Practical Electronics/Resistors/Parallel Proof

This page presents a proof that the resistance of a parallel network, R_tot of n resistors is the reciprocal of the sum of the reciprocals of the individual resistors:

First, consider that a current into (and out of) the whole network, i_tot, must be equal to the sum of the currents in each resistor (Kirchhoff's Current Law).

$i_{tot}=\sum \limits _{k=1}^{n}{i_{k}},$

where i_k is the current in the k^th resistor in the network. Now, from Ohm's Law, we can write:

${{v_{tot}} \over {R_{tot}}}=\sum \limits _{k=1}^{n}{{v_{k}} \over {R_{k}}},$

where:

• v_tot is the voltage across the entire network
• R_tot is the resistance of the entire network
• v_k is the voltage across the k^th resistor in the network
• R_k is the resistance of the k^th resistor in the network

Since we know that the voltage across each resistor is the same (a trivial case of Kirchhoff's Voltage Law), we can say

${{v} \over {R_{tot}}}=\sum \limits _{k=1}^{n}{{v} \over {R_{k}}},$

where v is the voltage across the network. Dividing through by v gives us our final answer:

${{1} \over {R_{tot}}}=\sum \limits _{k=1}^{n}{{1} \over {R_{k}}}.$

This is also sometimes equivalently written as:

$R_{tot}={1 \over {\sum \limits _{k=1}^{n}{1 \over {R_{k}}}}}$

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

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