voltages using
Gaussian Elimination.
Demonstrate during lecture using Figs. 17.22 through 17.25.
PC Resistor Board Lab, Due Wednesday, Nov. 5, 2014
Step 1. Solder resistors to board. First ohm color bar should be at the highest point.
Step 2. Measure the resistance from Node A to Node G and compare to calculations.
Measured results should be within 5% of calculated.
Step 3. Solder the power jack, energize with a 12Vdc wall wart, and
check that each resistor has at least 1/2 volt across it.
Step 4. Unplug the wall wart from your circuit.
Step 5. Measure the resistance from your assigned to every other node.
put the values in a matrix row which has columns A,B, …, through L.
Step 6. Compute the above resistances and place the calculations in a second row,
directly under the first row.
Step 7. A third row should show the error, (computed minus measured)/(measured)
multiplied by 100 to yield % error
Step 8. Plug in the wall wart to your circuit.
Step 9. Repeat Steps 5.6.7 using voltage instead of resistance. Your assigned row
is the voltage reference for your measurements.
Step 10. Using the right hand side of the circuit, measure and compute voltages
across individual resistors. Compare to calculations and compute the errors
as you did in previous steps.
A linear circuit can be represented by a Thevenin Equivalent
The three cases to consider are
 Case 1. All sources are independent
 Case 2. The circuit has dependent sources and independent
sources
 Case 3. The circuit has only dependent sources
Depending on the case, one or more of the following methods can be
used to find the Thevenin equivalent:
Direct Rth. (Applies only to Case 1).
 Turn off all independent sources (i.e., set V = 0 for voltage
sources, and I = 0 for current sources). Note - this is the
same thing as replacing voltage sources with short circuits,
and current sources with open circuits.
 Connect a fictitious ohmmeter across terminals a-b, and
“measure” Rth directly.
 Find Isc (or, alternatively, find Voc = Vth ).
 Compute Vth = Voc = Isc • Rth (or, alternatively, Isc = Voc /
Rth ).
 If time permits, find Vth (or, alternatively, Isc ) directly from
the circuit, and then double-check with the above.
Voc , Isc (Applies to Cases 1 and 2).
 Find Voc = Vth .
 Find Isc .
 Compute Rth = Vth / Isc .
Fictitious Source (Applies to all Cases)
 Attach a fictitious source Vab across terminals a-b. Find a
linear equation with the following form: Vab  A  BI ab .

By definition the linear equation must match Thevenin
equation Vab  Vth  Rth I ab , term by term. Thus,
matching the terms yields Vth  A, Rth  B .
Case
Direct Rth.
Voc , Isc
Case 1
Case 2
Case 3
OK
OK
OK
Fictitious
Source
OK
OK
OK
Iab
+
Vth
Rth
Vab  Vth  Rth I ab
–
Note – For Case 3, the Vth should be zero. Thus, for Case 3, you
can attach any voltage source Vab to the output (e.g., Vab = 1V),
find I ab , and compute Rth 
 Vab
.
I ab
Example: If you connect an
external circuit to terminals E
and F, the entirety of the PC
board and power supply can be
represented by a Thevenin
equivalent circuit.
Current
R
Rth
Thevenin
equivalent of
PC Board and
12V power
supply
E
Same current
R
Vth
F
12V
+
Vth = VEF voltmeter
reading (or calculation)
−
12V
Isc is short circuit
current through an
ammeter
12V
0V is having the
source turned off,
which is not the same
as source unplugged.
If the voltage source is
“stiff,” 0V is
approximated by
unplugging the source
and shorting
0V
connecting the
voltage terminals
together. If not “stiff,”
add the source
resistance to Rth.
+
Vth = VEF voltmeter
reading (or calculation)
−
Rth = E to F ohmmeter
reading (or calculation)
Part 1. 12V disconnected. For each of
the following, compute the resistance
displayed by an ohmmeter connected
Between B and D,
Between K and F,
Between D and E.
Part 2. 12V connected. For each of the
following, compute the voltage
displayed by a voltmeter connected
Red probe on B, black probe on D,
Red probe on K, black probe on F,
Red probe on D, black probe on E.
Part 3. Determine the Thevenin
equivalent circuit for the PC Board, with
12V connected, as seen by an external
circuit connected between D and E.
Name: ____________________________
Part 1. 12V disconnected. For each of
the following, compute the resistance
displayed by an ohmmeter connected
Between B and D,
Between K and F,
Between D and E.
Part 2. 12V connected. For each of the
following, compute the voltage
displayed by a voltmeter connected
Red probe on B, black probe on D,
Red probe on K, black probe on F,
Red probe on D, black probe on E.
Part 3. Determine the Thevenin
equivalent circuit for the PC Board, with
12V connected, as seen by an external
circuit connected between D and E.
Name: ____________________________
Part 1. 12V disconnected. For each of
the following, compute the resistance
displayed by an ohmmeter connected
Between B and D,
Between K and F,
Between D and E.
Part 2. 12V connected. For each of the
following, compute the voltage
displayed by a voltmeter connected
Red probe on B, black probe on D,
Red probe on K, black probe on F,
Red probe on D, black probe on E.
Part 3. Determine the Thevenin
equivalent circuit for the PC Board, with
12V connected, as seen by an external
circuit connected between D and E.
Name: ____________________________