DC circuit analysis

Circuit elements and steady state

A DC circuit is a network in which currents and voltages are constant in time after transients have ended. Common ideal elements include batteries, wires, resistors, switches, capacitors, and meters.

An ideal wire has negligible resistance and is treated as an equipotential connection. An ideal battery maintains a fixed potential difference, called emf, often written $mathcal{E}$.

Resistors in series

Resistors are in series when the same current passes through each one. The voltage drops add:

$V_{total}=V_1+V_2+cdots$

Using $V_i=IR_i$,

$V_{total}=I(R_1+R_2+cdots)$

Thus the equivalent resistance is

$R_{eq}=R_1+R_2+cdots$

Series resistances add because each resistor provides another obstacle to the same current.

Voltage division

For two series resistors connected across total voltage $V$, the current is

I=rac{V}{R_1+R_2}

The voltage across $R_1$ is

V_1=IR_1=Vrac{R_1}{R_1+R_2}

This is the voltage divider rule. Larger resistance gets a larger share of the voltage drop in series.

Resistors in parallel

Resistors are in parallel when they share the same two nodes and therefore the same voltage. The currents add:

$I_{total}=I_1+I_2+cdots$

Using $I_i=V/R_i$,

ight)$$ Thus $$rac{1}{R_{eq}}=rac{1}{R_1}+rac{1}{R_2}+cdots$$ Parallel resistors reduce equivalent resistance by providing additional paths for current. ## Current division For two parallel resistors, the current divides inversely with resistance. The current through $R_1$ is $$I_1=I_{total}rac{R_2}{R_1+R_2}$$ Lower resistance carries more current. ## Power in circuits Power dissipated in a resistor is $$P=IV=I^2R=rac{V^2}{R}$$ In series, the same current flows through each resistor, so larger resistance dissipates more power. In parallel, the same voltage appears across each resistor, so smaller resistance dissipates more power. ## Capacitors in DC steady state In a DC circuit after a long time, an ideal capacitor acts like an open circuit because no steady current flows through the insulating gap. During charging or discharging, current can flow temporarily, but steady DC current through an ideal capacitor is zero. The charge on a capacitor is $$Q=CDelta V$$ ## Strategy for circuit reduction For simple resistor networks, combine obvious series and parallel groups step by step. After finding total current from the source, work backward to find branch currents and voltage drops. Always check units and power conservation. If a circuit cannot be reduced by simple series-parallel combinations, Kirchhoff's rules are needed. ## The big idea DC circuit analysis relies on conservation of charge and energy. Series elements share current, parallel elements share voltage, and equivalent resistance simplifies networks. Voltage division, current division, and power formulas provide practical tools for analyzing electrical systems.