Basic Electricals - Section 2
Ohm's Law & Resistive Networks
Understanding voltage, current, and resistance provides the foundation for analysing electrical circuits as described in Section 1. Section 2 builds on these fundamentals by examining how resistive components behave when connected together in practical circuit networks.
The central relationship is Ohm's law: V=IR.
This equation connects voltage, current, and resistance and allows engineers to determine how a circuit will respond to a given electrical supply.
When multiple resistors are connected, their arrangement becomes important. In a series circuit, the same current flows through each component while the applied voltage is distributed among them. In a parallel circuit, each branch has the same voltage while the total current divides between the branches. These relationships allow complex networks to be simplified using the concept of equivalent resistance.
Two useful tools emerge from this analysis: the voltage divider, which determines how voltage is distributed in a series network, and the current divider, which determines how current is distributed between parallel branches.
These concepts are not limited to theoretical resistor circuits. They are used in the analysis of DC control circuits, battery systems, instrumentation, cable voltage drop, load sharing, electrical equipment, and protection circuits.
By the end of this section, you should be able to look at a resistive network, identify its series and parallel relationships, calculate the equivalent resistance, determine individual branch voltages and currents, and understand how changing one component can affect the entire circuit.
Q1. What is Ohm's Law?
A1. The electric current flowing through a conductor is directly proportional to the voltage applied across its ends, provided the temperature and physical conditions remain constant. Which implies – V ∝ I.
Ohm's law describes the relationship between voltage, current, and resistance in an ohmic component under constant physical conditions.
The fundamental equation is: V = I.R
where:
V = voltage in volts (V)
I = current in amperes (A)
R = resistance in ohms (Ω)
The equation can be rearranged into: I = V/R and: R = V/I
Key Rules
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Constant temperature: The law only works if the temperature and physical shape of the material do not change.
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Ohmic materials: Materials like copper and silver follow this law.
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Non-ohmic materials: Devices like light bulb filaments, diodes, and transistors do not follow Ohm's law because their resistance changes as they heat up.
Simple example
Suppose a 24 V supply is connected to a 12 Ω resistor.
So, current is given by I = V/R
-> I = 24/12
-> I = 2A.
If the resistance is increased to 24 Ω while the voltage remains 24 V:
-> I = 24/24
-> I = 1A.
Therefore, doubling the resistance halves the current when voltage remains constant.
Q2. What is equivalent resistance?
A2. A group of resistors can often be replaced by a single equivalent resistor that produces the same relationship between the source voltage and total current.
In other words: Req = Vsource / Itotal
This allows a complicated circuit to be simplified before analyzing individual components.
Q3. What is series resistance?
A3. Resistors are connected in series when they form a single continuous current path.
For example:
Source → R₁ → R₂ → R₃ → Source
There is only one path for current.
Therefore: I = I1 = I2 = I3
The same current flows through every resistor.
Equivalent resistance of series resistors
The total resistance is simply the sum: RT = R1 + R2 + ... + Rn
Example
Suppose:
R1 = 5 Ω, R2 = 10 Ω, R3 = 15 Ω
Then:
RT = 5 + 10 + 15 = 30 Ω
The three resistors can therefore be replaced by one 30 Ω resistor for the purpose of calculating the total circuit current.
Voltage distribution in a series circuit
Although current is the same through every resistor, voltage is not necessarily the same.
For each resistor:
V1 = I.R1, V2 = I.R2, V3 = I.R3
The total voltage is: VT = V1 + V2 + V3
This follows from the fact that the total resistance is: RT = R1 + R2 + R3
Therefore: VT = I.RT
In a series circuit:
Current is the same; voltage divides.
A larger resistor receives a larger voltage drop.
For example, if: R1 = 5 Ω, R2 = 10 Ω
then: V2 > V1
because the same current flows through both.
Q4. What is voltage divider rule?
A4. The voltage division rule states that the voltage drops across any resistor in a series circuit is proportional to that resistor's value divided by the total resistance, multiplied by the total source voltage.
The voltage-divider principle is a direct application of series resistance.
Suppose two resistors are connected in series:
Source → R₁ → R₂ → Source
The supply voltage is divided between the two resistors.
The voltage across R1 is: V1 = VT.R1 / (R1 + R2)
The voltage across R2 is: V2 = VT.R2 / (R1 + R2)
Why does voltage divide this way?
Start with Ohm's law: V1 = I.R1
The circuit current is: I = VT / (R1 + R2)
Therefore: V1 = VT.R1 / (R1 + R2)
So, the voltage across a resistor is proportional to its resistance.
For two or more series resistors:
Vi = V.Ri / (R1 + R2 + ... + Rn)
For example, if:
A 24 V supply is connected to: R1 = 4 Ω, R2 = 8 Ω
Total resistance: RT = 4 + 8 = 12 Ω
Current: I = 24/12 = 2A
Voltage across R1, V1 : 2*4 = 8 V,
Voltage across R2, V2 : 2*8 = 16 V.
Check: V1 + V2 = 8 + 16 = 24V = V.
Q5. What is parallel resistance?
A5. Resistors are connected in parallel when their terminals are connected to the same two nodes. There are multiple paths through which current can flow.
Voltage in a parallel circuit
The voltage across every branch is the same: V = V1 = V2 = V3
This is one of the most important rules to remember.
Current in a parallel circuit
The total current is the sum of the branch currents: IT = I1 + I2 + ... + In
Each branch current can be calculated using Ohm's law:
I1 = V/R1, I2 = V/R2
Therefore: IT = V/R1 + V/R2
Equivalent resistance of parallel resistors
For two resistors: RT = R1.R2 / (R1 + R2)
For three or more resistors: 1/RT = 1/R1 + 1/R2 + ... + 1/Rn
For example,
Two resistors: R1 = 6 Ω, R2 = 3 Ω are connected in parallel.
Then: RT = 6*3 / (6 + 3) = 18/9 = 2 Ω
Notice that the equivalent resistance is less than the smallest individual resistance.
This is always true for ordinary positive resistors connected in parallel.
Q6. What is current divider rule?
A6. The Current Divider Rule states that the total current entering a parallel circuit divide among the parallel branches inversely proportional to their resistances.
Current division is the parallel-circuit equivalent of voltage division.
In a parallel circuit:
Voltage is the same across each branch, while current divides between the branches.
Consider two parallel resistors: R1 and R2 with total current IT.
The current through R1 is: I1 = IT.R2 / (R1 + R2)
The current through R2 is: I2 = IT.R1 / (R1 + R2)
The current through a resistor depends inversely on its own resistance.
Therefore: if R1 < R2 , I1 > I2
The lower-resistance branch carries more current.
For example,
Two resistors: R1 = 4 Ω, R2 = 12 Ω are connected in parallel.
The total current is: IT = 8A.
Current through R1, I1 : 8*12/(4+12) = 8*12/16 = 6 A,
Voltage across R2, I2 : 8*4/(4+12) = 8*4/16 = 2 A.
Check: I1 + I2 = 6 + 2 = 8A = IT.
Notes:
1. Real circuits are often neither purely series nor purely parallel.
For example:
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R2 and R3 are parallel.
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Their equivalent resistance is then in series with R1 and R4.
First calculate: R23 = R2.R3 / (R2 + R3)
Then: Req = R1 + R23 + R4
This step-by-step reduction is the basic method for analysing many resistive networks.
2. The Two Rules You Should Never Forget.
If you remember only two relationships from this section -
For series circuits: Same Current -> Voltage Divider Rule
For parallel circuits: Same Voltage -> Current Divider Rule
And the deeper principle behind both is: V = IR.
Practice MCQ
1. A 10 Ω resistor is connected to a 20 V supply. The current is:
A. 0.5 A
B. 2 A
C. 10 A
D. 200 A
2. Three resistors of 5 Ω, 10 Ω and 15 Ω are connected in series. Their equivalent resistance is:
A. 5 Ω
B. 15 Ω
C. 20 Ω
D. 30 Ω
3. In a series circuit, which quantity is the same through every resistor?
A. Voltage
B. Resistance
C. Current
D. Power
4. Two resistors of 10 Ω and 20 Ω are connected in series across 60 V. Voltage across the 10 Ω resistor is:
A. 10 V
B. 20 V
C. 30 V
D. 40 V
5. Two 10 Ω resistors are connected in parallel. Their equivalent resistance is:
A. 20 Ω
B. 10 Ω
C. 5 Ω
D. 2 Ω
6. Three identical 12 Ω resistors are connected in parallel. Equivalent resistance is:
A. 36 Ω
B. 12 Ω
C. 6 Ω
D. 4 Ω
7. In a parallel circuit, which quantity is the same across every branch?
A. Current
B. Resistance
C. Voltage
D. Power
8. Two parallel resistors are 4 Ω and 12 Ω. If total current is 8 A, current through the 4 Ω resistor is:
A. 2 A
B. 4 A
C. 6 A
D. 8 A
9. Two resistors of 6 Ω and 3 Ω are connected in parallel. Equivalent resistance is:
A. 9 Ω
B. 3 Ω
C. 2 Ω
D. 1 Ω
10. Which statement about parallel resistors is correct?
A. Equivalent resistance is greater than the largest resistor
B. Equivalent resistance equals the sum of resistances
C. Equivalent resistance is less than the smallest resistor
D. Equivalent resistance is always zero
Answers:
‘1. B, 2. D, 3. C, 4. B, 5. C, 6. D, 7. C, 8. C, 9. C, 10. C.
Practice Problems
1. A 48 V DC supply is connected to a 16 Ω resistor.
Calculate: Current through the resistor.
2. Two resistors of 10 Ω and 20 Ω are connected in series across a 60 V DC supply.
Calculate:
a) Total resistance
b) Circuit current
c) Voltage across each resistor.
3. Two resistors of 6 Ω and 3 Ω are connected in parallel across a 24 V DC supply.
Calculate:
a) Equivalent resistance
b) Total supply current
c) Current through each resistor.
4. A 48 V DC supply feeds a circuit consisting of:
R1 = 4 Ω in series with a parallel combination of R2 = 12 Ω and R3 = 6 Ω.
Calculate:
a) Equivalent resistance of R2 and R3
b) Total circuit resistance
c) Total supply current
d) Voltage across R1
e) Voltage across the parallel combination
f) Current through R2 and R3.