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electronics fundamentals
circuits, devices, and applications
THOMAS L. FLOYD
DAVID M. BUCHLA
Chapter 5 – Parallel Circuits
Parallel Circuits
Resistors that are connected to the
same two points are said to be in
parallel.
A
R1
B
R2
R3
R4
Parallel Circuits
A parallel circuit occurs if:
1. more than one current path (branch) is
connected to a common voltage source.
2. The voltage between the two points appears
across each of the branches.
VS
+
R1
R2
R3
R4
Parallel Circuits
Parallel Circuits
Parallel Circuits
Parallel Circuits
Parallel Circuits
Parallel Circuits
Draw the parallel circuits shown below as
schematics
Parallel Circuits
Connect the groups of resistors to make
parallel circuits.
Parallel Circuits
VS
IT 
RT
VS  IT  RT
VS
RT 
IT
Parallel Circuits
Connecting resistors in parallel
1. reduces total resistance and
2. increases total current.
60 amp
24 v
VS
RT 
IT
120 amp
24 v
24
RT 
60
24
RT 
120
Parallel Circuits
From Chapter 2, what is Conductance and its unit
Conductance (C) is:
1. ease with which current can pass
2. the reciprocal of resistance,
3. Siemens
1
GT 
RT
Parallel Circuits
Parallel circuit rule for resistance
The total resistance of resistors in parallel is
the reciprocal of the sum of the reciprocals of
the individual resistors.
For example, the resistors in a parallel circuit are 680 W,
1.5 kW, and 2.2 kW. What is the total resistance? 386 W
VS
+
R1
680 W
R2
1.5 kW
R3
2.2 kW
Parallel Circuits
The total resistance of resistors in parallel is
the reciprocal of the sum of the reciprocals of
the individual resistors.
1
GT = G1 + G2 + G3 + … + GN RT  1
1
1
1


 ... 
R1 R 2 R 3
RN
Parallel Circuits
What is RT (REQuivalent) for this circuit
200
3500
50
RT = 39.55 ohms
Parallel Circuits
Special formula for calculating
resistance of two parallel resistors
The resistance of two parallel
resistors can be found by
either:
RT 
1
1 1

R1 R2
or
R1
R2
R1 R2
RT 
R1  R2
Total resistance of two resistors in parallel is equal the
product of the two resistors divided by the total resistance.
What is the total resistance if R1 = 27 kW and
R2 = 56 kW? 18.2 kW
Special formula for calculating
Parallel Circuits resistance for equal resistors in parallel
500
500
500
R
RT 
n
What is the value of RT 100 ohms
500
500
Special formula for calculating
Parallel Circuits resistance for equal resistors in parallel
500
500
R
RT 
n
500
RT 
What is the value of RT 55.56 ohms
R1 R2 R3 R4 R5
500
1
1
1

R1 R2
100
Parallel Circuits
Application of a parallel circuit for the exterior
light system of an automobile.
What happens if one headlight burns out?
What happens to the current drawn by the headlight circuit?
Parallel Circuits
Parallel circuits are used in residential wiring.
WHY?
Parallel Circuits
Draw the schematic for this circuit board.
Parallel Circuits
Parallel circuit rule for voltage
Voltage across parallel branches is the same.
Parallel Circuits
Parallel circuit rule for voltage
Because all components are connected across the same
voltage source, the voltage across each is the same.
The source voltage is 5.0 V. What will a volt- meter read if
it is placed across each of the resistors?
+5.0 V
+
VS
+5.0 V
+5.0 V
+
R1
680 W
+5.0 V
+
R2
1.5 kW
+5.0 V
+
R3
2.2 kW
Parallel Circuits
Parallel Circuits
24 v
What is the voltage reading for R1 and R2
Parallel Circuits
What is the voltage reading for R1 and R2
Parallel Circuits
Application of Ohms law
Calculate VS, I1, I2 and I3.
Calculate IS if R3 is shorted, if R3 is open and voltage remains the same
Vs = 5.047 volts, RT = 136.4 ohms, I1=0.023a, I2=0.009a, I3=.00505a
Ishort = Error; RT = 158 ohms, IS = 32mA
Parallel Circuits
Kirchhoff’s current law (KCL)
The sum of the currents into a node equals
the sum of the current out of that node.
Parallel Circuits
Kirchhoff’s current law (KCL)
The algebraic sum of the currents entering
and leaving a node equals zero.
Parallel Circuits
Proving Kirchhoff’s current law.
Parallel Circuits
Parallel circuits are know as current “dividers”
Parallel Circuits
Parallel Circuit is a Current divider
When current enters a node (junction)
it divides into currents with values that are inversely
proportional to the resistance values.
Parallel Circuits
The current IX through any branch equals
1. The total parallel resistance RT
2. Divided by the resistance RX for that branch
3. Multiplied by
R
T

4. The total current IT in the branch IX 

  IT
 RX 
Parallel Circuits
Current divider
For two resistor parallel circuits
 R2 
 R1 
I1  
 I T and I 2  
 IT
 R1  R2 
 R1  R2 
Notice the subscripts. The resistor in the numerator is
not the same as the one for which current is found.
Parallel Circuits
The branch with
the lowest
resistance has
the most
current,
The branch with
the highest
resistance has
the least
current.
Parallel Circuits
Current divider
Parallel Circuits
Solving for Current in a parallel circuit when the voltage is
unknown and the IT and the values of R1 and R2 are known.
 R2 
I1  
 IT
 R1  R 2 
 R1 
I2  
 IT
 R1  R 2 
I1  

Parallel Circuits
Current divider
Assume that R1is a 2.2 kW resistor that is in
parallel with R2, which is 4.7 kW. If the total
current into the resistors is 8.0 mA, what is
the current in each resistor?
 R2 
 4.7 kW 
I1  
 IT  
 8.0 mA = 5.45 mA
 6.9 kW 
 R1  R2 
 R1 
 2.2 kW 
I2  
 IT  
 8.0 mA = 2.55 mA
 6.9 kW 
 R1  R2 
Notice that the larger resistor has the smaller current.
Parallel Circuits
Power in parallel circuits
Power in each resistor can be calculated with any of the
standard power formulas.
P V I
P I R
T
S
T
2
T
T
T
V
P 
R
2
S
T
T
Just like a series circuit
PT = P1 + P2 + P3 + … + PN
What is the total power if 10 V is applied to the parallel
combination of R1 = 270 W and R2 = 150 W? 1.04 W
Parallel Circuits
What is the maximum power that the
amplifier must be able to deliver?
70 v
Power for one speaker =
612.5 watts
Parallel Circuits
Assume there are 8 resistive wires all with the same resistance that
form a rear window defroster for an automobile.
(a) If the defroster dissipates 90 W when connected to a 12.6 V source,
what power is dissipated by each resistive wire?
(b) What is the total resistance of the defroster?
(a) Each of the 8 wires will dissipate 1/8 of the total power
90 W
 11.25 W
8 wires
2
12.6 V 

V2

 1.76 W
(b) The total resistance is R 
P
90 W
What is the resistance of each wire?
R
R 
n
T
1.76 W x 8 = 14.1 W
Parallel Circuits
Open branches in parallel circuits
When an open condition occurs in a parallel circuit:
1. the total resistance increases and
2. the total current decreases and
3. the current through all other branches (i.e. R2)
remains unchanged.
Parallel Circuits
When a lamp filament opens (or the light is
turned off), the total current decreases by the
amount of current in the lamp that opened.
The other branch currents remain unchanged.
Parallel Circuits
Troubleshooting parallel circuits
Finding an open path by current measurement.
Parallel Circuits
Troubleshooting parallel circuits
Know how to determine if there is a open resistor
Parallel Circuits
Practice Exercise
VS
+
R1
680 W
R2
1.5 kW
Complete the parameters listed in the Table.
I1= 7.4 mA
I2= 3.3 mA
I3= 2.3 mA
IT= 13.0 mA
R1= 0.68 kW V1= 5.0 V P1= 36.8 mW
R2= 1.50 kW V2= 5.0 V P2= 16.7 mW
R3= 2.20 kW V3= 5.0 V P3= 11.4 mW
RT= 386 W VS= 5.0 V PT= 64.8 mW
R3
2.2 kW
Parallel Circuits
Key Terms
Parallel The relationship in electric circuits in which
two or more current paths are connected
between two separate points (nodes).
Branch One current path in a parallel circuit.
Kirchhoff’s A law stating the total current into a node
current law equals the total current out of the node.
Node A point or junction in a circuit at which two
or more components are connected.
Current divider A parallel circuit in which the currents divide
inversely proportional to the parallel branch
resistances.
Parallel Circuits
1. The total resistance of parallel resistors is equal to
a. the sum of the resistances
b. the sum of the reciprocals of the resistances
c. the sum of the conductances
d. none of the above
Parallel Circuits
Quiz
2. The number of nodes in a parallel circuit is
a. one
b. two
c. three
d. any number
Parallel Circuits
3. The total resistance of the parallel resistors is
a. 2.52 kW
b. 3.35 kW
c. 5.1 kW
d. 25.1 kW
R1
10 kW
R2
10 kW
R3
5.1 kW
Parallel Circuits
4. If three equal resistors are in parallel, the total
resistance is
a. one third the value of one resistor.
b. the same as one resistor.
c. three times the value of one resistor.
d. the product of the three resistors
Parallel Circuits
Quiz
5. In any circuit the total current entering a node is
a. less than the total current leaving the node.
b. equal to the total current leaving the node.
c. greater than the total current leaving the node.
d. can be any of the above, depending on the circuit.
Parallel Circuits
6. The current divider formula to find I1 for the special
case of two resistors is
 R1
I1  
 RT

 IT

 R2
b. I1  
 RT

 IT

a.

R2 
 IT
 R1  R2 
c. I1  

R1 
 IT
 R1  R2 
d. I1  
Parallel Circuits
7. The total current leaving the source is
a. 1.0 mA
b. 1.2 mA
c. 6.0 mA
d. 7.2 mA
VS +
12 V
R1
10 kW
R2
2.0 kW
Parallel Circuits
8. The current in R1 is
a. 6.7 mA
b. 13.3 mA
I = 20 mA
c. 20 mA
d. 26.7 mA
R1
100 W
R2
200 W
Parallel Circuits
9. The voltage across R2 is
a. 0 V
b. 0.67 V
c. 1.33 V
d. 4.0 V
I = 20 mA
R1
100 W
R2
200 W
Parallel Circuits
10. The total power dissipated in a parallel circuit is
equal to the
a. power in the largest resistor.
b. power in the smallest resistor.
c. average of the power in all resistors.
d. sum of the power in all resistors.
Parallel Circuits
Quiz
Answers:
1. d
6. c
2. b
7. d
3. a
8. b
4. a
9. c
5. b
10. d