# Circuits and DC Instruments

## Resistors in Series and Parallel


### Learning Objectives

By the end of this section, you will be able to:
1. Draw a circuit with resistors in parallel and in series.
2. Calculate the voltage drop of a current across a resistor using Ohm’s law.
3. Contrast the way total resistance is calculated for resistors in series and in parallel.
4. Explain why total resistance of a parallel circuit is less than the smallest resistance of any of the resistors in that circuit.
5. Calculate total resistance of a circuit that contains a mixture of resistors connected in series and in parallel.
Most circuits have more than one component, called a resistor that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections illustrated in . The total resistance of a combination of resistors depends on both their individual values and how they are connected.

### Resistors in Series

When are resistors in series? Resistors are in series whenever the flow of charge, called the current, must flow through devices sequentially. For example, if current flows through a person holding a screwdriver and into the Earth, then  in (a) could be the resistance of the screwdriver’s shaft,  the resistance of its handle,  the person’s body resistance, and  the resistance of her shoes.
 shows resistors in series connected to a voltage source. It seems reasonable that the total resistance is the sum of the individual resistances, considering that the current has to pass through each resistor in sequence. (This fact would be an advantage to a person wishing to avoid an electrical shock, who could reduce the current by wearing high-resistance rubber-soled shoes. It could be a disadvantage if one of the resistances were a faulty high-resistance cord to an appliance that would reduce the operating current.)
To verify that resistances in series do indeed add, let us consider the loss of electrical power, called a voltage drop, in each resistor in .
According to Ohm’s law, the voltage drop, , across a resistor when a current flows through it is calculated using the equation , where  equals the current in amps (A) and  is the resistance in ohms . Another way to think of this is that  is the voltage necessary to make a current  flow through a resistance .
So the voltage drop across  is , that across  is , and that across  is . The sum of these voltages equals the voltage output of the source; that is,
This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation , where  is the electric charge and  is the voltage. Thus the energy supplied by the source is , while that dissipated by the resistors is
These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, . The charge  cancels, yielding , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.)
Now substituting the values for the individual voltages gives
Note that for the equivalent single series resistance , we have
This implies that the total or equivalent series resistance  of three resistors is .
This logic is valid in general for any number of resistors in series; thus, the total resistance  of a series connection is
as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

### Resistors in Parallel

 shows resistors in parallel, wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.
Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building. (See (b).)
To find an expression for the equivalent parallel resistance , let us consider the currents that flow and how they are related to resistance. Since each resistor in the circuit has the full voltage, the currents flowing through the individual resistors are , , and . Conservation of charge implies that the total current  produced by the source is the sum of these currents:
Substituting the expressions for the individual currents gives
Note that Ohm’s law for the equivalent single resistance gives
The terms inside the parentheses in the last two equations must be equal. Generalizing to any number of resistors, the total resistance  of a parallel connection is related to the individual resistances by
This relationship results in a total resistance  that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

### Combinations of Series and Parallel

More complex connections of resistors are sometimes just combinations of series and parallel. These are commonly encountered, 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 . Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult.
The simplest combination of series and parallel resistance, shown in , is also the most instructive, since it is found in many applications. For example,  could be the resistance of wires from a car battery to its electrical devices, which are in parallel.  and  could be the starter motor and a passenger compartment light. We have previously assumed that wire resistance is negligible, but, when it is not, it has important effects, as the next example indicates.

### 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  drop in the wires can also be significant.
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 . The device represented by  has a very low resistance, and so when it is switched on, a large current flows. This increased current causes a larger  drop in the wires represented by , reducing the voltage across the light bulb (which is ), which then dims noticeably.

### Test Prep for AP Courses


### Section Summary

1. The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances: 
2. Each resistor in a series circuit has the same amount of current flowing through it.
3. The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
4. The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula:
    
5. Each resistor in a parallel circuit has the same full voltage of the source applied to it.
6. The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
7. If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

### Conceptual Questions


### Problem Exercises

Note: Data taken from figures can be assumed to be accurate to three significant digits.
    