Note that both the currents and powers in parallel connections are greater than for the same devices in series. 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 Figure 4. 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. Figure 4. This combination of seven resistors has both series and parallel parts. Each is identified and reduced to an equivalent resistance, and these are further reduced until a single equivalent resistance is reached.
The simplest combination of series and parallel resistance, shown in Figure 4, is also the most instructive, since it is found in many applications. For example, R 1 could be the resistance of wires from a car battery to its electrical devices, which are in parallel. R 2 and R 3 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.
Figure 5 shows the resistors from the previous two examples wired in a different way—a combination of series and parallel. We can consider R 1 to be the resistance of wires leading to R 2 and R 3. Figure 5. 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.
To find the total resistance, we note that R 2 and R 3 are in parallel and their combination R p is in series with R 1. Thus the total equivalent resistance of this combination is. First, we find R p using the equation for resistors in parallel and entering known values:. The total resistance of this combination is intermediate between the pure series and pure parallel values Thus its IR drop is.
We must find I before we can calculate V 1. The voltage applied to R 2 and R 3 is less than the total voltage 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. To find the current through R 2 , we must first find the voltage applied to it. We call this voltage V p , because it is applied to a parallel combination of resistors.
The voltage applied to both R 2 and R 3 is reduced by the amount V 1 , and so it is. The current is less than the 2. The power is less than the 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. 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 6. The device represented by R 3 has a very low resistance, and 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 6. 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 drop in the wires and reduces the voltage across the light. A switch has a variable resistance that is nearly zero when closed and extremely large when open, and it is placed in series with the device it controls. Explain the effect the switch in Figure 7 has on current when open and when closed. Figure 7. A switch is ordinarily in series with a resistance and voltage source.
Ideally, the switch has nearly zero resistance when closed but has an extremely large resistance when open. Note that in this diagram, the script E represents the voltage or electromotive force of the battery.
There is a voltage across an open switch, such as in Figure 7. Why, then, is the power dissipated by the open switch small? A student in a physics lab mistakenly wired a light bulb, battery, and switch as shown in Figure 8. Explain why the bulb is on when the switch is open, and off when the switch is closed.
Do not try this—it is hard on the battery! Figure 8. Knowing that the severity of a shock depends on the magnitude of the current through your body, would you prefer to be in series or parallel with a resistance, such as the heating element of a toaster, if shocked by it? Some strings of holiday lights are wired in series to save wiring costs.
An old version utilized bulbs that break the electrical connection, like an open switch, when they burn out. If one such bulb burns out, what happens to the others? If such a string operates on V and has 40 identical bulbs, what is the normal operating voltage of each? Newer versions use bulbs that short circuit, like a closed switch, when they burn out. To solve for the total current, you must first determine individual branch currents using Ohms law:.
Whenever more resistances are connected in parallel, they have the effect of reducing the overall circuit resistance. The net resistance of a parallel circuit is always less than any of the individual resistance values. Skip to content Electric Circuits. Previous: Series Circuits. This is analogous to the constant change in voltage across a parallel circuit. Voltage is the potential energy across each resistor. The analogy quickly breaks down when considering the energy. In the waterfall, the potential energy is converted into kinetic energy of the water molecules.
In the case of electrons flowing through a resistor, the potential drop is converted into heat and light, not into the kinetic energy of the electrons. The potential drop across each resistor in parallel is the same. 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. Figure summarizes the equations used for the equivalent resistance and equivalent capacitance for series and parallel connections. 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. 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. Notice that resistors and 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 and is. The circuit now reduces to three resistors, shown in Figure c. Redrawing, we now see that resistors and 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 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 resistors and are in series so the currents and are equal to.
The potential drops are and 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 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. Combining Series and Parallel Circuits Figure shows resistors wired in a combination of series and parallel.
We can consider to be the resistance of wires leading to and a Find the equivalent resistance of the circuit. Strategy a To find the equivalent resistance, first find the equivalent resistance of the parallel connection of and Then use this result to find the equivalent resistance of the series connection with.
The current through is equal to the current from the battery. The total resistance of this combination is intermediate between the pure series and pure parallel values and , respectively. The current through is equal to the current supplied by the battery: The voltage across is. The voltage applied to and is less than the voltage supplied by the battery by an amount When wire resistance is large, it can significantly affect the operation of the devices represented by and.
To find the current through , we must first find the voltage applied to it. The current is less than the 2. The power dissipated by is given by 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.
Check Your Understanding Consider the electrical circuits in your home. Give at least two examples of circuits that must use a combination of series and parallel circuits to operate efficiently. All the overhead lighting circuits are in parallel and connected to the main supply line, so when one bulb burns out, all the overhead lighting does not go dark.
Each overhead light will have at least one switch in series with the light, so you can turn it on and off. A refrigerator has a compressor and a light that goes on when the door opens.
There is usually only one cord for the refrigerator to plug into the wall. The circuit containing the compressor and the circuit containing the lighting circuit are in parallel, but there is a switch in series with the light. A thermostat controls a switch that is in series with the compressor to control the temperature of the refrigerator. 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. The device represented by 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 , reducing the voltage across the light bulb which is , which then dims noticeably. Problem-Solving Strategy: Series and Parallel Resistors Draw a clear circuit diagram, labeling all resistors and voltage sources. This step includes a list of the known values for the problem, since they are labeled in your circuit diagram. Identify exactly what needs to be determined in the problem identify the unknowns.
A written list is useful. Determine whether resistors are in series, parallel, or a combination of both series and parallel. Examine the circuit diagram to make this assessment. Resistors are in series if the same current must pass sequentially through them. Use the appropriate list of major features for series or parallel connections to solve for the unknowns. There is one list for series and another for parallel.
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