PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
3. Basic Electric Circuit Theory
3.1.
Understanding Potential, Potential Differences and Current
The potential of a point in a circuit has a meaning only when one point in a
circuit is earthed or grounded. The potential of the earthed/grounded point is
then taken as V=0. The potential of any points on the circuit is then the
potential relative to the earthed/grounded point.
The potential difference between two points in a circuit is the difference in
potential of the two points. The potential difference is taken as the final
potential minus the initial potential.
(Note: Some consider voltage across a component as the magnitude of the
potential difference (for example the voltage across a resistor), while change
in potential might be negative or positive depending on the direction taken ,
i.e. from higher to lower potential or from lower to higher potential points.)
The direction of the conventional current is taken from a point of high
potential to that of a lower potential.
Thus going through a resistor in the direction of the current gives a reduction
in potential (potential difference is negative), while going opposite the
direction of the current through a resistor gives an increase in potential
(potential difference is positive.)
3.2.
Kirchoff’s Laws
Kirchoff’s Laws are the law of conservation of energy and the law of
conservation of charges applied to the electic circuit.
Kirchoff’s Current Law (Junction rule)
The sum of all currents entering a junction is equal to the
sum of all currents leaving the same junction.
∑ I in = ∑ I out
I1
I3
I2
I4
Kirchoff’s Current Law is a restatement of the principle of conservation of
charges and mass.
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mohdnoormohdali
PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
Kirchoff’s Voltage Law (Loop Rule)
The algebraic sum of of the potential changes across all
elements in a closed circuit loop must be equal to zero.
∑ ∆V = 0
Kirchoff’s Voltage Law is a restatement of the principle of conservation
energy.
(Note: In this case the change in potential has to be carefully
considered. Going from the negative to the positive terminal
through a cell gives a positive change in potential. In the reverse
direction a negarive potential change is obtained)
3.3.
Resistance In Series And In Parallel
3.3.1. Series
When resistors are placed in series, the potential across the series connection
is the sum of the potential across each resistor.
V = ∑Vi
V = ∑ I i Ri
V = I ∑ Ri
as I is the common current through all resistors. Therefore the total resistance
is
RT = ∑ Ri
RT = R1 + R2 + R3
3.3.1.1.Voltage Divider Rule
When two resistors are in series, the voltage across a resistor is given as
R1
V1 =
× V , this is called voltage divider rule.
R1 + R2
3.3.2. Parallel
When resistors are placed in parallel arrangement, the current to the resistor
circuit is the sum of all currents through each resistor.
I = ∑ Ii
I =∑
Vi
Ri

1
I =  ∑
 Ri
V
I=
Req

V

1 
1
=  ∑
Req  Ri
 1
1
1
 = +
+ .. +
Rn
 R1 R2
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mohdnoormohdali
PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
This gives the equivalent resistance of resistors in parallel arrangement.
3.3.2.1.Current Divider Rule
When two resistors are in parallel arrangement, the current through a
resistor is given as
I1 =
3.4.
R2
I , this is called current divider rule.
R1 + R2
Branch Analysis
In branch analysis, each branch of the circuit has a unique label for the current
through that branch. Thus for a simple junction, there would be three unique
current labels. This makes branch analysis cumbersome when there is more
than one unique junction. However branch analysis is the basic method of
analyzing a circuit.
3.4.1. Writing The Equations
Each branch is labeled with a unique current label
Equation satisfying KCL for the junctions are written.
At junction B; I1+I2 = I3 …………..(1)
At junction E; I3 = I1+I2 , which is exactly as that at junction B.
Equations satisfying KVL for the loops are written.
Going around loop ABEFA and taking the potential change, ∆V;
- I1(3Ω) - I3(5Ω) + 9V = 0…………..(2)
Going around loop BCDEB and taking the potential change, ∆V;
I2(4Ω) – 12V + I3(5Ω) = 0………….(3)
Going around loop ABCDEFA and taking potential change, ∆V;
- I1(3Ω) + I2(4Ω) – 12V + 9V = 0
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mohdnoormohdali
PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
This final equation is actually the combination of the earlier two KVL
equations. Therefore, there are only two unique KVL equations.
A rule of thumb is that each component of the circuit has to be passed at least
once when taking the potential change.
3.4.2. Direct Substitution
Substituting (1) into (2) and (3) produce,
- I1(3Ω) – (I1 + I2)(5Ω) + 9V = 0…………..(2)
I2(4Ω) – 12V + (I1 + I2)( (5Ω) = 0………….(3)
After collecting similar terms give;
- I1(8Ω) – (I2)(5Ω) + 9V = 0…………..(2)
(I1)(5Ω) + I2(9Ω) – 12V = 0…………...(3)
- 8I1 – 5I2 + 9 = 0…………..(2) (dropping the Ω and V make for easier witing)
5I1 + 9I2 – 12 = 0…………...(3)
3.4.3. Method of Elimination
- 8I1 – 5I2 + 9 = 0…………....(2)
5I1 + 9I2 – 12 = 0…………...(3)
(2) x 5 : - 40I1 – 25I2 + 45 = 0
(3) x 8 : 40I1 + 72I2 – 96 = 0
Adding both up gives
(-40 + 40)I1 + (-25 + 72)I2 + (45 – 96) = 0
Thus I1 is eliminated from the equation and I2 can be calculated.
The value for I2 is the substituded into (2) or (3) to obtain I1.
3.5.Mesh Analysis
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mohdnoormohdali
PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
3.5.1. Writing The Equations
Going around loop ABEFA and taking the potential change, ∆V;
- I1(3Ω) –( I1 + I2 )(5Ω) + 9V = 0…………..(1)
Going around loop BCDEB and taking the potential change, ∆V;
I2(4Ω) – 12V + ( I1 + I2 )( (5Ω) = 0………….(2)
Note : KCL is applied at the junction while writing down KVL for the loop.
The total current through a branch is the algebraic sum of the loop currents
through the branch : The superposition of currents.
3.5.2. Method of Elimination
- I1(3Ω) –( I1 + I2 )(5Ω) + 9V = 0…………..(1)
I2(4Ω) – 12V + ( I1 + I2 )( (5Ω) = 0………….(2)
Expanding the equations give
- I1(3Ω) –( I1 )(5Ω) –( I2 )(5Ω) + 9V = 0…………..(1)
I2(4Ω) – 12V + ( I1)( (5Ω) + (I2 )( (5Ω) = 0………….(2)
Colecting the current terms gives
- I1(3Ω + 5Ω) – ( I2 )(5Ω) + 9V = 0…………..(1)
( I1)( (5Ω) + (I2 )(4Ω + 5Ω) - 12V = 0………….(2)
- I1(8Ω) – ( I2 )(5Ω) + 9V = 0…………..(1)
( I1)( (5Ω) + (I2 )( 9Ω) - 12V = 0………….(2)
- 8I1 – 5I2 + 9 = 0…………..(1)
5I1 + 9I2 - 12 = 0………….(2)
(1) x 5 : - 40I1 – 25I2 + 45 = 0…………..(1)
(2) x 8 : 40I1 + 82I2 - 96 = 0………….(2)
Using the method of elimination I1 and I2 can be obtained as in the previous
section.
3.5.3. Matrix Method
The matrix method is used to obtain the values of I1 and I2 without using the
method of elimination. Using either the Branch or Loop currents analysis the
equations are obtained as before.
- 8I1 – 5I2 + 9 = 0…………..(1)
5I1 + 9I2 - 12 = 0………….(2)
Rearranging the equations give
- 8I1 – 5I2 = - 9
5I1 + 9I2 = 12
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mohdnoormohdali
PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
Rearranging into matrix form will produce
-8 -5
5 9
I1
I2
=
-9
12
This can be easily solved using the built in matrix solution in a calculator (ex.
Casio FX 570MS), which can solve for 2 and 3 loop currents.
3.5.4. Cramer’s Rule / Method of Determinants
The method of determinants is especially used to solve for 3 loop currents in a
3 loop problem where the methods of substitution and elimination become
unyeilding.
Writing the equations in the matrix form give,
R11
R21
R31
R12
R22
R32
R13
R23
R33
Det R =
I1
I2
I3
R11
R21
R31
=
R12
R22
R32
R13
R23
R33
Then
I1 = 1 / Det R
V1
V2
V3
R12
R22
R32
R13
R23
R33
I2 = 1 / Det R
R11
R21
R31
V1
V2
V3
R13
R23
R33
I3 = 1 / Det R
R11
R21
R31
R12
R22
R32
V1
V2
V3
and
and
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V1
V2
V3
PHY 192- PHYSICS III
3. Basic Electric
Circuit Theory
Physics for Electrical Engineering
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