Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
http://pami.uwaterloo.ca/~akrem/
University of Waterloo,
Electrical and Computer Engineering Dep.
Basic Circuit-Analysis Techniques
Objective:
To study some basic techniques/principles commonly used to analyze electrical circuits.
(1) Equivalent Resistors:
Series Resistors
If two resistors are connected in series (as show in figure (a)) then the same current flows
through them.
Apply KVL around the loop of figure (a):
V V V 0
V V V
Apply ohm’s law:
V i R i R iR R 1
Apply KVL around the loop of figure (b):
V V 0
V V
Apply ohm’s law:
V i R 2
If figure (a) and (b) are equivalent, then from equation 1 and 2:
iR R i R R R R For n resistors connected in series:
R R R R … … … … R R Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Parallel Resistors
If two resistors are connected in parallel (as shown in figure (c)) then they have the same
voltage across them.
i
i
1
i1
Vs
+
-
R1
i2
Vs
R2
+
-
Figure (c)
Figure (d)
Apply the KCL at node#1 of figure (c):
i i i 0
i i i
Apply the Ohm’s law:
i
From figure (d)
Req
Vs Vs
R R 3
Vs
4
R If figure (c) and (d) are equivalent then they have the same current (i) and voltage (Vs).
Therefore, from equations 3 and 4:
Vs
Vs Vs
R R R 1
1
1
R R R For n resistors connected in parallel:
i
1
1
1
1
1
1
……..
R R R R R R Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Note: For only two resistors connected in parallel (Special case):
R 1
1
1
R R R R R for only two resistor connected in parellel
R R Example:
Find the equivalent resistor Req for the following circuit:
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Req=10 Ω
Circuits (MTE 120) (Spring 2010)
Example:
Find the equivalent resistor for the following circuit :
(a) Between A and B
(b) Between C and D
Solution
a) Between A and B
Akrem El-ghazal
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
R AB 5
Ω
6
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
b) Between C and D
R CD Ω
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Equivalent of sources:
Equivalent of Voltage Sources:
Equivalent of Current Sources:
a
Is1
Is
Is=Is1+Is2
Is2
Apply KCL at node (a) to find Is
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Voltage and Current-Divider Principle
Voltage-Divider Principle
When a voltage is applied to series combination of resistors, a fraction of the voltage
appears across each of the resistors. Consider two resistors connected in series as shown
below.
R1
+
Circuit
V
+
i
V1 +
V2
R2
-
The voltage across each resistor can be calculated as follows:
V
i
R R Then,
V i R
R
V V
voltage divider princible
R R V i R R
V V
voltage divider princible
R R Note:
If n resistors are connected in series as shown below, then the voltage across each one can be
calculated as:
V1 V
i=1,2,3,…….n.
R1
voltage divider princible
R R R 2 … . . R R Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Example:
Find the voltage across each resistor.
Solution:
Apply the voltage-divider principle
V 30
2
4V
267
V 30
6
12 V
267
V 30
7
14 V
267
Current-Divider Principle
If a current flows into a parallel combination of resistors, then the total current divides
and a fraction of the total current flows through each resistor. Consider two resistors
connected in parallel as shown below.
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
The current follows through each resistor can be determined as follows:
First apply the KCL at node#1 to find the total current as follows
i i i
a
Since the two resistors are connected in parallel they have the same voltage (V) across them.
In equation (a) use the Ohm’s law:
V
V
1
1
R R V 4 5V 4
5
R R R R R R Then, the common voltage for the two resistors:
R R Vi
R R Now, the common voltage (V) is calculated, we can use the common voltage(V) to calculate the
current in each resistor as follows:
V
i R
Then,
R
i i
current divider princible
R R i
i Then,
i i
V
R
R
current divider princible
R R Note: The above current-divider principle can be applied only for TWO resistors connected in
parallel. If you have more than two resistors in parallel, we should combine resistors so we only
have two resistors before applying the current-divider principle.
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Example:
Find i , i and i
i1
10 Ω
15 A
i2
30 Ω
i3
60 Ω
Solution:
The current-divider principle applies for TWO resistors in parallel .Therefore our first step is to
combine the 30Ω and 60Ω resistors (see the figure below), then use the current-divider
principle to find i and i89 :
i 15
20
10 A
10 20
i89 15
10
5A
10 20
Since the total current flows in both the 30Ω and 60Ω resistors is known (i89 5A (see the
figure below), we can use the current-divider principle to find i and i as follows:
i i89 i i89 60
60
10
5
A
30 60
30 60
3
30
30
5
5
A
30 60
30 60
3
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Solution #2
Another solution is to find the common voltage across the three resistors then use the common
voltage to calculate the current in each resistor.
To find the common voltage (V), first we find the equivalent resistor of the three resistors and
then multiply the equivalent resistor by the current (15 A).
Then, the common voltage(V):
20
100 V
3
Now, use the common voltage to find the current in each resistor (use the Ohm’s law):
100
100 10
100 5
i 10 A , i A,
i A
10
30
3
60
3
V 15 ;
SEE THE EXTRA-SESSION QUESTIONS
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Dependent Sources
A dependent or control voltage source is similar to an independent voltage source
exempt that the voltage across the source terminals is a function of other voltages or currents
in the circuit. Instead of a circle, it is customary to use a diamond to represent controlled
sources in circuit diagrams.
Dependent Voltage Sources
There are two types of dependent voltage sources:
Voltage-Controlled Voltage Source
The voltage across the dependent voltage source depends on a voltage that appears
elsewhere in the circuit.
Current-Controlled Voltage Source
The voltage across the dependent voltage source depends on a current that appears
elsewhere in the circuit.
Dependent Current Sources
There are two types of dependent current sources:
Voltage-Controlled current Source
The current through the dependent current source depends on a voltage that appears
elsewhere in the circuit.
Current-Controlled current Source
The current through the dependent current source depends on a current that appears
elsewhere in the circuit.
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
Example:
Find ix in the folllownf circuit.
Solution
By using KCL at node (a) we can find the current (i) flows in the 1Ω resistor.
i i< 3
Apply the KVL around the loop shown in the figure.
10 V V 2 i< 0
10 2 i< 1 i< 3 2 i< 0
i< 1.4 A