Resistors can be combined in parallel and in series to create new resistances. Resistors in series are pictured in this LTspice schematic. In combination, these resistors act like a single resistor with a value equal to the sum of the resistances:
|R1 + R2|
If you need a 3K resistor but only have 1K and 2K resistors, you can make an equivalent circuit by placing a 1K resistor in series with a 2K where the 3K resistor goes.
Kirchoff’s Voltage Law
Kirchoff’s Voltage Law: The sum of the electrical potential differences around a circuit must be zero.
The total potential difference across the resistors in series is the voltage supply VS. KVL says that the potential differences across the individual resistors, V1 and V2, sums to VS:
|VS = V1 + V2.|
To this relationship, we add that the currents through every point of the circuit are equal:
|IS = I1 = I2.|
You can think of this as KCL in action: the current flowing through a resistor equals the current flowing through its leads. So the current flowing through R1 must equal the current flowing through R2.
Combining these equations with Ohm’s law, we can predict the current, the voltages, and the effective resistance of two resistors in series. Ohm’s law and KCL predict that
|V1 = I1 ⋅ R1 = IS ⋅ R1||and||V2 = I2 ⋅ R2 = IS ⋅ R2.|
KVL adds that
|VS = IS ⋅ R1 + IS ⋅ R2 = IS ⋅ (R1 + R2)|
Rewritten, this equation tells us that
will be the current through the circuit.
This last equation can be reinterpreted as telling us the combined resistance of the resistors in series. If a single resistor were in the place of the series and we measured a voltage supply equal to VS and a current equal to IS then we could compute the value of that resistor using Ohm’s law as
|= R1 + R2.|
This is the resistance of two resistors in series predicted by Ohm’s law and KCL and KVL.
Notice that the order of the resistors does not matter. If R2 precedes R1 then the resistance of the series is still the sum R1 + R2. Ohm’s law, Kirchoff’s current law, and Kirchoff’s voltage law together predict that the voltage across any series of resistors is invariant to the ordering of the resistors within the series.
This invariance extends to a series of voltage sources and resistors. The current through the series is the same for every component by KCL. The voltage across the two terminals of the series is the sum of the component voltages no matter what the order. And Ohm’s law does not depend on order either.
This invariance applies to the LED supply example in my discussion of Ohm’s law. The 2V drop across the LED can occur before or after the current limiting resistor with the DC analysis unchanged. So the LED may precede or follow the current limiting resistor; the current is reduced by the resistor either way.
Resistors in series have another important feature and this feature does depend on the order of the resistors. R1 and R2 form a voltage divider at their junction:
|V2 = IS ⋅ R2 = VS||
where we have substituted in the expression for IS given above. In words, V2 is a fraction of VS determined by R1 and R2. If, for example, the resistances are equal then the voltage supply is “divided” in half.
Applying the voltage divider formula to the schematic at the top of this page, gives
|V2 = 9V||
LTspice produces this answer, as the figure shows. The message at the bottom appears when one places the mouse over the junction marked Vr after running a DC simulation. For more information on LTspice, see this introduction to LTspice. The divided voltage is often called the reference voltage and denoted by Vr on a schematic.
Voltage dividers appear frequently in stompbox circuits. On the left is a schematic for a basic fuzz face created in LTspice (see R.G. Keen’s article The Technology of the Fuzz Face). There are three voltage dividers in this schematic: the resistor pairs R3 and R4, R6 and R7, and R5 and R8.
Two of these pairs, R6 and R7 (out) and R5 and R8 (gain), are potentiometers (or pots) set at resistance mid-points. As components distinct from fixed resistors, pots are described on resistors: pots.
Thanks to idlechatterbox for finding a typo on this page!