Series/Parallel Combination
Circuits
Series and Parallel Elements,
Simplification, and Problem Solving
Strategies.
Kevin D. Donohue, University of Kentucky
1
Series and Parallel Elements

Two elements are connected in
series if and only if they have a
common node to which no
other elements are attached.
R1
R2
Is
R3

Two elements are connected in
parallel if and only if they
have common nodes at both
terminals.
Vs
Kevin D. Donohue, University of Kentucky
R1
R2
R3
2
Equivalent Resistance

Series and parallel resistors can be combined into one
element with an equivalent resistance.

Equivalent resistance refers to a single resistance replacing a
multiple resistor subcircuit such that it draws the same
current and produces the same voltage drop.

Apply Ohm’s law to derived formula for finding equivalent
resistance.
Kevin D. Donohue, University of Kentucky
3
Series and Parallel Equivalents
R1
Req  R1  R2    RN
R2
Req

RN
Req 
Req
R1
R2

RN
1
1
1
1


R1 R2
RN
Kevin D. Donohue, University of Kentucky
4
Series and Parallel Sources

Voltage sources in series can be combined into a single
voltage source:
Va
Vb
Va+Vb+Vc
Vc

Current sources in parallel can be combined into a single
current source:
Ia
Ib
Ic
Kevin D. Donohue, University of Kentucky
Ia + Ib + Ic
5
Examples
Simplify complex circuits containing
independent sources and resistors to a
minimum number of equivalent
elements.
Kevin D. Donohue, University of Kentucky
6
Examples

Find voltages and currents in all branches of
circuits containing resistors and independent
sources through simplifying and expanding the
circuit

Voltages and currents in the original circuit can be
determined by


Finding currents and voltages of the simplified circuit
Expanding back to the original configuration while
applying current and voltage division to determine their
current and voltage values.
Kevin D. Donohue, University of Kentucky
7
Single Loop Circuits with Dependent Sources
Since current is the the same throughout the entire
loop, express all unknowns in terms of the loop
current. Find Vo in circuit below:
2
100Vd
120 
+ Vd -
6V
3
Kevin D. Donohue, University of Kentucky
+
Vo
-
8
Single Node-Pair Circuits with Dependent
Sources
Since voltage is the the same across every element,
express all unknowns in terms of the node voltage
difference. Find Io in circuit below:
Id
2
Io
6A
0.8Id
3
Kevin D. Donohue, University of Kentucky
9