Series and Parallel Circuits
CIRCUITS 1- Chapter 7 [student]
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
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NODES, BRANCHES, AND LOOPS
A BRANCH represents a single element such as a voltage source or a resistor.
In other words, a branch represents any two-terminal
element. The circuit in the figure has five branches,
namely, the 10-V voltage source, the 2-A current
source, and the three resistors.
A NODE is the point of connection between two or
more branches.
A node is usually indicated by a dot in a circuit. If a
short circuit (a connecting wire) connects two nodes,
the two nodes constitute a single node. The circuit in
the figure has three nodes a, b, and c. Notice that the three
points that form node b are connected by perfectly
conducting wires and therefore constitute a single point. The
same is true of the four points forming node c.
A LOOP is any closed path in a circuit.
A loop is a closed path formed by starting at a node, passing
through a set of nodes, and returning to the starting node
without passing through any node more than once. A loop is
said to be independent if it contains at least one branch
which is not a part of any other independent loop.
Independent loops or paths result in independent sets of
equations.
It is possible to form an independent set of loops where one of the loops does not contain such a
branch. In the figure, abca with the 2 Ω resistor is independent. A second loop with the 3 Ω resistor
and the current source is independent. The third loop could be the one with the 2 Ω resistor in
parallel with the 3 Ω resistor. This does form an independent set of loops.
Problem:
1. Determine the number of branches and nodes in the circuit
shown in the figure. Identify which elements are in series
and which are in parallel.
2. How many branches and nodes does the circuit in the
figure have? Identify the elements that are in series and in
parallel.
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
Page 3 of 10
SERIES CIRCUITS
A circuit consists of any number of elements joined at terminal points, providing at least one
closed path through which charge can flow. The following circuit has three elements joined at three
terminal points (a, b, and c) to provide a closed path for the current I.
Two elements are in series if
1. They have only one terminal in common (i.e., one
lead of one is connected to only one lead of the
other).
2. The common point between the two elements is not
connected to another current-carrying element
In a series circuit, every component is connected wherein
there is only a single path for current to flow in the entire
circuit.
RESISTANCE
The total resistance of a series circuit is the sum of the resistance levels.
CURRENT
The current is the same through series elements.
VOLTAGE
POWER
The total power delivered to a resistive circuit is equal to the total power dissipated by the resistive
elements.
PROBLEM:
1. Find the following:
a. the total resistance for the series
circuit.
b. the source current I
c. the voltages V1, V2, and V3.
d. the power dissipated by R1, R2,
and R3.
e. the power delivered by the
source, and compare it to the
sum of the power levels of part
(d).
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
2. Determine RT, I, and V2 for the following
circuit.
3. Given RT and I, calculate R1 and E for the
circuit.
4. For the circuit shown in the figure,
determine
a. the battery voltage V,
b. the total resistance of the circuit,
and
c. the values of resistance of
resistors R1, R2 and R3,
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5. For the circuit shown in the figure,
determine the voltage across resistor R3.
If the total resistance of the circuit is
100 Ω, determine the current flowing
through resistor R1. Find also the value
of resistor R2.
6. A 12V battery is connected in a circuit
having three series-connected resistors
having resistances of 4 Ω, 9 Ω and 11 Ω.
Determine the current flowing through,
and the voltage across the 9 Ω resistor.
Find also the power dissipated in the 11
Ω resistor.
7. How much power is dissipated in the
circuit below?
Given that the voltages across R1, R2 and
R3 are 5V, 2V and 6V respectively.
8. Calculate P2 of the circuit below.
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
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MULTIPLE VOLTAGE SOURCES
Voltage sources can be connected in series to increase or decrease the total voltage applied to a
system. The net voltage is determined simply by summing the sources with the same polarity and
subtracting the total of the sources with the opposite “pressure.” The net polarity is the polarity of
the larger sum.
----------------------------------------------------If the voltage sources cause current to flow in
the same direction, then the effective voltage is
the sum of the individual voltage sources.
----------------------------------------------------If the same and opposing voltage sources are
present, then the effective voltage is the
algebraic sum of the individual voltages.
Example: What is the effective applied voltage
for the circuit below?
Example: What is the effective applied voltage
for the circuit below?
----------------------------------------------------If the voltage sources cause current to flow in
opposite directions, then the effective voltage is
the difference between the individual voltage
sources with the polarity being the same as the
larger of the two.
Example: What is the effective applied voltage
for the circuit below?
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
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VOLTAGE DIVIDER RULE
Single-Source, Two-Resistor Network
In a series circuit, the voltage across the resistive elements will divide as the magnitude of the
resistance levels.
In other words, the VOLTAGE DIVIDER RULE states that the voltage across a resistor in a
series circuit is equal to the value of that resistor times the total impressed voltage across the
series elements divided by the total resistance of the series elements.
PROBLEM:
9. Determine the value of voltage V shown
in the figure.
10. Two resistors are connected in series
across a 24V supply and a current of 3A
flows in the circuit. If one of the resistors
has a resistance of 2 Ω, determine (a)
the value of the other resistor, and (b)
the voltage across the 2 Ω resistor. If
the circuit is connected for 50 hours,
how much energy is used?
11. Determine the voltage V1 for the circuit.
12. Using the voltage divider rule, determine
the voltages V1 and V3 for the series
circuit.
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
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PARALLEL CIRCUITS
In a parallel circuit, all components are connected directly across every other component. All the
elements have terminals a and b in common.
RESISTANCE
For parallel resistors, the total resistance will always decrease as additional elements are added in
parallel.
VOLTAGE
The voltage across parallel elements is the same.
CURRENT
For single-source parallel networks, the source current (Is) is equal to the sum of the individual
branch currents.
POWER
The total power delivered to a resistive circuit is equal to the total power dissipated by the resistive
elements.
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
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PROBLEM:
1. For the parallel network of the following
figure:
a. Calculate RT.
b. Determine I.
c. Calculate I1 and I2, and
demonstrate that Is = I1 + I2.
d. Determine the power to each
resistive load.
e. Determine the power delivered by
the source, and compare it to the
total power dissipated by the
resistive elements.
4. Two resistors, of resistance 3 Ω and 6 Ω,
are connected in parallel across a battery
having a voltage of 12V. Determine (a)
the total circuit resistance and (b) the
current flowing in the 3 Ω resistor.
5. For the circuit shown in the figure, find
(a) the value of the supply voltage V and
(b) the value of current I.
2. Given the information provided in the
figure:
a. Determine R3.
b. Calculate E.
c. Find Is
d. Find I2.
e. Determine P2.
6. How much power is dissipated in the
circuit below?
3. For the circuit shown in the figure,
determine (a) the reading on the
ammeter, and (b) the
value of resistor R2.
7. How much power is dissipated in the
circuit below?
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
Page 9 of 10
CURRENT DIVIDER RULE
Single-Source, Two-Resistor Network
Current divider rule will determine how the current
entering a set of parallel branches will split between the
elements.
•
For two parallel elements of equal value, the current
will divide equally.
•
For parallel elements with different values, the
smaller the resistance, the greater the share of input
current.
•
For parallel elements of different values, the current will split with a ratio equal to the
inverse of their resistor values.
PROBLEM:
8. Determine the current I2 for the circuit
using the current divider rule.
10. Determine the magnitude of the currents
I1, I2, and I3 for the circuit.
11. Determine the resistance R1 to effect the
division of current in the figure.
9. Find the current I1 for the circuit.
CIRCUITS 1: SERIES AND PARALLEL CIRCUITS
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SERIES-PARALLEL CIRCUITS
A firm understanding of the basic principles associated with series and parallel circuits is a
sufficient background to begin an investigation of any single-source dc network having a
combination of series and parallel elements or branches. In general,
SERIES-PARALLEL NETWORKS are networks that contain both series and parallel circuit
configurations.
One can become proficient in the analysis of series-parallel networks only through exposure,
practice, and experience. In time the path to the desired unknown becomes more obvious as one
recalls similar configurations and the frustration resulting from choosing the wrong approach.
PROBLEMS:
1. Solve for the voltage V1 across the 5 Ω
resistor, with the polarity shown in the
figure.
3. Find the current I4 and the voltage V2 for
the network.
2. Solve for I1 through the 40 kΩ resistor in
the circuit.
4. Find the indicated currents and voltages
for the network.