Practical Electronics/Resistors/Series Proof

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This page presents a proof that the resistance of a series network, Rtot of n resistors is the sum of the resistance of each resistor in the network:

Resistors in Series.svg

First, consider that a voltage across the whole network must be equal to the sum of the voltages across each resistor (Kirchhoff's Voltage Law).

v_{tot}  = \sum\limits_{k = 1}^n {v_k },

where vk is the voltage across the kth resistor in the network. Now, from Ohm's Law, we can write:

i_{tot} R_{tot}  = \sum\limits_{k = 1}^n {i_k R_k },

where:

  • itot is the current through the entire network
  • Rtot is the resistance of the entire network
  • ik is the current in the kth resistor in the network
  • Rk is the resistance of the kth resistor in the network

Since from Kirchhoff's Current Law we know that the current in each resistor is the same, we can write

iR_{tot}  = i\sum\limits_{k = 1}^n {R_k },

where i is the current through the network. Dividing through by i gives us our final answer:

R_{tot}  = \sum\limits_{k = 1}^n {R_k}

This is a vital result and should be remembered. Notice that the proof for parallel networks is essentially the same, but uses KCL first and KVL second.

Other interesting points are that this result describes behaviour of parallel capacitive networks and series inductive networks.