MECH1100
Chapter 5 Parallel Circuits
Topics
Resistors in Parallel
Total Parallel Resistance
Voltage in a Parallel Circuit
Ohm’s Law
Kirchhoff’s Current Law
Current Dividers
Power in Parallel Circuits
MECH1100
Chapter 5 Parallel Circuits
Resistors in parallel
Resistors that are connected to the same two points
are said to be in parallel.
A
R1
B
R2
R3
R4
MECH1100
Chapter 5 Parallel Circuits
A parallel circuit is identified by the fact that it has
more than one current path (branch) connected to a
common voltage source.
VS
+
R1
R2
R3
R4
MECH1100
Chapter 5 Parallel Circuits
Parallel circuit rule for voltage
Because all components are connected across the same
voltage source, the voltage across each is the same.
For example, the source voltage is 5.0 V. What will a voltmeter read if it is placed across each of the resistors?
+5.0 V
+
VS
+5.0 V
+5.0 V
+
R1
680 
+5.0 V
+
R2
1.5 k
+5.0 V
+
R3
2.2 k
Chapter 5 Parallel Circuits
Current in Parallel circuits:
When resistors are connected in parallel, the current has
More than one path. The number of current paths is equal
To the number of parallel branches.
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Chapter 5 Parallel Circuits
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Furthermore with current in a parallel circuit:
Current increase as paths are added in parallel circuits.
*Assuming the voltage is constant, an increase in total
Current from the source means that the total resistance
Has decreased, in accordance with Ohm’s Law.
Additional resistors connected in parallel will further
Reduce the resistance and increase the total current.
Chapter 5 Parallel Circuits
Resistance in Parallel Circuits:
Important: When resistors are connected in
parallel, the total resistance of the circuit
decreases! The total resistance of a parallel
circuit is always less than the value of the
smallest resistor.
For example: if you have a 10 ohm resistor and
100 resistor connected in parallel, the total
resistance is less than 10 ohms.
MECH1100
Chapter 5 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.
Remember back in chapter 2.5 of conductance
(G) -
MECH1100
Chapter 5 Parallel Circuits
MECH1100
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 (S)
1
GT 
RT
MECH1100
Chapter 5 Parallel Circuits
For example, the resistors in a parallel circuit are 680
, 1.5 k, and 2.2 k. What is the total resistance?
VS
+
R1
680 
R2
1.5 k
R3
2.2 k
386 
Chapter 5 Parallel Circuits
Let’s calculate total Resistance again
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MECH1100
Chapter 5 Parallel Circuits
Special case for resistance
of two parallel resistors
R1
R2
The resistance of two parallel resistors can be found by
1
R

either: T 1 1

R1 R2
R1 R2
or RT 
R1  R2
What is the total resistance if R1 = 27 k and
R2 = 56 k? 18.2 k
MECH1100
Chapter 5 Parallel Circuits
Calculate this one:
RT 
R1 R2
R1  R2
Chapter 5 Parallel Circuits
Another special case
total resistance
• If you have a parallel circuit with all
resistors having the same value.
• To find the total resistance by:
• Example: a circuit with 12 resistors
that has all the same value of 100
ohms. The total resistance is
•
100/ 12 = 8.33 ohms
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MECH1100
Chapter 5 Parallel Circuits
Parallel circuit
VS
+
R1
680 
R2
1.5 k
R3
2.2 k
Tabulating current, resistance, voltage and power is a
useful way to summarize parameters in a parallel circuit.
Continuing with the previous example, 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 k V1= 5.0 V P1= 36.8 mW
R2= 1.50 k V2= 5.0 V P2= 16.7 mW
R3= 2.20 k V3= 5.0 V P3= 11.4 mW
RT= 386  VS= 5.0 V PT= 64.8 mW
MECH1100
Chapter 5 Parallel Circuits
Kirchhoff’s current law
Kirchhoff’s current law (KCL) is generally stated as:
The sum of the currents entering a node is equal to the
sum of the currents leaving the node.
Notice in the previous example that the current from
the source is equal to the sum of the branch currents.
I1= 7.4 mA
I2= 3.3 mA
I3= 2.3 mA
IT= 13.0 mA
R1= 0.68 k V1= 5.0 V P1= 36.8 mW
R2= 1.50 k V2= 5.0 V P2= 16.7 mW
R3= 2.20 k V3= 5.0 V P3= 11.4 mW
RT= 386  VS= 5.0 V PT= 64.8 mW
Chapter 5 Parallel Circuits
Current divider
When current enters a node (junction) it divides into
currents with values that are inversely proportional to the
resistance values.
The most widely used formula for the current divider
is the two-resistor equation. For resistors R1 and R2,
 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.
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Chapter 5 Parallel Circuits
Current divider
Assume that R1is a 2.2 k resistor that is in
parallel with R2, which is 4.7 k. If the total
current into the resistors is 8.0 mA, what is
the current in each resistor?
 R2 
 4.7 k 
I1  
 IT  
 8.0 mA = 5.45 mA
 6.9 k 
 R1  R2 
 R1 
 2.2 k 
I2  
 IT  
 8.0 mA = 2.55 mA
 6.9 k 
 R1  R2 
Notice that the larger resistor has the smaller current.
MECH1100
Chapter 5 Parallel Circuits
Power in parallel circuits
Power in each resistor can be calculated with any of the
standard power formulas. Most of the time, the voltage is
V2
known, so the equation P 
is most convenient.
R
As in the series case, the total power is the sum
of the powers dissipated in each resistor.
What is the total power if 10 V is applied to the parallel
combination of R1 = 270  and R2 = 150 ? 1.04 W
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Chapter 5 Parallel Circuits
Assume there are 8 resistive wires 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 or 90 W  11.25 W
8 wires
2
12.6 V 

V2

 1.76 
(b) The total resistance is R 
P
90 W
What is the resistance of each wire? 1.76  x 8 = 14.1 
MECH1100
Chapter 5 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.
MECH1100
Chapter 5 Parallel Circuits
Quiz
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
MECH1100
Chapter 5 Parallel Circuits
Quiz
2. The number of nodes in a parallel circuit is
a. one
b. two
c. three
d. any number
MECH1100
Chapter 5 Parallel Circuits
Quiz
3. The total resistance of the parallel resistors is
a. 2.52 k
b. 3.35 k
c. 5.1 k
d. 25.1 k
R1
10 k
R2
10 k
R3
5.1 k
MECH1100
Chapter 5 Parallel Circuits
Quiz
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
MECH1100
Chapter 5 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.
MECH1100
Chapter 5 Parallel Circuits
Quiz
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  
MECH1100
Chapter 5 Parallel Circuits
Quiz
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 k
R2
2.0 k
MECH1100
Chapter 5 Parallel Circuits
Quiz
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 
R2
200 
MECH1100
Chapter 5 Parallel Circuits
Quiz
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 
R2
200 
MECH1100
Chapter 5 Parallel Circuits
Quiz
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.
MECH1100
Chapter 5 Parallel Circuits
Quiz
Answers:
1. d
6. c
2. b
7. d
3. a
8. b
4. a
9. c
5. b
10. d