Monday, January 5, 2015

Parallel circuits : voltage sources in parallel, a practical application—kitchen range heating element, current dividers and summary of parallel circuits

11–4 VOLTAGE SOURCES IN PARALLEL

Voltage sources, such as batteries and generators, can be interconnected in a parallel arrangement in order to increase the current output capability. Voltage sources connected in parallel must have the same voltage rating, and care must be taken to match the polarities—positive to positive and negative to negative.

Figure 11–11 shows three 12-volt batteries properly connected in a parallel arrangement. Compare this with the series connection described in the summary to Chapter 10, and note that the parallel connection involves one voltage only.

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The advantage gained with a parallel connection is based on current considerations. For example, if each battery in this illustration is capable of delivering 100 amperes of current, it is true that three batteries in parallel can deliver as much as 300 amperes. This is the same as saying that the system can deliver three times more power, or watts, than can be obtained from one battery alone. In the event that the current demand is limited to 100 amperes, the battery system would last three times as long as would one battery before it needed to be replaced or recharged.

11–5 A PRACTICAL APPLICATION—KITCHEN RANGE HEATING ELEMENT

Note: In the past, earlier designs of kitchen range heating elements lent themselves well to a demonstration of the use of series and parallel connections with a dual-voltage source.

Dual Voltage and the Three-Wire Supply

It is commonly assumed that American household appliances operate on 120 volts. However, it is economically advantageous to operate larger heating devices, such as electric kitchen ranges, water heaters, clothes dryers, and high-Btu air conditioners on a higher

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voltage, namely, 240 V. Since power is equal to the product of voltage and current (P 5 E 3 I), it stands to reason that by doubling the voltage we cut the current in half. This makes for great savings of resources because with only half the current, the circuit wiring can be installed with smaller conductors.

The common distribution system brings two voltages on three wires into a home. Figure 11–12 illustrates such a system as if it came from two series-connected generators. Each generator develops 120 volts; the two in series give 240 volts. The two outside wires at top and bottom provide the 240-volt supply for ranges and large heaters. The middle wire, in practice, is grounded securely to a cold-water pipe. The neutral wire carries only the difference in current carried by the two outside wires and therefore need not be as large as the outside wires.

These relationships of voltages and currents hold true whether the source voltage is DC or AC. On alternating current, the two circles marked G in Figure 11–12 represent the two halves of a 240-volt winding on a distribution transformer that supplies AC energy to a house. Such a wiring arrangement for AC is called three-wire, single-phase.

The Range Top Heating Element

Electric ranges connect to the neutral as well as to the 240-volt wires, thus pro- viding a dual voltage. It will be shown that this voltage of 120/240 V can be applied many different ways to a heating element similar to the one shown in Figure 11–13. Notice that this heating element is composed of two series-connected resistors with three terminals. With both 120 and 240 volts available, a multiple-contact switch allows as many as eight different heating rates to be obtained from various combinations of voltages on the two resistors of the cooking unit.

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For example, one manufacturer uses a six-position switch to obtain five different heat settings from the two heater elements in Figure 11–13, as follows:

Position #1

Off

not shown

Position #2

137 watts

with both resistors in series on 120 V

Position #3

313 watts

with the 46-ohm resistor on 120 V

Position #4

557 watts

with both resistors in parallel on 120 V

Position #5

1,252 watts

with the 46-ohm resistor on 240 V

Position #6

2,228 watts

with both resistors in parallel on 240 V

You are encouraged to confirm the information above by making the appropriate calculations.

11–6 CURRENT DIVIDERS

All parallel circuits are current dividers. As discussed previously, the sum of the currents through each branch must equal the total circuit current. More current will flow through a branch having a low resistance than a branch having a high resistance. The amount of current flowing through each branch is inversely proportional to the resistance of that branch.

In a parallel circuit, the voltage across each branch is the same. Therefore, the cur- rent flow through any branch can be computed by dividing the source voltage (ET) by the resistance of that branch.

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EXAMPLE 11–8

Three resistors having values of 40 ohms, 60 ohms, and 120 ohms are connected in parallel. The total circuit current is 3 amperes. Find the current through each branch using the current divider formula.

Solution

First find the total circuit resistance.

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SUMMARY

• There is only one voltage in parallel circuits.

• In parallel circuits, each device has its own current, which is independent of other devices.

• In parallel circuits, the total current is equal to the sum of all the individual branch currents.

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• In parallel circuits, the total circuit resistance is always smaller than any of the branch resistors.

• The total resistance of a parallel circuit can be calculated by the reciprocal formula.

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• Parallel circuits with only two resistors can be solved by the product over the sum equation.

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• Parallel circuits with all equal resistors have a total circuit resistance equal to the value of one resistor divided by the number of branches.

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• Voltage sources in parallel must have the same voltage rating.

• Voltage sources in parallel can yield more current and increased power output.

• The current divider formula can be used to determine the current through each branch of a parallel circuit.

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Achievement Review

1. Determine the total resistance of a 5-ohm resistor and a 15-ohm resistor connected in parallel.

2. Four 12-ohm resistors are connected in parallel. Calculate the total circuit resistance.

3. A 4-ohm, 12-ohm, and 16-ohm resistor are connected in parallel. Calculate the total circuit resistance by using the reciprocal formula.

4. The group of resistors mentioned in question 3 is connected in parallel across a 120-volt DC supply. Calculate the current through each resistor. Find the total current and the total circuit resistance by applying Ohm’s law.

5. Five lamps of equal resistance are connected in parallel across a 120-volt line.

If the total current supplied measures 3 amperes, what is the resistance of each

lamp?

6. The combined resistance (RT) of two lamps in parallel is 35 ohms. If the resistance of one lamp is 105 ohms, what is the resistance of the other?

7. Three equal resistors are connected in series and have a total resistance of 45 ohms. What would be their combined resistance if they were connected in parallel?

8. Find R1 and RT.

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18. Show, by calculation, how eight different heating rates can be obtained from a heating element, as shown in Figure 11–13, consisting of two resistors in series with values of 50 ohms and 65 ohms, respectively.