22.9

Transformers

One of the most important applications of mutual induction and self-induction takes place in a transformer. A transformer is a device for increasing or decreasing an ac voltage. For instance, whenever cordless devices (e.g., a cell phone) are plugged into a wall receptacle to recharge the batteries, a transformer plays a role in reducing the 120-V ac voltage to a much smaller value. Typically, between 3 and 9 V are needed to energize batteries. In another example, a picture tube in a television set needs about 15 000 V to accelerate the electron beam, and a transformer is used to obtain this high voltage from the 120 V at a wall socket.

The physics of transformers.

Figure 22-29 shows a drawing of a transformer. The transformer consists of an iron core on which two coils are wound: a primary coil with Np turns and a secondary coil with Ns turns. The primary coil is the one connected to the ac generator. For the moment, suppose the switch in the secondary circuit is open, so there is no current in this circuit.

Figure 22-29 A transformer consists of a primary coil and a secondary coil, both wound on an iron core. The changing magnetic flux produced by the current in the primary coil induces an emf in the secondary coil. At the far right is the symbol for a transformer.

The alternating current in the primary coil establishes a changing magnetic field in the iron core. Because iron is easily magnetized, it greatly enhances the magnetic field relative to that in an air core and guides the field lines to the secondary coil. In a well-designed core, nearly all the magnetic flux that passes through each turn of the primary also goes through each turn of the secondary. Since the magnetic field is changing, the flux through the primary and secondary coils is also changing, and consequently an emf is induced in both coils. In the secondary coil the induced emf arises from mutual induction and is given by Faraday’s law as

In the primary coil the induced emf is due to self-induction and is specified by Faraday’s law as

The term is the same in both of these equations, since the same flux penetrates each turn of both coils. Dividing the two equations shows that

In a high-quality transformer the resistances of the coils are negligible, so the magnitudes of the emfs, and , are nearly equal to the terminal voltages, Vs and Vp, across the coils (see Section 20.9 for a discussion of terminal voltage). The relation is called the transformer equation and is usually written in terms of the terminal voltages:

| (22.12)

Power distribution stations use high-voltage transformers similar to these to step-up or step-down voltages. (Lester Lefkowitz/Photographer’s Choice/Getty Images)

According to the transformer equation, if Ns is greater than Np, the secondary (output) voltage is greater than the primary (input) voltage. In this case we have a step-up transformer. On the other hand, if Ns is less than Np, the secondary voltage is less than the primary voltage, and we have a step-down transformer. The ratio Ns/Np is referred to as the turns ratio of the transformer. A turns ratio of 8/1 (often written as 8:1) means, for example, that the secondary coil has eight times more turns than does the primary coil. Conversely, a turns ratio of 1:8 implies that the secondary coil has one-eighth as many turns as the primary coil.
A transformer operates with ac electricity and not with dc. A steady direct current in the primary coil produces a flux that does not change in time, and thus no emf is induced in the secondary coil. The ease with which transformers can change voltages from one value to another is a principal reason why ac is preferred over dc.
With the switch in the secondary circuit of Figure 22-29 closed, a current Is exists in the circuit and electrical energy is fed to the TV tube. This energy comes from the ac generator connected to the primary coil. Although the secondary voltage Vs may be larger or smaller than the primary voltage Vp, energy is not being created or destroyed by the transformer. Energy conservation requires that the energy delivered to the secondary coil must be the same as the energy delivered to the primary coil, provided no energy is dissipated in heating these coils or is otherwise lost. In a well-designed transformer, less than 1% of the input energy is lost in the form of heat. Noting that power is energy per unit time, and assuming 100% energy transfer, the average power delivered to the primary coil is equal to the average power delivered to the secondary coil: . But (Equation 20.15a), so IpVp = IsVs, or

| (22.13)

Observe that Vs/Vp is equal to the turns ratio Ns/Np, while Is/Ip is equal to the inverse turns ratio Np/Ns. Consequently, a transformer that steps up the voltage simultaneously steps down the current, and a transformer that steps down the voltage steps up the current. However, the power is neither stepped up nor stepped down, since . Example 14 emphasizes this fact.

Problem solving insight

| Example 14 | | | A Step-Down Transformer

A step-down transformer inside a stereo receiver has 330 turns in the primary coil and 25 turns in the secondary coil. The plug connects the primary coil to a 120-V wall socket, and there is a current of 0.83 A in the primary coil while the receiver is turned on. Connected to the secondary coil are the transistor circuits of the receiver. Find (a) the voltage across the secondary coil, (b) the current in the secondary coil, and (c) the average electric power delivered to the transistor circuits.
Reasoning The transformer equation, Equation 22.12, states that the secondary voltage Vs is equal to the product of the primary voltage Vp and the turns ratio Ns/Np. On the other hand, Equation 22.13 indicates that the secondary current Is is equal to the product of the primary current Ip and the inverse turns ratio Np/Ns. The average power delivered to the transistor circuits is the product of the secondary current and the secondary voltage.
Solution

a. | The voltage across the secondary coil can be found from the transformer equation:

| (22.12)

b. | The current in the secondary coil is

| (22.13)

c. | The average power delivered to the secondary is the product of Is and Vs:

| (20.15a)

As a check on our calculation, we verify that the power delivered to the secondary coil is the same as that sent to the primary coil from the wall receptacle: .

Transformers play an important role in the transmission of power between electrical generating plants and the communities they serve. Whenever electricity is transmitted, there is always some loss of power in the transmission lines themselves due to resistive heating. Since the resistance of the wires is proportional to their length, the longer the wires the greater is the power loss. Power companies reduce this loss by using transformers that step up the voltage to high levels while reducing the current. A smaller current means less power loss, since P = I2R, where R is the resistance of the transmission wires (see problem 59). Figure 22-30 shows one possible way of transmitting power. The power plant produces a voltage of 12 000 V. This voltage is then raised to 240 000 V by a 20:1 step-up transformer. The high-voltage power is sent over the long-distance transmission line. Upon arrival at the city, the voltage is reduced to about 8000 V at a substation using a 1:30 step-down transformer. However, before any domestic use, the voltage is further reduced to 240 V (or possibly 120 V) by another step-down transformer that is often mounted on a utility pole. The power is then distributed to consumers.

Figure 22-30 Transformers play a key role in the transmission of electric power.

Check Your Understanding 5

A transformer that stepped up the voltage and the current simultaneously would (a) produce less power at the secondary than was supplied at the primary, (b) produce more power at the secondary than was supplied at the primary, (c) produce the same amount of power at the secondary that was supplied at the primary, (d) violate the law of conservation of energy. Choose one or more. [The answer(s) is (are) given at the end of the book.]
Background: In a transformer, mutual induction allows the primary and secondary coils to interact. Depending on the turns ratio of the transformer, the voltages across the primary and secondary are not the same, and the currents in the primary and the secondary are not the same. However, for both coils the power is given by the product of current and voltage.
For similar questions (including calculational counterparts), consult Self-Assessment Test 22.2 , which is described next.

Test your understanding of the material in Sections 22.7, 22.8 and 22.9 :

· | Generators
· | Back Emf
· | Mutual Inductance
· | Self-Inductance
· | Transformers

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