Series and Parallel Circuits 2: Resistors

  1. Series and Parallel Circuits 1: The Basics
  2. Series and Parallel Circuits 2: Resistors (this article)
  3. Series and Parallel Circuits 3: Capacitors
  4. AC Series and Parallel Circuits 4: Inductors and Capacitors
  5. Series and Parallel Circuits 5: More About Circuits

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Resistors act differently when placed in series or in parallel configuration, as shown below.  In series configuration, the current through R1 is the same as the current through R2, R3 and R4, while in parallel configuration, the current through R1 is I = V/R1, where V is the voltage across the two common circuit nodes.   Similarly, the current through R2 is V/R2, the current through R3 is V/R3, and the current through R4 is V/R4.

So, for the series circuit:

where:

I is the current through the series circuit (A)

VT is total voltage drop across the leftmost and rightmost nodes of the series circuit (V)

RT is the total resistance = R1 + R2 + R3 + R4 (Ω)

and in the series circuit, the voltage drops across each resistor such that:

and in the series circuit, the current (I) is the same throughout the circuit:

where:

                                   IT is the current throughout the series circuit (A)

                                   Ii is the current through component i of the series circuit (A)

                                   n is the nth component of the series circuit

and in the series circuit, the total voltage (VT) is the sum of the voltages across each component:

where:

                                   Vi is the voltage across component i in the series circuit (V)

                                   I is the current throughout the entire series circuit (A)

                                   Ri is the resistance of component i in the series circuit (Ω)

                                   n is the total number of components in the series circuit

and the total resistance (RT) of the series circuit is the sum of the resistances (Ri) of all of the components in the circuit:

For the parallel circuit, voltage (V) is constant across the legs of the circuit and the total current (IT) is the sum of the currents through each of the legs of the circuit (which is proportional to the inverse of the resistance of that particular leg of the circuit) such that:

where:

                             IT is the total current through all legs of the circuit (A)

                             Ii is the current through leg i of the circuit (A)

                             V is the voltage across all legs of the circuit (V)

                             Ri is the resistance of leg i of the circuit (Ω)

and the resistance of the entire parallel circuit follows the relation:

and for a parallel circuit in general:

and for the simplified case, where the resistances in each leg of a parallel circuit are all the same (RL), the total resistance (RT) is:

where:

                                   RT is the total resistance of the parallel circuit (Ω)

RL is the resistance of leg L of the parallel circuit where all legs have the same resistance (Ω)

                                   n is the number of legs of the parallel circuit

Thus, in a series circuit, voltages and resistances are additive, while current is the same throughout.  And in a parallel circuit, the voltage drop across each parallel leg is the same, while the current though each leg is proportional to the inverse of the resistance of that leg.  Further, in a parallel circuit, the currents through each leg of the parallel circuit add together to achieve the total current of the circuit.  And the inverse of the total resistance of the parallel circuit is proportional to the sum of the inverse of the resistances of each of the legs of the circuit.  In the case where the resistances of all legs of a parallel circuit are the same, the total resistance of the parallel circuit is the resistance of each leg (RL) divided by the number of legs of the parallel circuit (n).   

The above discussion applies to resistors as well as to all devices that flow current such as inductors (including ferrites, chokes, coils, etc.), diodes, LEDs and even capacitors when they flow current (e.g., ripple currents under AC, etc.).  We will discuss capacitances (capacitors) and capacities (batteries) in series and parallel configurations in the next post. 

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