Basic Electrical
Engineering
Lecture # 04
Simple Resistive Circuits
Course Instructor:
Engr. Sana Ziafat
Agenda
• Series / parallel (how voltage sources/ current
sources in series/parallel may be combined).
• Resistance in series
• Resistance in parallel
• Voltage divider
• Current divider
Voltages Sources and Current
Sources
Series and Parallel Combinations
•Two Types of Current:
•DC—Direct Current—
produced by solar cells and
chemical cells (batteries)
•Current only flows in one
direction.
•AC—Alternating Current
•Current flows back and
forth (alternates)
•Found in homes
•Generators produce AC
current
Voltage Sources in Series
 DC voltage sources in series can be combined and
replaced with a single source.
 AC voltage sources in series can be combined and
replaced with a single source only if the angular
frequency of operation w are identical.
 DC and AC voltage sources can be added together
when calculating a total voltage.
 AC voltage sources operating at different frequencies
can be added together.
 The current flowing through one voltage source must
be equal to the current flowing through the other
voltage source.
Example 1
Example 2
Or
Example 3
Or
Or
Example 4
Or
Or
DC and AC sources
• A 5V dc voltage source in series
a 2V sin(10t) ac voltage source
has a total combined voltage of
5V+2Vsin(10t).
▫ However, we do not have a
symbol for a single voltage
source that can replace the
symbols for the dc and ac
sources.
AC sources with Different w
• A 2V ac voltage source operating at
10 rad/s in series with a 2V ac
voltage source operating at 10.5
rad/s has a total voltage of
2Vsin(10t)+2Vsin(10t).
▫ Again, there is not a symbol for a
single ac voltage source that can
replace the symbols for the two ac
sources operating at different
frequencies.
Voltage Sources in Parallel
• Since the voltage sources share common nodes,
the only time two or more voltage sources are
allowed in parallel is when they have exactly the
same voltage, polarity, and frequency of
operation (if ac sources).
▫ The multiple voltage sources can be replaced by a
single source with the same voltage, polarity, and
frequency of operation (if ac sources).
Example 5
Allowed
Not Allowed
Example 6
Allowed
Not Allowed
Current Sources in Parallel
 DC current sources in parallel can be combined and
replaced with a single source.
 AC current sources in parallel can be combined and
replaced with a single source only if the angular
frequency of operation w are identical.
 DC and AC current sources in parallel can be added
together when calculating a total current.
 AC current sources operating at different frequencies
can be added together.
 The voltage drop across one current source must be
equal to the voltage dropped across the other current
sources in parallel.
Example 7
Example 8
Example 9
Or
DC and AC Current Sources
• A 5A dc current source in parallel a 2A sin(10t)
ac current source means that they are
contributing a total current of 5A+2Asin(10t) at
that node.
▫ However, we do not have a symbol for a single
current source that can replace the symbols for the
dc and ac sources.
AC Sources with Different w
• A 2A ac current source operating at 10 rad/s in
parallel with a 2V ac current operating at 10.5
rad/s means that they are contributing a total
current of 2Asin(10t)+2Asin(10t) at that node.
▫ Again, there is not a symbol for a single ac current
source that can replace the symbols for the two ac
sources operating at different frequencies.
Current Sources in Series
• Since components in series must have the same
current flowing through each component, the
only time two or more currents sources are
allowed in series is when they have exactly the
same magnitude current, the current is flowing
in the same direction, and frequency of
operation (if ac sources).
▫ The multiple current sources in series can be
replaced by a single source with the same
magnitude, direction of current flow, and
frequency of operation (if ac sources).
Example 10
Allowed
Not Allowed
Summary
 Voltage sources in series can be added.
 Current sources in parallel can be added.
 Only in the case where the magnitude, polarity, and
frequency of operation are identical can multiple voltage
sources be in parallel.
 They can be replaced with a single voltage source of the
same magnitude, polarity, and frequency of operation.
 Only in the case where the magnitude, direction of
current flow, and frequency of operation are identical can
multiple currents sources be in series.
 They can be replaced with a single current source of the
same magnitude, direction of current flow, and frequency of
operation.
Batteries in Series and Parallel
•In series—The voltage is
increased.
•In parallel—No change in
voltage; these batteries will
last longer!
Circuit diagrams
• Minimum Three elements:
-Source of electricity (battery)
-Path or conductor on which electricity flows (wire)
-Electrical resistor (lamp) which is any device that requires electricity to operate
• Pictorial way of showing circuits
This is the Ammeter symbol
This is the Voltmeter symbol.
This is the resistor symbol.
This is the switch symbol.
This is the battery symbol.
Resistors in Series & Parallel
Series Circuit
• Series circuit - has only one path
through which the electricity can flow.
• When two circuit elements connect at
single point
• In the above diagram, the electricity
flows through both loads.
• In series circuit current will remain
same.
Series circuit rule for current:
Because there is only one path, the current everywhere
is the same.
For example, the reading on the first ammeter
is 2.0 mA, What do the other meters read?
+ 2.0 mA _
R1
+ 2.0 mA _
R2
VS
_
2.0 mA +
_
2.0 mA +
Equivalent Resistance:
We know the following for series resistors:
R1
R2
. . .
Req
RN
. . .
Figure : Resistors in series.
Req = R1 + R2 + . . . + RN
1
Can you prove HOW????
Equivalent Resistance
• For the case of series circuit
equivalent resistance is larger than
largest resistance in a series
connection.
Parallel Circuit
• Parallel circuit -When two circuit
elements connect at single Node pair.
• In parallel circuit voltage will remain
same will remain same across their
terminals.
Parallel Circuits
• A parallel circuit has multiple paths through
which the electricity can flow.
• In a parallel circuit, the current though one path
may be different than the current through the
other path.
Equivalent Resistance:
We know the following for parallel resistors:
. . .
R eq
R1
RN
R2
. . .
Figure : Resistors in parallel.
1
1
1
1


 . . . 
Req
R1 R2
RN
Equivalent Resistance:
For the special case of two resistors in parallel:
R eq
R1
R2
Figure: Two resistors in parallel.
Req
R1 R2

R1  R2
Equivalent Resistance:
Resistors in combination.
By combination we mean we have a mix of series and
Parallel. This is illustrated below.
R1
Req
R3
R2
R4
R5
Figure : Resistors In Series – Parallel Combination
To find the equivalent resistance we usually start at
the output of the circuit and work back to the input.
Equivalent Resistance:
Resistors in combination.
R1
R3
R2
Req
Rx
R4 R5
Rx 
R4  R5
R1
R eq
R2
Figure : Resistance reduction.
Ry
R y  Rx  R3
Equivalent Resistance:
Resistors in combination.
R1
RZ
Req
Req
RZ 
R2 RY
R2  RY
Req  RZ  R1
Figure : Resistance reduction, final steps.
Equivalent Resistance:
Resistors in combination.
It is easier to work the previous problem using numbers than to
work out a general expression. This is illustrated below.
Example : Given the circuit below. Find Req.
10 
R eq
8
10 
Figure : Circuit
3
6
Equivalent Resistance:
Resistors in combination.
Example : Continued
. We start at the right hand side
of the circuit and work to the left.
10 
Req
10 
8
10 
2
Figure: Reduction steps
Ans:
Req  15 
Req
5
Series - Parallel Circuits
Current and Resistance in Series
Circuits
• For the series circuit the same current flows
through both loads.
• The loads can be added together to calculate the
total load.
• Rtot = R1 + R2, where Rtot is the total
resistance, R1 is the resistance of one load, and
R2 is the resistance of the other.
• The total load (resistance) in a series circuit with
“n” loads is the sum of the resistance of the “n”
objects. Rtot = R1 + R2 + … + Rn.
Total Voltage in a Series Circuit
• Ohm’s Law can be used to calculate the total
voltage in a series circuit by calculating the sum
of the voltage parts.
• V = V1 + V2, where V is the total voltage (battery
voltage), V1 is the voltage at the first load, and
V2 is the voltage at the other load.
Current in a Parallel Circuit
• The total current in a parallel circuit is the sum
of the two parts.
• I = I1 + I2, where I is the total current, I1 is the
current through one load, and I2 is the current
through the other load.
I1
I2
Parallel Circuits
Parallel Circuits
Resistance in Parallel Circuits
• Using Ohm’s Law you can derive a formula for
the equivalent resistance of two resistors in
parallel.
• I1 = V/R1
• I2 = V/R2
• I = I1 + I2 = V/R1 + V/R2
• = (VR2 + VR1)/R1R2 = V(R2+R1)/R1R2
• Rtot = V/(V(R2+R1)/ R1R2 = R1R2/(R1+R2)
Voltage Divider & Current
Divider Circuits
Summary
Voltage divider rule
The voltage drop across any given resistor in a series
circuit is equal to the ratio of that resistor to the total
resistance, multiplied by source voltage.
VS
Assume R1 is twice the size of
R2. What is the voltage across
R1? 8 V
12 V
R1
R2
Summary
R1
Voltage divider
15 k
VS +
20 V
R2
10 k
What is the voltage across R2?
Notice that 40% of
The total resistance is 25 k.
the source voltage is
Applying the voltage divider formula:
across R2, which
 R2 
 10 k 
represents 40% of
V2  VS    20 V 
 8V
 25 k 
 RT 
the total resistance.
Summary
Voltage divider
Voltage dividers can be set up for a variable output using
a potentiometer. In the circuit shown, the output voltage
is variable.
VS +
15 V
What is the largest output
voltage available? 5.0 V
R1
20 k
R2
10 k
VOUT
Summary
Power in Series Circuits
R1
470 
Applying the voltage
divider rule:
 470  
V1  20 V 
  11.75 V
 800  
 330  
V2  20 V 
  8.25 V
 800  
R2
330 
VS +
20 V
Use the voltage divider rule to
find V1 and V2. Then find the
power in R1 and R2 and PT.
The power dissipated by each
resistor is:
11.75 V 

P
2
 0.29 W
470  2
8.25 V 

P2 
 0.21 W
330 
1
}
PT =
0.5 W
Current Divider Rule
• Allows us to determine how the current flowing
into a node is split between the various parallel
resistors
Current Divider Rule
• If current enters a parallel network with a
number of equal resistors, current will split
equally between resistors
• In a parallel network, the smallest value resistor
will have the largest current
• Most of the current will follow the path of least
resistance
▫ Largest resistor will have the least current
Readings
• Chapter 3: 3.1, 3.2, 3.3, 3.4 (Electric Circuits)
▫ By James W. Nilson
Quiz
1. In a series circuit with more than one resistor, the
current is
a. larger in larger resistors
b. smaller in larger resistors
c. always the same in all resistors
d. there is not enough information to say
Quiz
2. In a series circuit with more than one resistor, the
voltage is
a. larger across larger resistors
b. smaller across larger resistors
c. always the same across all resistors
d. there is not enough information to say
Quiz
3. If three equal resistors are in series, 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. there is not enough information to say
Quiz
4. A series circuit cannot have
a. more than two resistors
b. more than one voltage source
c. more than one path
d. all of the above
Quiz
5. In a closed loop, the algebraic sum of all voltages (both
sources and drops)
a. is 0
b. is equal to the smallest voltage in the loop
c. is equal to the largest voltage in the loop
d. depends on the source voltage
Quiz
6. The current in the 10 k resistor is
a. 0.5 mA
b. 2 mA
c. 2.4 mA
d. 10 mA
VS +
24 V
R1
10 k
R2
2.0 k
Quiz
7. The output voltage from the voltage divider is
a. 2 V
b. 4 V
c. 12 V
d. 20 V
VS +
24 V
R1
10 k
R2
2.0 k
VOUT
Quiz
8. The smallest output voltage available from the voltage
divider is
a. 0 V
b. 1.5 V
c. 5.0 V
d. 7.5 V
VS +
15 V
R1
10 k 
R2
10 k
VOUT
Quiz
9. The total power dissipated in a series 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
Quiz
10. The meaning of the voltage VAB is the voltage at
a. Point A with respect to ground
b. Point B with respect to ground
c. The average voltage between points A and B.
d. The voltage difference between points A and B.
Quiz
Answers:
1. c
6. b
2. a
7. b
3. c
8. a
4. c
9. d
5. a
10. d
Q&A