Motors, Generators, and Transformers

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Section Key Terms

Electric Motors, Generators, and Transformers

As we learned previously, a current-carrying wire in a magnetic field experiences a force—recall $F=I\ell B\sin \theta$ . Electric motors, which convert electrical energy into mechanical energy, are the most common application of magnetic force on current-carrying wires. Motors consist of loops of wire in a magnetic field. When current is passed through the loops, the magnetic field exerts a torque on the loops, which rotates a shaft. Electrical energy is converted to mechanical work in the process. Figure_20_03_elecmotor shows a schematic drawing of an electric motor.

Let us examine the force on each segment of the loop in Figure_20_03_elecmotor to find the torques produced about the axis of the vertical shaft—this will lead to a useful equation for the torque on the loop. We take the magnetic field to be uniform over the rectangular loop, which has width w and height $\ell \text{,}$ as shown in the figure. First, consider the force on the top segment of the loop. To determine the direction of the force, we use the right-hand rule. The current goes from left to right into the page, and the magnetic field goes from left to right in the plane of the page. Curl your right fingers from the current vector to the magnetic field vector and your right thumb points down. Thus, the force on the top segment is downward, which produces no torque on the shaft. Repeating this analysis for the bottom segment—neglect the small gap where the lead wires go out—shows that the force on the bottom segment is upward, again producing no torque on the shaft.

Consider now the left vertical segment of the loop. Again using the right-hand rule, we find that the force exerted on this segment is perpendicular to the magnetic field, as shown in Figure_20_03_elecmotor. This force produces a torque on the shaft. Repeating this analysis on the right vertical segment of the loop shows that the force on this segment is in the direction opposite that of the force on the left segment, thereby producing an equal torque on the shaft. The total torque on the shaft is thus twice the toque on one of the vertical segments of the loop.

To find the magnitude of the torque as the wire loop spins, consider Figure_20_03_toploop, which shows a view of the wire loop from above. Recall that torque is defined as $\tau =rF\sin \theta \text{,}$ where F is the applied force, r is the distance from the pivot to where the force is applied, and θ is the angle between r and F. Notice that, as the loop spins, the current in the vertical loop segments is always perpendicular to the magnetic field. Thus, the equation $F=I\ell B\sin \theta$ gives the magnitude of the force on each vertical segment as $F=I\ell B\text{.}$ The distance r from the shaft to where this force is applied is w/2, so the torque created by this force is

Because there are two vertical segments, the total torque is twice this, or

If we have a multiple loop with N turns, we get N times the torque of a single loop. Using the fact that the area of the loop is $A=w\ell \text{;}$ the expression for the torque becomes

This is the torque on a current-carrying loop in a uniform magnetic field. This equation can be shown to be valid for a loop of any shape.

From the equation $\tau =NIAB\sin \theta \text{,}$ we see that the torque is zero when $\theta =0\text{.}$ As the wire loop rotates, the torque increases to a maximum positive torque of $w\ell B$ when $\theta =\mathrm{90°}\text{.}$ The torque then decreases back to zero as the wire loop rotates to $\theta =\mathrm{180°}\text{.}$ From $\theta =\mathrm{180°}$ to $\theta =\mathrm{360°}\text{,}$ the torque is negative. Thus, the torque changes sign every half turn, so the wire loop will oscillate back and forth.

For the coil to continue rotating in the same direction, the current is reversed as the coil passes through $\theta =0\text{and}\theta =\mathrm{180°}$ using automatic switches called brushes, as shown in Figure_20_03_brushes.

Consider now what happens if we run the motor in reverse; that is, we attach a handle to the shaft and mechanically force the coil to rotate within the magnetic field, as shown in Figure_20_03_generator. As per the equation $F=qvB\sin \theta$ —where $\theta$ is the angle between the vectors $\overrightarrow{v}$ and $\overrightarrow{B}\text{—charges}$ in the wires of the loop experience a magnetic force because they are moving in a magnetic field. Again using the right-hand rule, where we curl our fingers from vector $\overrightarrow{v}$ to vector $\overrightarrow{B}$, we find that charges in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. However, charges in the vertical wires experience forces parallel to the wire, causing a current to flow through the wire and through an external circuit if one is connected. A device such as this that converts mechanical energy into electrical energy is called a generator.

Because current is induced only in the side wires, we can find the induced emf by only considering these wires. As explained in Induced Current in a Wire, motional emf in a straight wire moving at velocity v through a magnetic field B is $\epsilon =B\ell v\text{,}$ where the velocity is perpendicular to the magnetic field. In the generator, the velocity makes an angle $\theta$ with B (see Figure_20_03_3dgen), so the velocity component perpendicular to B is $v\sin \theta \text{.}$ Thus, in this case, the emf induced on each vertical wire segment is $\epsilon =B\ell v\sin \theta \text{,}$ and they are in the same direction. The total emf around the loop is then

Although this expression is valid, it does not give the emf as a function of time. To find how the emf evolves in time, we assume that the coil is rotated at a constant angular velocity $\omega \text{.}$ The angle $\theta$ is related to the angular velocity by $\theta =\omega t\text{,}$ so that

Recall that tangential velocity v is related to angular velocity $\omega$ by $v=r\omega \text{.}$ Here, $r=w\mathrm{/2}$, so that $v=\left(w/2\mathrm{}\right)\omega$ and

Noting that the area of the loop is $A=\ell w$ and allowing for N wire loops, we find that

is the emf induced in a generator coil of N turns and area A rotating at a constant angular velocity $\omega$ in a uniform magnetic field B. This can also be expressed as

where

is the maximum (peak) emf.

Figure_20_03_Light shows a generator connected to a light bulb and a graph of the emf vs. time. Note that the emf oscillates from a positive maximum of ${\epsilon }_0$ to a negative maximum of ${\mathrm{-E}}_0\text{.}$ In between, the emf goes through zero, which means that zero current flows through the light bulb at these times. Thus, the light bulb actually flickers on and off at a frequency of 2f, because there are two zero crossings per period. Since alternating current such as this is used in homes around the world, why do we not notice the lights flickering on and off? In the United States, the frequency of alternating current is 60 Hz, so the lights flicker on and off at a frequency of 120 Hz. This is faster than the refresh rate of the human eye, so you don’t notice the flicker of the lights. Also, other factors prevent various different types of light bulbs from switching on and off so fast, so the light output is smoothed out a bit.

In real life, electric generators look a lot different than the figures in this section, but the principles are the same. The source of mechanical energy that turns the coil can be falling water—hydropower—steam produced by the burning of fossil fuels, or the kinetic energy of wind. Figure_20_03_Turbine shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.

Another very useful and common device that exploits magnetic induction is called a transformer. Transformers do what their name implies—they transform voltages from one value to another; the term voltage is used rather than emf because transformers have internal resistance. For example, many cell phones, laptops, video games, power tools, and small appliances have a transformer built into their plug-in unit that changes 120 V or 240 V AC into whatever voltage the device uses. Figure_20_03_Trans shows two different transformers. Notice the wire coils that are visible in each device. The purpose of these coils is explained below.

Figure_20_03_Lam_coil shows a laminated-coil transformer, which is based on Faraday’s law of induction and is very similar in construction to the apparatus Faraday used to demonstrate that magnetic fields can generate electric currents. The two wire coils are called the primary and secondary coils. In normal use, the input voltage is applied across the primary coil, and the secondary produces the transformed output voltage. Not only does the iron core trap the magnetic field created by the primary coil, but also its magnetization increases the field strength, which is analogous to how a dielectric increases the electric field strength in a capacitor. Since the input voltage is AC, a time-varying magnetic flux is sent through the secondary coil, inducing an AC output voltage.

For the transformer shown in Figure_20_03_Lam_coil, the output voltage $V_{\text{S}}$ from the secondary coil depends almost entirely on the input voltage $V_{\text{P}}$ across the primary coil and the number of loops in the primary and secondary coils. Faraday’s law of induction for the secondary coil gives its induced output voltage $V_{\text{S}}$ to be

where $N_{\text{S}}$ is the number of loops in the secondary coil and $\text{Δ}\text{Φ}/\text{Δ}t$ is the rate of change of magnetic flux. The output voltage equals the induced emf $\text{(}{V}_{\text{S}}={\epsilon }_{\text{S}}\text{),}$ provided coil resistance is small—a reasonable assumption for transformers. The cross-sectional area of the coils is the same on each side, as is the magnetic field strength, and so $\text{Δ}\text{Φ/Δ}t$ is the same on each side. The input primary voltage $V_{\text{P}}$ is also related to changing flux by

Taking the ratio of these last two equations yields the useful relationship

This is known as the transformer equation. It simply states that the ratio of the secondary voltage to the primary voltage in a transformer equals the ratio of the number of loops in secondary coil to the number of loops in the primary coil.

Transmission of Electrical Power

Transformers are widely used in the electric power industry to increase voltages—called step-up transformers—before long-distance transmission via high-voltage wires. They are also used to decrease voltages—called step-down transformers—to deliver power to homes and businesses. The overwhelming majority of electric power is generated by using magnetic induction, whereby a wire coil or copper disk is rotated in a magnetic field. The primary energy required to rotate the coils or disk can be provided by a variety of means. Hydroelectric power plants use the kinetic energy of water to drive electric generators. Coal or nuclear power plants create steam to drive steam turbines that turn the coils. Other sources of primary energy include wind, tides, or waves on water.

Once power is generated, it must be transmitted to the consumer, which often means transmitting power over hundreds of kilometers. To do this, the voltage of the power plant is increased by a step-up transformer, that is stepped up, and the current decreases proportionally because

The lower current $I_{\text{transmitted}}$ in the transmission wires reduces the Joule losses, which is heating of the wire due to a current flow. This heating is caused by the small, but nonzero, resistance $R_{\text{wire}}$ of the transmission wires. The power lost to the environment through this heat is

which is proportional to the current squared in the transmission wire. This is why the transmitted current $I_{\text{transmitted}}$ must be as small as possible and, consequently, the voltage must be large to transmit the power $P_{\text{transmitted}}\text{⋅}$

Voltages ranging from 120 to 700 kV are used for transmitting power over long distances. The voltage is stepped up at the exit of the power station by a step-up transformer, as shown in Figure_20_03_Grid.

Once the power has arrived at a population or industrial center, the voltage is stepped down at a substation to between 5 and 30 kV. Finally, at individual homes or businesses, the power is stepped down again to 120, 240, or 480 V. Each step-up and step-down transformation is done with a transformer designed based on Faradays law of induction.

Check Your Understanding

Section Summary

• Electric motors contain wire loops in a magnetic field. Current is passed through the wire loops, which forces them to rotate in the magnetic field. The current is reversed every half rotation so that the torque on the loop is always in the same direction.
• Electric generators contain wire loops in a magnetic field. An external agent provides mechanical energy to force the loops to rotate in the magnetic field, which produces an AC voltage that drives an AC current through the loops.
• Transformers contain a ring made of magnetic material and, on opposite sides of the ring, two windings of wire wrap around the ring. A changing current in one wire winding creates a changing magnetic field, which is trapped in the ring and thus goes through the second winding and induces an emf in the second winding. The voltage in the second winding is proportional to the ratio of the number of loops in each winding.
• Transformers are used to step up and step down the voltage for power transmission.
• Over long distances, electric power is transmitted at high voltage to minimize the current and thereby minimize the Joule losses due to resistive heating.

Concept Items

Critical Thinking

Test Prep Multiple Choice

Test Prep Short Answer

Performance Task

Test Prep Extended Response

Glossary

electric motor

device that transforms electrical energy into mechanical energy

generator

device that transforms mechanical energy into electrical energy

transformer

device that transforms voltages from one value to another