21.1 Resistors in Series and Parallel

Resistor circuits are building blocks of electrical systems. They come in two main configurations: series and parallel. Each type affects voltage, current, and resistance differently, so analyzing a DC circuit starts with identifying how its resistors are connected.

Ohm's law ties voltage, current, and resistance together, while the rules for combining resistors differ between series and parallel setups. Once you've got these principles down, you can break apart even complex circuits into manageable pieces.

Resistor Circuits

Circuits with resistors

Series circuits connect resistors end-to-end so that current flows through each one in sequence. Think of old-style Christmas lights strung on a single loop of wire. Because there's only one path, the same current passes through every resistor. Each resistor causes a voltage drop, and those drops add up to the total source voltage.

Parallel circuits connect resistors side-by-side, giving current multiple paths to follow. Home wiring works this way: every outlet sees the full 120 V regardless of what's plugged in elsewhere. The voltage across each branch is the same, but the current splits among the branches. The total current drawn from the source is the sum of the branch currents.

Ohm's law for voltage drops

Ohm's law relates voltage, current, and resistance:

$V = IR$

• $V$ = voltage (volts, V)
• $I$ = current (amperes, A)
• $R$ = resistance (ohms, Ω)

In a series circuit, current is the same through every resistor, so you apply Ohm's law to each one individually:

• $V_1 = IR_1$, $V_2 = IR_2$, and so on
• The voltage drops must add up to the source voltage: $V_{source} = V_1 + V_2 + \dots + V_n$

In a parallel circuit, voltage is the same across every branch, so you use Ohm's law to find each branch current:

• $I_1 = V/R_1$, $I_2 = V/R_2$, and so on
• The branch currents add up to the total current: $I_{total} = I_1 + I_2 + \dots + I_n$

Resistance calculation in series vs parallel

Series: Total (equivalent) resistance is the straight sum of individual resistances.

$R_{total} = R_1 + R_2 + \dots + R_n$

This makes intuitive sense: the current has to push through each resistor one after another, so every resistor adds to the total opposition.

Parallel: The reciprocals of the individual resistances add up to the reciprocal of the total resistance.

$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$

Why reciprocals? In parallel, you're really adding conductances (the ease of current flow, $G = 1/R$). More paths mean more total conductance, which means less total resistance. The total resistance of a parallel combination is always less than the smallest individual resistor in the group.

Parallel circuits and total resistance

Each resistor you add in parallel opens another path for current. More paths means less overall opposition, so the equivalent resistance drops.

Here's a concrete example with two 10 Ω resistors in parallel:

1. Write the reciprocal formula: $\frac{1}{R_{total}} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10}$
2. Flip to get $R_{total}$: $R_{total} = \frac{10}{2} = 5 \; \Omega$

The result (5 Ω) is half of either individual resistor. For the special case of two resistors in parallel, there's a handy shortcut:

$R_{total} = \frac{R_1 \cdot R_2}{R_1 + R_2}$

This "product over sum" formula saves time and avoids the reciprocal step when you only have two resistors.

Complex circuit resistance calculations

Most real circuits mix series and parallel sections. The strategy is to simplify from the inside out:

1. Identify which resistors are in series and which are in parallel.
2. Reduce parallel groups first using the reciprocal formula. For example, if $R_4$ and $R_5$ are in parallel: $\frac{1}{R_{parallel}} = \frac{1}{R_4} + \frac{1}{R_5}$
3. Replace that parallel group with its single equivalent resistance.
4. Add series resistances that are now in a straight line: $R_{series} = R_1 + R_2 + R_{parallel}$
5. Repeat steps 1-4 until the entire circuit is reduced to one equivalent resistance.

Once you have the single equivalent resistance, use $V = IR$ with the source voltage to find the total current. Then work back outward through the circuit to find individual voltage drops and branch currents.

Circuit Analysis Principles

Kirchhoff's Current Law (KCL): The total current entering any junction equals the total current leaving it. This is just conservation of charge: charge can't pile up at a point in the wire.

Kirchhoff's Voltage Law (KVL): The sum of all voltage gains and drops around any closed loop in a circuit equals zero. This is conservation of energy: a charge that travels around a complete loop returns to the same potential it started at.

Power dissipation in a resistor is calculated with:

$P = I^2 R$

Power is measured in watts (W). You can also write equivalent forms using Ohm's law: $P = IV$ or $P = V^2/R$.

A few terms worth knowing:

• Branch current: the current flowing through one specific path in a parallel circuit.
• Junction (or node): a point where three or more conductors meet. This is where you apply KCL.