Resistor Combinations

Resistance is the opposition to the flow of current.

Combinations of Resistors
Resistors do not occur in isolation. They are almost always part of a larger circuit, and frequently that larger circuit contains many resistors. It is often the case that resistors occur in combinations that repeat.
Series Combinations of Resistors

Two elements are said to be in series whenever the same current physically flows through both of the elements. The critical point is that the same current flows through both resistors when two are in series. The particular configuration does not matter. The only thing that matters is that exactly the same current flows through both resistors. Current flows into one element, through the element, out of the element into the other element, through the second element and out of the second element. No part of the current that flows through one resistor "escapes" and none is added. This figure shows several different ways that two resistors in series might appear as part of a larger circuit diagram.

You might wonder just how often you actually find resistors in series. The answer is that you find resistors in series all the time.

An example of series resistors is in house wiring. The leads from the service entrance enter a distribution box, and then wires are strung throughout the house. The current flows out of the distribution box, through one of the wires, then perhaps through a light bulb, back through the other wire. We might model that situation with the circuit diagram shown below.

In many electronic circuits series resistors are used to get a different voltage across one of the resistors. We'll look at those circuits, called voltage dividers, in a short while. Here's the circuit diagram for a voltage divider.

Besides resistors in series, we can also have other elements in series - capacitors, inductors, diodes. These elements can be in series with other elements. For example, the simplest form of filter, for filtering low frequency noise out of a signal, can be built just by putting a resistor in series with a capacitor, and taking the output as the capacitor voltage.

As we go along you'll have lots of opportunity to use and to expand what you learn about series combinations as you study resistors in series.

Let's look at the model again. We see that the wires are actually small resistors (small value of resistance, not necessarily physically small) in series with the light bulb, which is also a resistor. We have three resistors in series although two of the resistors are small. We know that the resistors are in series because all of the current that flows out of the distribution box through the first wire also flows through the light bulb and back through the second wire, thus meeting our condition for a series connection. Trace that out in the circuit diagram and the pictorial representation above.

Let us consider the simplest case of a series resistor connection, the case of just two resistors in series. We can perform a thought experiment on these two resistors. Here is the circuit diagram for the situation we're interested in.

Imagine that they are embedded in an opaque piece of plastic, so that we only have access to the two nodes at the ends of the series connection, and the middle node is inaccessible. If we measured the resistance of the combination, what would we find? To answer that question we need to define voltage and current variables for the resistors. If we take advantage of the fact that the current through them is the same (Apply KCL at the interior node if you are unconvinced!) then we have the situation below.

Note that we have defined a voltage across each resistor (V_a and V_b) and current that flows through both resistors (I_s) and a voltage variable, V_s, for the voltage that appears across the series combination.

Let's list what we know.

• The current through the two resistors is the same.
• The voltage across the series combination is given by:
• V_s= V_a + V_b
• The voltages across the two resistors are given by Ohm's Law:
• V_a = I_s R_a
• V_b = I_s R_b

We can combine all of these relations, and when we do that we find the following.

• V_s= V_a + V_b
• V_s= I_s R_a + I_s R_b
• V_s= I_s (R_a + R_b)
• V_s= I_s R_series

Here, we take R_series to be the series equivalent of the two resistors in series, and the expression for R_series is: R_series = R_a + R_b

What do we mean by series equivalent? Here are some points to observe.

• If current and voltage are proportional, then the device is a resistor.
• We have shown thatV_s= I_s R_series, so that voltage is proportional to current, and the constant of proportionality is a resistance.
• We will call that the equivalent series resistance.

There is also a mental picture to use when considering equivalent series resistance. Imagine that you have two globs of black plastic. Each of the globs of black plasic has two wires coming out. Inside these two black plastic globs you have the following.

• In the first glob you have two resistors in series. Only the leads of the series combination are available for measurement externally. You have no way to penetrate the box and measure things at the interior node.
• In the second box you have a single resistor that is equal to the series equivalent. Only the leads of this resistor are available for measurement externally.

Then, if you measured the resistance using the two available leads in the two different cases you would not be able to tell which black plastic glob had the single resistor and which one had the series combination.

Parallel Resistors

The other common connection is two elements in parallel. Two resistors or any two devices are said to be in parallel when the same voltage physically appears across the two resistors. Schematically, the situation is as shown below.

Note that we have defined the voltage across both resistor (V_p) and the current that flows through each resistor (I_a and I_b) and a voltage variable, V_p, for the voltage that appears across the parallel combination.

Let's list what we know.

• The voltage across the two resistors is the same.
• The current through the parallel combination is given by:
• I_p= I_a + I_b
• The currents through the two resistors are given by Ohm's Law:
• I_a = V_p /R_a
• I_b = V_p /R_b

We can combine all of these relations, and when we do that we find the following.

• I_p= I_a + I_b
• I_p= V_p /R_a + V_p /R_b
• I_p= V_p[ 1/R_a + 1/R_b]
• I_p= V_p/R_parallel

Here, we take R_parallel to be the parallel equivalent of the two resistors in parallel, and the expression for R_parallel is: 1/R_parallel = 1/R_a + 1/R_b

There may be times when it is better to rearrange the expression for R_parallel. The expression can be rearranged to get:

R_parallel = (R_a*R_b)/(R_a + R_b)

Either of these expressions could be used to compute a parallel equivalent resistance. The first has a certain symmetry with the expression for a series equivalent resistance.

Parallel Resistors - A Point to Remember

• It is important to note that the equivalent resistance of two resistors in parallel is always smaller than either of the two resistors.