AcademyDC Circuits
Academy
Series and Parallel Resistor Networks
Level 1 - Physics topic page in DC Circuits.
Principle
A resistor network can be reduced when the circuit geometry forces a simple constraint. Resistors in series carry the same current. Resistors in parallel share the same potential difference.
Notation
\(R_{\mathrm{eq}}\)
equivalent resistance at two terminals
\(\mathrm{\Omega}\)
\(R_i\)
individual resistance
\(\mathrm{\Omega}\)
\(I\)
current through a branch
\(\mathrm{A}\)
\(V\)
potential difference across a network
\(\mathrm{V}\)
\(V_i\)
potential difference across one resistor
\(\mathrm{V}\)
\(P\)
electrical power dissipated
\(\mathrm{W}\)
Method
Derivation 1: Series resistors
In a series chain, charge has only one path, so the current through every resistor is the same. The total potential difference is the sum of the drops.
Potential drops add
\[V=V_1+V_2+\cdots\]
Use Ohm's law
\[V=IR_1+IR_2+\cdots\]
Factor the common current
\[V=I(R_1+R_2+\cdots)\]
Equivalent resistance
\[R_{\mathrm{eq}}=R_1+R_2+\cdots\]
Derivation 2: Parallel resistors
In a parallel group, each branch connects to the same two nodes, so every branch has the same potential difference. The total current is the sum of branch currents.
Currents add
\[I=I_1+I_2+\cdots\]
Use Ohm's law in each branch
\[I=\frac{V}{R_1}+\frac{V}{R_2}+\cdots\]
Factor the common voltage
\[I=V\left(\frac{1}{R_1}+\frac{1}{R_2}+\cdots\right)\]
Equivalent resistance
\[\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots\]
Derivation 3: Divider relations
Once a network is reduced, the unreduced circuit still contains the individual currents and voltages. Dividers recover them from the common constraint.
Series voltage divider
\[V_i=IR_i=\frac{R_i}{R_{\mathrm{eq}}}V\]
Two-branch current divider
\[I_1=\frac{V}{R_1}=\frac{R_2}{R_1+R_2}I\]
Rules
Ohm's law
\[V=IR\]
Series resistors
\[R_{\mathrm{eq}}=R_1+R_2+\cdots\]
Parallel resistors
\[\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots\]
Two resistors in parallel
\[R_{\mathrm{eq}}=\frac{R_1R_2}{R_1+R_2}\]
Power in a resistor
\[P=IV=I^2R=\frac{V^2}{R}\]
Examples
Question
A
\[12\,\mathrm{V}\]
source is connected to \[3.0\,\Omega\]
and \[5.0\,\Omega\]
in series. Find the current.Answer
\[R_{\mathrm{eq}}=8.0\,\Omega,\qquad I=\frac{12}{8.0}=1.5\,\mathrm{A}\]
Checks
- A series equivalent resistance is larger than any individual resistor in the chain.
- A parallel equivalent resistance is smaller than the smallest branch resistance.
- In series, larger resistance gets a larger voltage drop.
- In parallel, smaller resistance gets a larger current.
- Equivalent resistance preserves the terminal current for the same terminal voltage; it does not erase the internal branch behavior.