Series and parallel circuits

Real circuits contain more than one load. The receptacles on a kitchen circuit, the lights on a switch, the elements in a heater and the resistors in a control circuit are connected either one after another (in series), side by side (in parallel), or in combinations of both. How they are connected decides how voltage and current divide among them, what happens when one fails, and how much current the circuit draws. Building wiring is almost entirely parallel; series connections appear in switches, fuses, overloads, control circuits and some lighting.

This chapter gives the rules for series and parallel circuits, shows how to solve combination circuits step by step, introduces Kirchhoff's two laws, and explains how opens and shorts behave in each type of circuit.

Series circuits

In a series circuit, components are connected end to end so there is only one path for current. The same current flows through every component.

The rules for series circuits:

• Current is the same through every component: \(I_T = I_1 = I_2 = I_3\).
• Resistances add: \(R_T = R_1 + R_2 + R_3 + \cdots\).
• The voltage drops add up to the source voltage: \(V_T = V_1 + V_2 + V_3 + \cdots\).
• Powers add: \(P_T = P_1 + P_2 + P_3 + \cdots\).

Voltage dividers

Because voltage divides in proportion to resistance, two resistors in series make a voltage divider, used to produce a lower voltage for a sensor or control input:

Series circuits have one serious drawback for building loads: if any component opens (a lamp burns out or a connection fails), current stops everywhere and every load goes off. Old strings of holiday lights wired in series showed this: one dead bulb darkened the whole string. Fifty bulbs in series on 120 V each get only 2.4 V.

Series connection is exactly what you want for control and protection. A switch, fuse, breaker or motor overload is placed in series with the load so that opening it stops the current. Safety interlocks are wired in series so that any one of them can stop a machine.

Parallel circuits

In a parallel circuit, every component is connected directly across the same two points, so each has its own path. The same voltage appears across every branch.

The rules for parallel circuits:

• Voltage is the same across every branch: \(V_T = V_1 = V_2 = V_3\).
• Branch currents add: \(I_T = I_1 + I_2 + I_3 + \cdots\).
• Total resistance is found from the reciprocals: \(\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots\). It is always less than the smallest branch resistance.
• Powers add.

Two shortcuts: for two resistors, \(R_T = \frac{R_1 R_2}{R_1 + R_2}\) ("product over sum"); for \(n\) equal resistors \(R\), \(R_T = R/n\).

Building wiring is parallel. Every receptacle and light on a branch circuit sees the full 120 V, each load works independently, and switching one off does not affect the others. The price is that the currents add up in the shared conductors and breaker.

Combination circuits

Most real circuits combine series and parallel parts. Solve them by simplifying step by step: replace each group of parallel resistors with its equivalent, then add series resistances, until one equivalent resistance remains. Then work back out to find each current and voltage.

Kirchhoff's laws

The series and parallel rules are special cases of two general laws, published by Gustav Kirchhoff in 1845, that apply to every circuit. Lesson 6 uses them to solve circuits that cannot be reduced to series and parallel combinations:

• Kirchhoff's current law (KCL): the total current flowing into any junction equals the total current flowing out. Charge does not pile up anywhere.
• Kirchhoff's voltage law (KVL): around any closed loop, the voltage rises (sources) equal the voltage drops (loads). Energy given to each coulomb is all used up by the time it returns.

Opens and shorts

Knowing how faults behave in each kind of circuit is the basis of troubleshooting (Lesson 12).

• An open in a series circuit stops current everywhere. The full source voltage appears across the open point, which is how you find it: measure voltage across each component in turn, and the one showing the full supply voltage is open.
• An open in one branch of a parallel circuit stops only that branch. Total current falls by that branch's current; the other loads keep working.
• A short across one component of a series circuit removes its resistance: current rises and the other components get more voltage.
• A short across a parallel branch is a short across the supply. Current is limited only by the wiring and source resistance and can be hundreds or thousands of amperes, which is what the overcurrent device must interrupt.