10.2 Resistors in Series and Parallel

Resistors in Series

Resistors are said to be in series whenever the current flows through the resistors sequentially. Consider Figure 10.12, which shows three resistors in series with an applied voltage equal to $V_{ab}.$ Since there is only one path for the charges to flow through, the current is the same through each resistor. The equivalent resistance of a set of resistors in a series connection is equal to the algebraic sum of the individual resistances.

Figure 10.12 (a) Three resistors connected in series to a voltage source. (b) The original circuit is reduced to an equivalent resistance and a voltage source.

In Figure 10.12, the current coming from the voltage source flows through each resistor, so the current through each resistor is the same. The current through the circuit depends on the voltage supplied by the voltage source and the resistance of the resistors. For each resistor, a potential drop occurs that is equal to the loss of electric potential energy as a current travels through each resistor. According to Ohm’s law, the potential drop V across a resistor when a current flows through it is calculated using the equation $V=IR,$ where I is the current in amps (A) and R is the resistance in ohms $\left(\text{Ω}\right).$ Since energy is conserved, and the voltage is equal to the potential energy per charge, the sum of the voltage applied to the circuit by the source and the potential drops across the individual resistors around a loop should be equal to zero:

This equation is often referred to as Kirchhoff’s loop law, which we will look at in more detail later in this chapter. For Figure 10.12, the sum of the potential drop of each resistor and the voltage supplied by the voltage source should equal zero:

Since the current through each component is the same, the equality can be simplified to an equivalent resistance, which is just the sum of the resistances of the individual resistors.

Any number of resistors can be connected in series. If N resistors are connected in series, the equivalent resistance is

One result of components connected in a series circuit is that if something happens to one component, it affects all the other components. For example, if several lamps are connected in series and one bulb burns out, all the other lamps go dark.

Example 10.2

Equivalent Resistance, Current, and Power in a Series Circuit

A battery with a terminal voltage of 9 V is connected to a circuit consisting of four $20\text{-}\text{Ω}$ and one $10\text{-}\text{Ω}$ resistors all in series (Figure 10.13). Assume the battery has negligible internal resistance. (a) Calculate the equivalent resistance of the circuit. (b) Calculate the current through each resistor. (c) Calculate the potential drop across each resistor. (d) Determine the total power dissipated by the resistors and the power supplied by the battery.

Figure 10.13 A simple series circuit with five resistors.

Strategy

In a series circuit, the equivalent resistance is the algebraic sum of the resistances. The current through the circuit can be found from Ohm’s law and is equal to the voltage divided by the equivalent resistance. The potential drop across each resistor can be found using Ohm’s law. The power dissipated by each resistor can be found using $P=I^{2}R$, and the total power dissipated by the resistors is equal to the sum of the power dissipated by each resistor. The power supplied by the battery can be found using $P=I\epsilon$.

Solution

1. The equivalent resistance is the algebraic sum of the resistances:
   $$R_{\text{S}}=R_1+R_2+R_3+R_4+R_5=20\text{Ω}+20\text{Ω}+20\text{Ω}+20\text{Ω}+10\text{Ω}=90\text{Ω}.$$
2. The current through the circuit is the same for each resistor in a series circuit and is equal to the applied voltage divided by the equivalent resistance:
   $$I=\frac{V}{R_{\text{S}}}=\frac{9\text{V}}{90\text{Ω}}=0.1\text{A.}$$
3. The potential drop across each resistor can be found using Ohm’s law:
   $$\begin{array}{c}{V}_1=V_2=V_3=V_4=\left(0.1\text{A}\right)20\text{Ω}=2\text{V},\hfill \\ V_5=\left(0.1\text{A}\right)10\text{Ω}=1\text{V},\hfill \\ V_1+V_2+V_3+V_4+V_5=9\text{V}.\hfill \end{array}$$
   Note that the sum of the potential drops across each resistor is equal to the voltage supplied by the battery.
4. The power dissipated by a resistor is equal to $P=I^{2}R$, and the power supplied by the battery is equal to $P=I\epsilon$:
   $$\begin{array}{}\\ P_1=P_2=P_3=P_4={\left(0.1\text{A}\right)}^2\left(20\text{Ω}\right)=0.2\text{W},\hfill \\ P_5={\left(0.1\text{A}\right)}^2\left(10\text{Ω}\right)=0.1\text{W},\hfill \\ P_{\text{dissipated}}=0.2\text{W}+0.2\text{W}+0.2\text{W}+0.2\text{W}+0.1\text{W}=0.9\text{W},\hfill \\ P_{\text{source}}=I\epsilon =\left(0.1\text{A}\right)\left(9\text{V}\right)=0.9\text{W}.\hfill \end{array}$$

Significance

There are several reasons why we would use multiple resistors instead of just one resistor with a resistance equal to the equivalent resistance of the circuit. Perhaps a resistor of the required size is not available, or we need to dissipate the heat generated, or we want to minimize the cost of resistors. Each resistor may cost a few cents to a few dollars, but when multiplied by thousands of units, the cost saving may be appreciable.

Let’s briefly summarize the major features of resistors in series:

1. Series resistances add together to get the equivalent resistance:
   $$R_{\text{S}}=R_1+R_2+R_3+\text{⋯}+R_{N-1}+R_N={\displaystyle \sum _{i=1}^N{R}_i}.$$
2. The same current flows through each resistor in series.
3. Individual resistors in series do not get the total source voltage, but divide it. The total potential drop across a series configuration of resistors is equal to the sum of the potential drops across each resistor.

Resistors in Parallel

Figure 10.14 shows resistors in parallel, wired to a voltage source. Resistors are in parallel when one end of all the resistors are connected by a continuous wire of negligible resistance and the other end of all the resistors are also connected to one another through a continuous wire of negligible resistance. The potential drop across each resistor is the same. Current through each resistor can be found using Ohm’s law $I=V\text{/}R,$ where the voltage is constant across each resistor. For example, an automobile’s headlights, radio, and other systems are wired in parallel, so that each subsystem utilizes the full voltage of the source and can operate completely independently. The same is true of the wiring in your house or any building.

Figure 10.14 (a) Two resistors connected in parallel to a voltage source. (b) The original circuit is reduced to an equivalent resistance and a voltage source.

The current flowing from the voltage source in Figure 10.14 depends on the voltage supplied by the voltage source and the equivalent resistance of the circuit. In this case, the current flows from the voltage source and enters a junction, or node, where the circuit splits flowing through resistors $R_1$ and $R_2$. As the charges flow from the battery, some go through resistor $R_1$ and some flow through resistor $R_2.$ The sum of the currents flowing into a junction must be equal to the sum of the currents flowing out of the junction:

This equation is referred to as Kirchhoff’s junction rule and will be discussed in detail in the next section. In Figure 10.14, the junction rule gives $I=I_1+I_2$. There are two loops in this circuit, which leads to the equations $V=I_1{R}_1$ and $I_1{R}_1=I_2{R}_2$. Note the voltage across the resistors in parallel are the same $\left(V=V_1=V_2\right)$ and the current is additive:

Generalizing to any number of N resistors, the equivalent resistance $R_{\text{P}}$ of a parallel connection is related to the individual resistances by

This relationship results in an equivalent resistance $R_{\text{P}}$ that is less than the smallest of the individual resistances. When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, so the total resistance is lower.

Let us summarize the major features of resistors in parallel:

1. Equivalent resistance is found from
   $$R_{\text{P}}={\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\text{⋯}+\frac{1}{R_{N-1}}+\frac{1}{R_N}\right)}^{\mathrm{-1}}={\left({\displaystyle \sum _{i=1}^N\frac{1}{R_i}}\right)}^{\mathrm{-1}},$$
   and is smaller than any individual resistance in the combination.
2. The potential drop across each resistor in parallel is the same.
3. Parallel resistors do not each get the total current; they divide it. The current entering a parallel combination of resistors is equal to the sum of the current through each resistor in parallel.

In this chapter, we introduced the equivalent resistance of resistors connect in series and resistors connected in parallel. You may recall that in Capacitance, we introduced the equivalent capacitance of capacitors connected in series and parallel. Circuits often contain both capacitors and resistors. Table 10.1 summarizes the equations used for the equivalent resistance and equivalent capacitance for series and parallel connections.

Combinations of Series and Parallel

More complex connections of resistors are often just combinations of series and parallel connections. Such combinations are common, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel.

Combinations of series and parallel can be reduced to a single equivalent resistance using the technique illustrated in Figure 10.15. Various parts can be identified as either series or parallel connections, reduced to their equivalent resistances, and then further reduced until a single equivalent resistance is left. The process is more time consuming than difficult. Here, we note the equivalent resistance as $R_{\text{eq}}.$

Figure 10.15 (a) The original circuit of four resistors. (b) Step 1: The resistors $R_3$ and $R_4$ are in series and the equivalent resistance is $R_{34}=10\text{Ω}.$ (c) Step 2: The reduced circuit shows resistors $R_2$ and $R_{34}$ are in parallel, with an equivalent resistance of $R_{234}=5\text{Ω}.$ (d) Step 3: The reduced circuit shows that $R_1$ and $R_{234}$ are in series with an equivalent resistance of $R_{1234}=12\text{Ω},$ which is the equivalent resistance $R_{\text{eq}}.$ (e) The reduced circuit with a voltage source of $V=24\text{V}$ with an equivalent resistance of $R_{\text{eq}}=12\text{Ω}.$ This results in a current of $I=2\text{A}$ from the voltage source.

Notice that resistors $R_3$ and $R_4$ are in series. They can be combined into a single equivalent resistance. One method of keeping track of the process is to include the resistors as subscripts. Here the equivalent resistance of $R_3$ and $R_4$ is

The circuit now reduces to three resistors, shown in Figure 10.15(c). Redrawing, we now see that resistors $R_2$ and $R_{34}$ constitute a parallel circuit. Those two resistors can be reduced to an equivalent resistance:

This step of the process reduces the circuit to two resistors, shown in in Figure 10.15(d). Here, the circuit reduces to two resistors, which in this case are in series. These two resistors can be reduced to an equivalent resistance, which is the equivalent resistance of the circuit:

The main goal of this circuit analysis is reached, and the circuit is now reduced to a single resistor and single voltage source.

Now we can analyze the circuit. The current provided by the voltage source is $I=\frac{V}{R_{\text{eq}}}=\frac{24\text{V}}{12\text{Ω}}=2\text{A}.$ This current runs through resistor $R_1$ and is designated as $I_1.$ The potential drop across $R_1$ can be found using Ohm’s law:

Looking at Figure 10.15(c), this leaves $24\text{V}-14\text{V}=10\text{V}$ to be dropped across the parallel combination of $R_2$ and $R_{34}.$ The current through $R_2$ can be found using Ohm’s law:

The resistors $R_3$ and $R_4$ are in series so the currents $I_3$ and $I_4$ are equal to

Using Ohm’s law, we can find the potential drop across the last two resistors. The potential drops are $V_3=I_3{R}_3=6\text{V}$ and $V_4=I_4{R}_4=4\text{V}.$ The final analysis is to look at the power supplied by the voltage source and the power dissipated by the resistors. The power dissipated by the resistors is

The total energy is constant in any process. Therefore, the power supplied by the voltage source is $P_s=IV=(2\text{A})(24\text{V})=48\text{W}.$ Analyzing the power supplied to the circuit and the power dissipated by the resistors is a good check for the validity of the analysis; they should be equal.

Example 10.4

Combining Series and Parallel Circuits

Figure 10.16 shows resistors wired in a combination of series and parallel. We can consider $R_1$ to be the resistance of wires leading to $R_2$ and $R_3.$ (a) Find the equivalent resistance of the circuit. (b) What is the potential drop $V_1$ across resistor $R_1$? (c) Find the current $I_2$ through resistor $R_2$. (d) What power is dissipated by $R_2$?

Figure 10.16 These three resistors are connected to a voltage source so that $R_2$ and $R_3$ are in parallel with one another and that combination is in series with $R_1.$

Strategy

(a) To find the equivalent resistance, first find the equivalent resistance of the parallel connection of $R_2$ and $R_3.$ Then use this result to find the equivalent resistance of the series connection with $R_1.$

(b) The current through $R_1$ can be found using Ohm’s law and the voltage applied. The current through $R_1$ is equal to the current from the battery. The potential drop $V_1$ across the resistor $R_1$ (which represents the resistance in the connecting wires) can be found using Ohm’s law.

(c) The current through $R_2$ can be found using Ohm’s law $I_2=\frac{V_2}{R_2}.$ The voltage across $R_2$ can be found using $V_2=V-V_1.$

(d) Using Ohm’s law $\left(V_2=I_2{R}_2\right)$, the power dissipated by the resistor can also be found using$P_2=I_2^2{R}_2=\frac{V_2^2}{R_2}$.

Solution

1. To find the equivalent resistance of the circuit, notice that the parallel connection of $R_2$ and $R_3$ is in series with $R_1$, so the equivalent resistance is
   $$R_{\text{eq}}=R_1+{\left(\frac{1}{R_2}+\frac{1}{R_3}\right)}^{\mathrm{-1}}=1.00\text{Ω}+{\left(\frac{1}{6.00\text{Ω}}+\frac{1}{13.00\text{Ω}}\right)}^{\mathrm{-1}}=5.10\text{Ω}.$$
   The total resistance of this combination is intermediate between the pure series and pure parallel values ($20.0\text{Ω}$ and $0.804\text{Ω}$, respectively).
2. The current through $R_1$ is equal to the current supplied by the battery:
   $$I_1=I=\frac{V}{R_{\text{eq}}}=\frac{12.0\text{V}}{5.10\text{Ω}}=2.35\text{A}.$$
   The voltage across $R_1$ is
   $$V_1=I_1{R}_1=(2.35\text{A})(1\text{Ω})=2.35\text{V}.$$
   The voltage applied to $R_2$ and $R_3$ is less than the voltage supplied by the battery by an amount $V_1.$ When wire resistance is large, it can significantly affect the operation of the devices represented by $R_2$ and $R_3$.
3. To find the current through $R_2$, we must first find the voltage applied to it. The voltage across the two resistors in parallel is the same:
   $$V_2=V_3=V-V_1=12.0\text{V}-2.35\text{V}=9.65\text{V}.$$
   Now we can find the current $I_2$ through resistance $R_2$ using Ohm’s law:
   $$I_2=\frac{V_2}{R_2}=\frac{9.65\text{V}}{6.00\text{Ω}}=1.61\text{A}.$$
   The current is less than the 2.00 A that flowed through $R_2$ when it was connected in parallel to the battery in the previous parallel circuit example.
4. The power dissipated by $R_2$ is given by
   $$P_2=I_2^2{R}_2={(1.61\text{A})}^2(6.00\text{Ω})=15.5\text{W}.$$

Significance

The analysis of complex circuits can often be simplified by reducing the circuit to a voltage source and an equivalent resistance. Even if the entire circuit cannot be reduced to a single voltage source and a single equivalent resistance, portions of the circuit may be reduced, greatly simplifying the analysis.

Practical Implications

One implication of this last example is that resistance in wires reduces the current and power delivered to a resistor. If wire resistance is relatively large, as in a worn (or a very long) extension cord, then this loss can be significant. If a large current is drawn, the IR drop in the wires can also be significant and may become apparent from the heat generated in the cord.

For example, when you are rummaging in the refrigerator and the motor comes on, the refrigerator light dims momentarily. Similarly, you can see the passenger compartment light dim when you start the engine of your car (although this may be due to resistance inside the battery itself).

What is happening in these high-current situations is illustrated in Figure 10.17. The device represented by $R_3$ has a very low resistance, so when it is switched on, a large current flows. This increased current causes a larger IR drop in the wires represented by $R_1$, reducing the voltage across the light bulb (which is $R_2$), which then dims noticeably.

Figure 10.17 Why do lights dim when a large appliance is switched on? The answer is that the large current the appliance motor draws causes a significant IR drop in the wires and reduces the voltage across the light.