Lecture 2:Circuit Theory (1)
Ohm’s law, Kirchhoff’s voltage law, examples,
Kirchhoff’s current law, Mesh Analysis, Voltage
Dividers, Current Dividers, Superposition Theorem
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Ohm’s Law
Ohm's law states that the current through
a conductor between two points is
directly proportional to the potential
difference across the two points. The constant of
proportionality is the resistance
wikipedia
I=V/R or V=IR
What is the origin of this law in electrmagnetics?
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Kirchhoff’s Current Law
For any junction:  I in   I out
wikipedia
 For junction B: I AB  I BD  I BC
 For junction D: I BD  I BC  I AB (dependent)
 Number of independent KCL equations 
number of junctions  1
Analogy with water flow in pipes?
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Kirchhoff’s Voltage Law
For any closed loop:  Vrise  0,
or  Vrise  0
For the left-hand loop: 6  2 I AB  3I BD  0
For the right-hand loop: 4  3I BD  4 I BC  0
For the outter loop: 4  6  2 I AB  4 I BC  0 (dependent)
A new loop equation is independent on the previous if it
contains new current(s) and/or battery(s).
origin of this law in electrmagnetics?
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Mesh-Current Analysis
every loop is assigned one
current flowing in the clockwise
directions
Kirchhoff’s voltage and current
laws are then used to solve for
these currents
Writing Kirchhoff's voltage laws:
6  2 I1  3I1  3I 2  0
4  3I1  3I 2  4 I 2  0
 I1   5 3 
 I    3 7 

 2 
1

 5 3   I1   6 
 3 7   I    4 

  2  
 6   2.077 
 4   1.462  A
  

Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Voltage Divider
R1
The voltage divider is a series
combination of two resistors.
The total voltage V is divided between
these resistors according to the direct
ratio of their resistances:
I
I
V1
V
V
R1  R2
V R1
V R2
V1 R1
V1  I R1 
, V2  I R2 
,

R1  R2
R1  R2 V2 R2
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
V2
R2
Current Divider
The current divider is a parallel
combination of two resistors.
I
The total current I is divided
between these resistors according to
the inverse ratio of their resistances:
R1 R2
V I
R1  R2
I R2
I R1
I1 R2
V
V
I1 

, I2 

,

R1 R1  R2
R2 R1  R2 I 2 R1
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
I2
I1
R1
V
R2
Superposition Theorem
A circuit which contains more than one voltage source can be
considered as the superposition of a number of circuits, each has
only one voltage source only, while the rest of the voltage
sources are replaced by short circuits.

Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Superposition (Cont’d)
Original Circuit
First Sub-Circuit
This circuit has been solved
using Kirchhoff's laws:
I1 
 I1   2.077 
 I    0.615  A
 2 

 I 3  1.462 
4
I 2  I1
 0.923 A
3 4
I 3  I1  I 2  0.692 A
6
 1.615 A
2  3 4
I1  I1  I1  1.615  0.461  2.076 A
I 2  I 2  I 2  0.923  0.308  0.615 A
I 3  I 3  I 3  0.692  0.769  1.461 A
Dr. Mohamed Bakr, ENGINEER 3N03, 2015
Second Sub-Circuit
I 3 
4
 0.769 A
4  3 2
2
 0.308 A
3 2
I1  I 3  I 2  0.461 A
I 2  I 3