Chapter 05
Series Circuits
Source:
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Tsai
Circuit Analysis: Theory and Practice Delmar Cengage Learning
Series Circuits
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Two elements in a series
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Connected at a single point
No other current-carrying
connections at this point
A series circuit is constructed by
connecting various elements in
series
Current will leave the
positive terminal of a
voltage source, Move through
the resistors, and return to
negative terminal of the
source
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Series Circuits

Current is similar to water flowing through a
pipe
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Current leaving the element must be the same as
the current entering the element
Same current passes through every element
of a series circuit
The laws, theorems, and rules that you apply to
DC or AC circuits
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The Voltage Polarity
Example 1
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Example 2
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Kirchhoff’s Voltage Law (KVL)
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The algebraic sum of the voltage that rises and
drops around a closed loop is equal to zero
 E  0 or ET - V1 - V2 - V3 - ∙∙∙ - Vn = 0
Another way of stating KVL is:

Summation of voltage rises is equal to the
summation of voltage drops around a closed loop
ET = V1 + V2 + V3 + ∙∙∙ + Vn
For example：
E - V1 - V2 - V3 = 0
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Examples of Verifying KVL
Example 1
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Example 2
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Resistors in Series
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Most complicated circuits can
be simplified
For a series circuit
V1 + V2 + V3 = E
IR1 + IR2 + IR3 = E
I (R1 + R2 + R3 )= E
I (R1 + R2 + R3 )= I RT
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Total resistance RT in a
series circuit is the sum of
all the resistor values
RT = R1 + R2 + R3
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Examples of Resistors in Series
Example 1
RT =?
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Example 2
RT =?
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Power in a Series Circuit
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Power dissipated by each resistor is determined by the
power formulas:
P = VI = V 2/R = I 2R
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Since energy must be conserved, power delivered by
voltage source is equal to total power dissipated by
resistors
PT = P1 + P2 + P3 + ∙∙∙ + Pn
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Examples of Power in a Series Circuit
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RT =? , I =? , and Voltage across each resistor
Power dissipated by each resistor
Power delivered by the voltage source
Verify that total power dissipated by resistors is
equal to power delivered by the voltage source
Example 1
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Example 2
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Voltage Sources in Series
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In a circuit with more than one
source in series
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Example 1
Sources can be replaced by a
single source having a value that
is the sum or difference of the
individual sources
Polarities must be taken into
account
Resultant source
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Sum of the rises in one
direction minus the sum of the
voltages in the opposite
direction
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Example: Voltage Sources in Series
Example 2
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Interchanging Series Components
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Order of series components
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May be changed without affecting operation of circuit
Sources may be interchanged, but their
polarities cannot be reversed
After circuits have been redrawn, it may become
easier to visualize circuit operation
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Example: Interchanging Series Components
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The Voltage Divider Rule
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Voltage applied to a series circuit
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Will be dropped across all the resistors
in proportion to the magnitude of
the individual resistors
Rx
Vx 
E
RT
Example 1 Voltage across each resistor
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Examples: Voltage Divider Rule
Example 2
Voltage across each resistor
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Example 3
Voltage across each resistor
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Voltage Divider Rule Application
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If a single resistor is very large compared to the
other series resistors, the voltage across that resistor will
be the source voltage
If the resistor is very small, the voltage across it will
be essentially zero
Example 4
Voltage across each resistor
Example 5
Voltage across each resistor
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Circuit Ground
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Ground
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Point of reference or a common point in a circuit
for making measurements
One type of grounding is chassis ground
In this type of grounding
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Common point of circuit is often the metal chassis of
the piece of equipment
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Circuit Ground
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Chassis ground
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Earth ground
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Often connected to Earth Ground
Physically connected to the earth by
a metal pipe or rod
If a fault occurs within a circuit,
the current is redirected to the
earth
Voltages are often measured
with respect to ground
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Voltage Subscripts
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For the circuit shown, we can express the
voltage between any two node points (a
and b) as Vab .
If a is at a higher potential than b, then Vab is
positive
If b is at a higher potential than a, then Vab is
negative
Vab = Va – Vb
Vba = Vb - Va
Vab = -Vba
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Example: Double Subscripts
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Be sure to take the sign of the polarity on the
same side of the source or element as you go
around the circuit

Vab = -50V
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Vbd =?

Vac =?
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Example: Double Subscripts
Vbd = ? Vea = ?
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Example: Double Subscripts
(2) Vab = ?
(1) Vab = ?
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Single Subscripts
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In a circuit with a ground reference point
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Voltages may be expressed with respect to that
reference point
Voltage Vb at point b with respect to ground
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Ground represents a point of zero reference potential
Vb = ?
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Single Subscripts
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If voltages at various points in a circuit are known
with respect to ground, then the voltage
between points is easily determined
Vab = Va – Vb = ?
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Point Sources
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Voltage source given with respect to ground
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May be represented by a voltage at a single point (node) in the
circuit
This voltage may be referred to as a point source
Voltages at these points represent voltages with
respect to ground, even if ground is not shown
I=?
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Internal Resistance of Voltage Sources
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Ideal sources have no internal resistance
In an ideal source
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Terminal voltage does not change when the load
changes
For a practical source, there is internal resistance
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Internal Resistance of Voltage Sources
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As the internal resistance changes
Drop across the load changes
Vab= 10.0V
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Vab= 0.012V
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Ammeter Loading Effects
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An ammeter is placed in a circuit to make a
measurement
 Ammeter resistance will affect the circuit
Amount of loading depends on instrument and
circuit
If the resistance of Ammeter is small
compared to the resistance of the circuit, the
loading effect will be small
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Ammeter Loading Effects
Case1: RA (250 Ω) << R1 (20kΩ)
I1= 0.5mA
Measure I1 = 0.494mA
Load effect
= (0.5 - 0.494) / 0.5
= 1.23%
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Ammeter Loading Effects
Case2: RA (250 Ω) > R1 (100Ω)
I2= 0.5mA
Measure I2 = 0.143mA
Load effect
= (0.5 - 0.143) / 0.5
= 71.4%
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Circuit Analysis using Multisim
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Kernel abilities
1. What is total resistance Rt of resistors R1~Rn in
series? Please give an example.
2. What is KVL? Please give an example.
3. What is voltage divider? Please give an example.
4. What is loading effect of measuring current? Please
give an example.
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Problem 7
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Solve unknown voltages
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Problem 22
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Solve unknown voltage source
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