Equivalent Resistance (Symbol: R_eq or r_eq): Series and Parallel Connection

• The term equivalent resistance in electrical circuits denotes the overall resistance that a configuration of resistors offers to a voltage source. The configuration of resistors—whether in series or parallel—impacts the overall resistance.

1. Series Connection:

• In a series connection, resistors are connected end-to-end, meaning the current flows through each resistor sequentially. The total (or equivalent) resistance of resistors in series is simply the sum of their individual resistances.

Fig. 1: Equivalent resistance when resistors are connected in series.

The formula for the equivalent resistance, R_eq in a series connection is:

R_eq = R₁ + R₂ + R₃ + … + R_n

Example 1: Let’s calculate the equivalent resistance for four resistors connected in series with these values:   R1​=1Ω, R2=3 Ω,        R3=5 Ω,          R4=7 Ω

Solution:

The equivalent resistance is:

R_eq = R₁ + R₂ + R₃ + R₄

Substitute the values:

R_eq = 1Ω + 3Ω + 5Ω + 7Ω = 16Ω

So, the equivalent resistance is 16 Ω

Key Points for Series Connection:

• The current through each resistor is the same.
• The voltage across each resistor is different unless all resistances are equal.
• The total voltage is the sum of the voltages across all resistors.

2. Parallel Connection:

• In a parallel connection, all resistors are connected across the same two points, so the voltage across each resistor is the same, but the current is divided between them.

Fig. 2: Equivalent resistance when resistors are connected in parallel.

The formula for the equivalent resistance, R_eq in a parallel connection is:

Or, for two resistors: 

Example 2: Let’s calculate the equivalent resistance for four resistors connected in parallel with these values:   R1​=10 Ω,  R2=20 Ω,        R3=30 Ω,            R4=40 Ω

Solution:

The formula is:

Now, take the reciprocal to find R_eq:

So, the equivalent resistance is approximately 4.8 Ω

Key Points for Parallel Connection:

• The voltage across each resistor is the same.
• The current through each resistor is different unless all resistances are equal.
• The equivalent resistance is always smaller than the smallest resistor in the network.