electronics fundamentals
circuits, devices, and applications
THOMAS L. FLOYD
DAVID M. BUCHLA
Chapter 6 – Series and Parallel Combination Circuits
Series and Parallel Circuits Identifying series-parallel relationships
Most practical circuits have combinations of series and
parallel components.
From Chapters 4 and 5
Components that are connected in
series will share a common path.
Components that are connected in
parallel will be connected across
the same two nodes.
1
2
Series and Parallel Circuits
Combination circuits
Circuits containing both series and parallel
circuits are called COMBINATION circuits
You can frequently simplify analysis by combining
series and parallel components.
Solve by forming the simplest equivalent circuit possible.
An equivalent circuit is one that has:
1. characteristics that are electrically the same as
another circuit but
2. is generally simpler.
Series and Parallel Circuits
Equivalent circuits
R1
1.0 k
R2
is equivalent to
R1
2.0 k
1.0 k
There are no electrical measurements that can
distinguish the boxes.
Series and Parallel Circuits
Equivalent circuits
Another example:
is equivalent to
R1
R2
1.0 k 1.0 k
R1,2
500 
There are no electrical measurements that can
distinguish the boxes.
Series and Parallel Circuits
Equivalent circuits
is equivalent to
R1
1.0 k
R3
R 1,2
R3
R2
4.7 k
3.7 k
4.7 k
2.7 k
is equivalent to
R1,2,3
2.07 k
There are no electrical
measurements that can
distinguish between the
three boxes.
Series and Parallel Circuits
What Do We Know
1. For Series Circuits:
a. Current at all points is the same
IT = IR1 = IR2
b. Voltage drops across each resistor
VT = V 1 + V2
2. For Parallel Circuit
a. Current divides across each resistor in the branch
IT = IR1 + IR2
b. Voltage at all nodes is the same
VT = V1 = V2
Series and Parallel Circuits Seven Step Process for Solving a
Combination Circuit
1. Simplify the circuit to a series circuit by finding the effective
equivalent resistance (REQ) of each parallel section in the
circuit. Redraw the simplified circuit.
2. Calculate the total resistance (RT) of the circuit by adding all
REQ’s to the other series resistances.
3. Calculate the total current (IT) using RT in Ohm’s law.
4. Calculate the voltage drop across any series resistances or
REQ’s using Ohm’s law.
5. Calculate the branch currents in all parallel sections of the
circuit using the voltage drop across REQ and Ohm’s law.
6. Use the branch currents and resistance values to calculate
the voltage of the parallel resistances.
7. Make a summary of the voltage drops and currents for each
resistance to make sure they total correctly.
Series and Parallel Circuits
A simple series-parallel resistive
circuit.
Series and Parallel Circuits
R4 is added to the circuit in
series with R1.
Series and Parallel Circuits
R5 is added to the circuit in
series with R2.
Series and Parallel Circuits
R6 is added to the circuit in
parallel with the series
combination of R1 and R4.
Series and Parallel Circuits
Sketch the circuits for nodes
A to B
A to C
B to C
Series and Parallel Circuits
Sketch the circuits for nodes
A to B
RT  R1 ( R2  R3)
Series and Parallel Circuits
Sketch the circuits for nodes
A to C
RT  R3 ( R1  R2)
Series and Parallel Circuits
Sketch the circuits for nodes
B to C
RT  R2 ( R1  R3)
Series and Parallel Circuits
R ( AB)  ( R5 ( R 6  R 3)) R 2 ( R1  R 4)  R 7
Series and Parallel Circuits
Reduce the following circuit to REQ
Series and Parallel Circuits
REQ( R3 R4) 
6Ω
Series and Parallel Circuits
REQ( R3 R4)  R6 
10 Ω
Series and Parallel Circuits
( REQ( R3 R4)  R6) R5  5 Ω
Series and Parallel Circuits
IT
((REQ( R3 R4)  R6) R5)  R1  R2 
IT = 2 amps
50 Ω
Series and Parallel Circuits
Review of voltage relationships.
Series and Parallel Circuits
Kirchoff’s current law
What are the readings for node A?
I
+
I
-
26.5 mA
+
A
I
VS
5.0 V
+
+
8.0 mA
R2
470 
R4
-
18.5 mA
100 
R1
270 
R3
330 
R6
100 
R5
100 
Series and Parallel Circuits
The voltage-divider equation
was developed for a series
circuit. Recall that the output
voltage is given by
 R2
V2  
 RT
Loaded voltage divider
+
R1
A
R2
R3

 VS

1. A voltage-divider with a resistive load forms a
combination (parallel) circuit.
2. The voltage divider is said to be LOADED.
3. The loading reduces the total resistance from node A to
ground.
Series and Parallel Circuits
Loaded voltage divider
What is the voltage
across R3?
VS =
+15 V
Form an equivalent series
circuit by combining R2 and
R3; then apply the voltagedivider formula to the
equivalent circuit:
R1
330 
R2
470 
A
R3
2.2 k
R2,3  R2 R3  470  2.2 k = 387 
V3  V2,3
 R2,3 
387 



VS  

15 V  8.10 V
R R 
 330   387  
2,3 
 1
Series and Parallel Circuits
The loading effect of a voltmeter.
Series and Parallel Circuits
Given: VS = 10 V and R1
and R2 are not defective but
the meter reads only 4.04 V
when it is across either R1
or R2.
What is a possible
explanation of the meter
not displaying 10 volts?
Loading effect of
a voltmeter
VS +
10 V
R1
470 k
+
R2
470 k
+
4.04
10 VV
4.04 V
1. A voltmeter has internal resistance
2. This RINT can change the resistance of the circuit under test.
3. A 1 M internal resistance of the meter accounts for the
readings.
.
VS
R3
R1
Output
-
• The Wheatstone bridge
consists of:
1. a dc voltage source and
2. four resistive arms
forming two voltage
dividers.
• The output is taken between
the dividers.
• Frequently, one of the bridge
resistors is adjustable. (R2)
Wheatstone bridge
+
Series and Parallel Circuits
R2
R4
Series and Parallel Circuits
Balanced Wheatstone bridge
When the
R
bridge is
V
Output
balanced, the
R
output voltage
is zero
The products of resistances in the
opposite diagonal arms are equal.
1
+
-
S
R3
2
R4
Series and Parallel Circuits
Wheatstone bridge.
Voltage
Divider 1
Voltage
Divider 2
Series and Parallel Circuits
 R2 
R1  R3 
 R4 
Wheatstone bridge.
V1=V2
I1=I3
V3=V4
I2=I4
Series and Parallel Circuits
12
V
470 
330 
Output
-
 R2 
R1  R3 
 R4 
VS
R3
R1
+
What is the
value of R2 if
the bridge is
balanced? 384 
Wheatstone bridge
R2
R4
270 
Series and Parallel Circuits
Finding an Unknown Resistance
 R2 
RX  RV  
 R4 
Scale Factor
Series and Parallel Circuits
Unbalanced Wheatstone Bridge
• Unbalance occurs when
VOUT ≠ 0
• Used to measure
1. Mechanical Strain
2. Temperature
3. Pressure
• VOUT is converted to a
digital output indicating
the value of the reading.
Measuring a physical parameter using a transducer.
Series and Parallel Circuits
VOUT = VA-VB = 0
 RX 
VX   VS
 RT 
Series and Parallel Circuits
10
10
10
10
10
Using the Thermistor chart,
what is VOUT when the
temperature is 50o C
VA=8.8v
VB=6.0v
VA-B=2.8v
Series and Parallel Circuits
Example of a load cell.
Series and Parallel Circuits
Wheatstone Bridge
1. Remove RL to make an
open circuit between A & B
2. Calculate R1||R3 = 165 
3. Calculate R2||R4 = 179 
4. Calculate the voltage from
A to ground 7.5 V
5. Calculate the voltage from
B to ground 6.87 V
VS
+15 V
+
-
R1
R2
330 
390 
RL
A
B
150 
R3
R4
330 
330 
Series and Parallel Circuits
Maximum power transfer Theorem
The maximum power is transferred from a source to a
load when: RL = RS(resistance of the voltage source)
RS
VS +
RL
The maximum power transfer theorem assumes the
source voltage and resistance are fixed.
Series and Parallel Circuits
Maximum power transfer Theorem
What is the power delivered to the load?
RS
VS +
10 V
The voltage to the
load is 5.0 V
The power delivered is:
5.0 V 

V
PL 

= 0.5 W
RL
50 
2
2
50 
RL
50 
Series and Parallel Circuits
Superposition theorem
A way to determine currents and voltages in a
linear circuit that has multiple sources
1. Take one source at a time and
2. Algebraically summing the results.
R3
R1
+
-
VS2
18 V
R2
6.8 k
-
+
-
VS1
12 V
6.8 k
I2
+
2.7 k
Series and Parallel Circuits
Superposition theorem
Four Step Process
1. Leave one voltage or current source at a
time in the circuit (circuit 1) and replace
the other (circuit 2) with a short.
2. Calculate REQ/Total and then calculate the
voltage or current for the resistor(s).
3. Repeat steps 1 and 2 for the other circuit
(circuit 2).
4. Algebraically add the results for all
sources.
Series and Parallel Circuits
Series and Parallel Circuits
Summary
-
VS1
12 V
6.8 kk
k

6.8
VS2S2
-++1.56 mA
18 V
RR222
6.8 kk
k

6.8
II222
-
Source 1:
Source 2:
2.7

2.7 kkk
+
Set up a table of
pertinent information
and solve for each
quantity listed:
RR333
RR111
++
What does the ammeter
read for I2?
RT(S1)= 6.10 k I1= 1.97 mA I2= 0.98 mA
RT(S2)= 8.73 k I3= 2.06 mA I2= 0.58 mA
Both sources
The total current is the algebraic sum.
I2= 1.56 mA
Series and Parallel Circuits
Thevenin’s theorem and Wheatstone Bridge
RTH A RL
VTH
7.5 V
165 
150 
B RTH'
179 
VTH'
6.87 V
Putting the load on the Thevenin
circuits and applying the superposition
theorem allows you to calculate the load
current. The load current is: 1.27 mA
.0152-.0191
Series and Parallel Circuits
Troubleshooting
The effective troubleshooter must think logically
about circuit operation.
Understand normal circuit operation
and find out the symptoms of the failure.
Decide on a logical set of steps to find the
fault.
Following the steps in the plan, make
measurements to isolate the problem.
Modify the plan if necessary.
Series and Parallel Circuits
The output of the voltagedivider is 6.0 V. Describe
how you would use
analysis and planning in
finding the fault.
Troubleshooting
VS =
+15 V
R1
330 
R2
470 
A
R3
2.2 k
Decide
From an
onearlier
a logical
calculation,
set of steps
V3 to
should
locateequal
the fault.
8.10 V. A
You
low could
voltage
decide
is most
to 1)
likely
check
caused
the source
by a low
voltage,
source2)
disconnect
voltage or incorrect
the load and
resistors
check(possibly
the outputR1voltage,
and R2 and
ifreversed).
it is correct,
If the
3) circuit
check the
is new,
loadincorrect
resistance.
components
If R3 is
correct,
are possible.
check other resistors.
Series and Parallel Circuits
FIGURE 6–97
Series and Parallel Circuits
Selected Key Terms
Loading The effect on a circuit when an element that
draws current from the circuit is connected
across the output terminals.
Load current The output current supplied to a load.
Bleeder The current left after the load current is
current subtracted from the total current into the circuit.
Wheatstone A 4-legged type of bridge circuit with which an
bridge unknown resistance can be accurately measured
using the balanced state. Deviations in resistance
can be measured using the unbalanced state.
Series and Parallel Circuits
Selected Key Terms
Thevenin’s A circuit theorem that provides for reducing
theorem any two-terminal resistive circuit to a single
equivalent voltage source in series with an
equivalent resistance.
Maximum power The condition, when the load resistance
transfer equals the source resistance, under which
maximum power is transferred to the load.
Superposition A method for analyzing circuits with two or
more sources by examining the effects of each
source by itself and then combining the
effects.
Series and Parallel Circuits
1. Two circuits that are equivalent have the same
a. number of components
b. response to an electrical stimulus
c. internal power dissipation
d. all of the above
Series and Parallel Circuits
2. If a series equivalent circuit is drawn for a
complex circuit, the equivalent circuit can be
analyzed with
a. the voltage divider theorem
b. Kirchhoff’s voltage law
c. both of the above
d. none of the above
Series and Parallel Circuits
3. For the circuit shown,
a. R1 is in series with R2
d. R2 is in parallel with R3
-
c. R2 is in series with R3
VS
+
b. R1 is in parallel with R2
R1
R2
R3
Series and Parallel Circuits
4. For the circuit shown,
R4
a. R1 is in series with R2
R1
b. R4 is in parallel with R1
-
d. none of the above
+
c. R2 is in parallel with R3
VS
R2
R3
Series and Parallel Circuits
5. A signal generator has an output voltage of 2.0 V with no
load. When a 600  load is connected to it, the output
drops to 1.0 V. The Thevenin resistance of the generator is
a. 300 
b. 600 
c. 900 
d. 1200 .
Series and Parallel Circuits
6. For the circuit shown, Kirchhoff's voltage law
a. applies only to the outside loop
b. applies only to the A junction.
c. can be applied to any closed path.
d. does not apply.
VS +
10 V
R1
270 
A
R2
330 
R3
470 
Series and Parallel Circuits
7. The effect of changing a measured quantity due to
connecting an instrument to a circuit is called
a. loading
b. clipping
c. distortion
d. loss of precision
Series and Parallel Circuits
8. An unbalanced Wheatstone bridge has the voltages
shown. The voltage across R4 is
a. 4.0 V
d. 7.0 V
-
c. 6.0 V
VS
12 V
+
b. 5.0 V
R1
7.0 V
R2
R3
+ RL 1.0 V
R4
Series and Parallel Circuits
9. Assume R2 is adjusted until the Wheatstone bridge is
balanced. At this point, the voltage across R4 is measured
and found to be 5.0 V. The voltage across R1 will be
a. 4.0 V
d. 7.0 V
+ RL -
-
c. 6.0 V
VS
12 V
+
b. 5.0 V
R3
R1
R2
R4
5.0 V
Series and Parallel Circuits
10. Maximum power is transferred from a fixed source
when
a. the load resistor is ½ the source resistance
b. the load resistor is equal to the source resistance
c. the load resistor is twice the source resistance
d. none of the above
Series and Parallel Circuits
Quiz
Answers:
1. b
6. c
2. c
7. a
3. d
8. a
4. d
9. d
5. b
10. b