Learning Goal: To understand the concepts underlying the operation of transforme

Question

Learning Goal:

To understand the concepts underlying the operation of transformers.

One of the advantages of alternating current (ac) over direct current (dc) is the ease with which voltage levels can be increased or decreased. Such a need is always present due to the practical requirements of energy distribution. On the one hand, the voltage supplied to end users must be reasonably low for safety reasons (depending on the country, that voltage may be 110 volts, 220 volts, or some other value of that order). On the other hand, the voltage used in transmitting electric energy must be as high as possible to minimize i2R losses in the transmission lines. A device that uses the principle of electromagnetic induction to increase or decrease the voltage by a certain factor is called a transformer.

The main components of a transformer are two coils (windings) that are electrically insulated from each other. The coils are wrapped around the same core, which is typically made of a material with a very large relative permeability to ensure maximum mutual inductance. One coil, called the primary coil, is connected to a voltage source; the other, thesecondary coil, delivers the power. The alternating current in the primary coil induces the changing magnetic flux in the core that creates the emf in the secondary coil. The magnitude of the emf induced in the secondary coil can be controlled by the design of the transformer. The key factor is the number of turns in each coil.

Consider an ideal transformer, that is, one in which the coils have no ohmic resistance and the magnetic flux ?B is the same for each turn of both the primary and secondary coils. If the number of turns in the primary coil is N1 and that in the secondary coil is N2, then the emfs induced in the coils can be written as

E1=?N1??B?tandE2=?N2??B?t,

and therefore,

E2E1=N2N1.

If the coils have zero resistance (as we have assumed), then for each coil the terminal voltage will be equal to the induced emf. We can then write

V2V1=N2N1.

Note that if N2>N1, then V2>V1. In this case we have a step-up transformer. Conversely, if N2<N1, then V2<V1, and we are dealing with astep-down transformer. Without energy losses, the power in the primary and secondary coils is the same:

V1I1=V2I2.

Part A

The primary coil of a transformer contains 100 turns; the secondary has 200 turns. The primary coil is connected to a size-AA battery that supplies a constant voltage of 1.5 volts. What voltage would be measured across the secondary coil?

Part B

A transformer is intended to decrease the value of the alternating voltage from 500 volts to 25 volts. The primary coil contains 200 turns. Find the necessary number of turns N2 in the secondary coil.

Part C

A transformer is intended to decrease the value of the alternating current from 500 amperes to 25 amperes. The primary coil contains 200 turns. Find the necessary number of turns N2 in the secondary coil.

Part D

In a transformer, the primary coil contains 400 turns, and the secondary coil contains 80 turns. If the primary current is 2.5 amperes, what is the secondary current I2?

Express your answer numerically in amperes.

Part E

The primary coil of a transformer has 200 turns and the secondary coil has 800 turns. The power supplied to the primary coil is 400 watts. What is the power generated in the secondary coil if it is terminated by a 20-ohm resistor?

Part F

A transformer supplies 60 watts of power to a device that is rated at 20 volts. The primary coil is connected to a 120-volt ac source. What is the current I1 in the primary coil?

Express your answer in amperes.

Part G

The voltage and the current in the primary coil of a nonideal transformer are 120 volts and 2.0 amperes. The voltage and the current in the secondary coil are 19.4 volts and 11.8 amperes. What is the efficiency e of the transformer? The efficiency of a transformer is defined as the ratio of the output power to the input power, expressed as a percentage: e=100Pout/Pin.

Express your answer as a percentage

Explanation / Answer

A) There would be no voltage across the secondary.

A DC current does not cause an EMF

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B) N2 = [E2/E1] N1 =  [25 / 500] 200

N2 = 10

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C) Since the power going in equals the power going out, the ratio of the currents is the inverse of the ratio of the voltages.

N2 = [I2/I1] N1 =  [500 / 25] 200

N2 = 4000

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D)This is a step-down transformer with a ratio of 5:1. Assuming the primary current is AC, the secondary current would be

I2 = [N1/N2] I1 =  [400 / 80] 2.5

I2 = 12.5 A

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E) Since the power in the primary and secondary is the same

The power supplied to the resistor = 400 W.

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F) The power is the same on both sides of the transformer, so the current is

I1 = P / V1 = 60 / 120

I1 = 0.5 A

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G) e = 100% [(19.4 * 11.8) / (120 * 2)]

= 95.4%

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