Resistors

Fig 1

The ideal resistor is a two terminal device that strictly follows Ohm's Law (R=V/I) for any given voltage across its terminals and current observed at its terminals.

Resistors have two basic functions in circuit design: 1) to limit current, and 2) to divide voltage.

When practical resistors resist the flow of a current, they create a measurable "voltage drop," and the electrical energy is converted into waste heat. The amount of heat a real resistor can convert before exploding into flames is specified by its power rating. When selecting resistors for either new designs, or when replacing bad ones for repair, the two most important characteristics are the resistance value, measured in ohms, and the power rating, given in watts.

The power rating is generally restricted to the size of resistor. Larger resistors tend to have higher power ratings than smaller resistors. When selecting a replacement for a failed resistor, a larger resistor with a high power rating is a good bet, unless its size makes it impractical for installation. This is especially true for any circuit that contains significant current. Such circuits normally have "power" in the description, like "power amp" or "power supply." Low current circuits, like signal processing circuits (preamps, EQs, effects, etc) may have 1/4W resistors simply due to the 1/4W resistor being fairly standard, but the actual circuits may only require 1/8W or less, so substituting down may be OK as long as you double check your math. Since you know the resistance value, install the new resistor, measure the voltage drop, solve for current I=V/R, then solve for P=IV. Make sure P < power rating of the resistor.

Fig 2

Series and Parallel

In Fig 2, each of the three 2 terminal networks have a total resistance of 50Ω.

R3 and R4 and in series. Resistances simply add together when in series. R3 + R4 = Rtotal.

R1 and R2 are in parallel. We use a special notation when we want to say "something in parallel with something else," and it looks like this: R1 || R2

Resistance don't add together when in parallel. The total resistance will always be less than the smallest resistance paralleled.

The reciprocal of resistance is conductance. A high resistance doesn't conduct very well, and a low resistance conducts very well... right? Mathematically, a reciprocal is just 1/x, so in this case conductance is equal to 1/R.

In parallel, conductances add together.

Expressing conductances as reciprocals of resistance, we get this formula:

1/R1 + 1/R2 = 1/Rtotal

When R1 and R2 are different values you often have to resort to a calculator or some tedious long division to get exact values. Thankfully, after a little practice you can usually get pretty close with guessing, or by following the rule that the total is always less than the smallest resistance. Finally, if R1 = R2, then Rtotal will be just R/2, i.e. 100Ω || 100Ω = 50Ω.

If there are many resistors, then the formula is R/n where n is the number of resistors and R must the same value for all of them.

Fig 3

Speakers

For a practical example of how resistor networks combine, here is a typical arrangement used in 4 speaker cabinets. The is a series-parallel arrangement. This is a popular arrangement as it has two convenient features: 1) the total impedance is the same as each single speaker's impedance, 2) if 1 speaker blows, all remaining 3 will still work.

Note that each parallel pair of speakers combines to 4Ω. Then those two pairs of 4Ω are wired in series for a final impedance of 8Ω.