50 Electric Generators

Electric Generators

Electric generators convert mechanical energy into electrical energy using the principle of electromagnetic induction. As discussed in the previous chapter, a changing magnetic flux through a conducting loop induces an electromotive force (emf). In a generator, this change in magnetic flux is produced by continuously rotating a coil in a magnetic field.

Electric generators are found throughout modern society. Large generators in power plants provide electricity for homes, hospitals, schools, and industries. Smaller generators are used as backup power sources during emergencies, in portable medical equipment, and in ambulances and disaster-response systems where a reliable electrical supply is essential.

Although commercial generators have complex designs with many coils and sophisticated control systems, they all operate according to the same physical principle: a changing magnetic flux induces an emf according to Faraday's law. The following example illustrates how to calculate the average emf produced by a rotating coil.

Example 50.1: Calculating the Emf Induced in a Generator Coil

The generator coil shown in Figure 50.1 is rotated through one-fourth of a revolution (from [latex]\theta =0^\circ[/latex] to [latex]\theta =90^\circ[/latex]) in 15.0 ms. The 200-turn circular coil has a radius of 5.00 cm and rotates in a uniform magnetic field of 1.25 T. What is the average emf induced?

Strategy

We use Faraday's law of electromagnetic induction to calculate the average induced emf over the time interval [latex]\Delta t[/latex]:

[latex]\text{emf}=-N\frac{\Delta\Phi}{\Delta t}.[/latex]

Since we are asked for the magnitude of the induced emf, we ignore the negative sign, which only indicates the direction required by Lenz's law.

The number of turns is [latex]N=200[/latex], and the rotation takes [latex]\Delta t=15.0\ \text{ms}[/latex]. We first determine the change in magnetic flux.

Solution

Because both the magnetic field strength and the coil area remain constant, the change in magnetic flux is

[latex]\Delta\Phi=\Delta\left(BA\cos\theta\right)=AB\,\Delta\left(\cos\theta\right).[/latex]

As the coil rotates from [latex]0^\circ[/latex] to [latex]90^\circ[/latex], the cosine changes from 1 to 0, giving

[latex]\Delta(\cos\theta)=0-1=-1.[/latex]

Therefore,

[latex]\Delta\Phi=-AB.[/latex]

The magnitude of the induced emf is therefore

[latex]\text{emf}=N\frac{AB}{\Delta t}.[/latex]

The area of the circular coil is

[latex]A=\pi r^2=\pi(0.0500\ \text{m})^2=7.85\times10^{-3}\ \text{m}^2.[/latex]

Substituting the known values gives

[latex]\text{emf}=200\frac{(7.85\times10^{-3}\ \text{m}^2)(1.25\ \text{T})}{15.0\times10^{-3}\ \text{s}}=131\ \text{V}.[/latex]

Discussion

The average induced emf is

[latex]\boxed{\text{emf}=131\ \text{V}}[/latex]

This voltage is comparable to the household electrical supply used in many countries (typically about 120 V or 230 V, depending on the region). In practical generators, larger voltages are produced by increasing the number of turns in the coil, using stronger magnetic fields, increasing the coil area, or rotating the generator at a higher angular speed.

Healthcare Connection: Hospitals depend on electric generators to provide uninterrupted power during outages. Emergency generators automatically supply electricity to life-support systems, ventilators, operating rooms, imaging equipment, refrigerators for medications and vaccines, and many other critical medical devices. Although these generators are much larger and more sophisticated than the one shown here, they operate using exactly the same physical principle illustrated in this example.

The average emf calculated in Example 50.1 is useful, but a generator produces a voltage that changes continuously as the coil rotates. To understand how generators supply alternating current (AC), we must determine the emf at every instant during the rotation.

Consider the rectangular coil shown in Figure 50.2. As the coil rotates in a uniform magnetic field, the angle between the magnetic field and the normal (perpendicular) to the coil changes continuously. Because the magnetic flux depends on this angle, the induced emf also changes continuously with time.

As the coil rotates, the charges inside the wire move through the magnetic field and experience the magnetic force. The vertical sides of the coil move across the magnetic field lines, so the magnetic force pushes charges along the wire, producing an electric current. The top and bottom segments, however, experience forces perpendicular to the wire and therefore do not contribute to the induced emf.

For one vertical side of the coil, the motional emf is

where [latex]v\sin\theta[/latex] is the component of the wire's velocity perpendicular to the magnetic field.

Since both vertical sides contribute equally and their emfs add together, the total induced emf around the loop is

This equation shows how the emf depends on the orientation of the rotating coil, but we would like an expression that describes how the emf changes with time.

If the coil rotates with a constant angular velocity [latex]\omega[/latex], then the angle changes according to

Substituting this relationship gives

The linear speed of each side of the coil is related to the angular speed by

where the radius of rotation is one-half the coil width:

Therefore,

Substituting this expression into the equation for the emf gives

Since the area of the rectangular coil is

and a practical generator usually contains many turns of wire, the induced emf becomes

This is the general expression for the instantaneous emf produced by a generator containing [latex]N[/latex] turns of area [latex]A[/latex] rotating with angular speed [latex]\omega[/latex] in a uniform magnetic field [latex]B[/latex].

It is often written in the simpler form

where

is the peak emf, or maximum voltage produced by the generator.

The frequency of the alternating voltage is related to the angular speed by

and the period of one complete cycle is

The sinusoidal shape of the voltage is one of the defining characteristics of alternating current. As the coil rotates, the magnetic flux changes smoothly from a maximum positive value to zero, then to a maximum negative value, and back again. Because the induced emf is proportional to the rate at which the magnetic flux changes, the voltage naturally follows a sine wave.The expression for the peak emf,

[latex]\text{emf}_0=NAB\omega,[/latex]

also provides valuable engineering insight. A generator can produce a larger voltage by increasing any of four quantities:

• Increasing the number of turns [latex]N[/latex] in the coil.
• Increasing the area [latex]A[/latex] of each loop.
• Using a stronger magnetic field [latex]B[/latex].
• Rotating the generator more rapidly (increasing [latex]\omega[/latex]).

This dependence is evident in many everyday generators. For example, bicycle generators produce a brighter light as the bicycle moves faster because increasing the rotational speed increases the induced emf. Modern power plants use the same principle on a much larger scale: turbines driven by steam, flowing water, wind, or combustion rotate massive generator coils or magnets at carefully controlled speeds to produce a stable AC voltage for the electrical grid.

Healthcare Connection

Nearly every hospital relies on AC generators during power outages. Backup generators must rotate at a precisely controlled speed so that the frequency of the electricity remains constant (typically 60 Hz in North America and 50 Hz in many other countries). Maintaining the correct voltage and frequency is essential for sensitive medical equipment such as MRI systems, ventilators, infusion pumps, patient monitors, and laboratory analyzers.

Although the generator described so far produces alternating current (AC), it is also possible to produce a form of direct current (DC) by replacing the slip rings with a device called a commutator. A commutator consists of two conducting half-rings that reverse the connection to the external circuit every half rotation. As a result, the current delivered to the external circuit always flows in the same direction, even though the current inside the rotating coil reverses direction.

The output of this arrangement is not a constant DC voltage but a pulsed DC voltage, as illustrated in Figure 50.4. The voltage rises and falls in magnitude but never changes polarity. Historically, commutators were widely used in electric motors, generators, and battery-powered equipment before the development of modern solid-state electronics.

Although the commutator prevents the voltage from becoming negative, the output still contains large fluctuations known as ripple. Many electrical devices require a much steadier voltage than this. Modern electronic systems therefore use semiconductor components, such as diodes and voltage regulators, to convert AC into nearly constant DC with very little ripple.
Today, mechanical commutators are used much less frequently because electronic rectifiers are more efficient, require less maintenance, and can produce smoother DC voltages. Nevertheless, the commutator remains an elegant application of electromagnetic induction and demonstrates how the same rotating coil can generate either alternating or direct current depending on how it is connected to the external circuit.

Healthcare Connection

Many medical devices—including portable patient monitors, infusion pumps, implant programmers, and defibrillators—operate internally using direct current supplied by batteries or electronic power supplies. Although hospitals receive alternating current from the electrical grid, power supplies inside medical equipment convert that AC into stable DC using electronic rectifiers and filtering circuits. Modern electronics have therefore largely replaced mechanical commutators in healthcare technology.

Example 50.2: Calculating the Maximum Emf of a Generator

Calculate the maximum emf, [latex]\text{emf}_0[/latex], produced by the generator described in Example 50.1.

Strategy

To determine the maximum emf, we first calculate the angular velocity of the rotating coil. We then substitute this value into the equation

[latex]\text{emf}_0=NAB\omega.[/latex]

All the remaining quantities were determined in Example 50.1.

Solution

The angular velocity is defined as the angle rotated per unit time:

[latex]\omega=\frac{\Delta\theta}{\Delta t}.[/latex]

During one-fourth of a revolution, the coil rotates through an angle of

[latex]\Delta\theta=\frac{\pi}{2}\ \text{rad},[/latex]

in a time of

[latex]\Delta t=0.0150\ \text{s}.[/latex]

Therefore,

[latex]\begin{aligned} \omega &=\frac{\pi/2}{0.0150\ \text{s}}\\ &=104.7\ \text{rad/s}. \end{aligned}[/latex]

This corresponds to approximately 1000 revolutions per minute (rpm).
Substituting the known values into the equation for the peak emf gives

[latex]\begin{aligned} \text{emf}_0 &=NAB\omega\\ &=(200)(7.85\times10^{-3}\ \text{m}^2)(1.25\ \text{T})(104.7\ \text{rad/s})\\ &=206\ \text{V}. \end{aligned}[/latex]

Discussion

The maximum induced emf is

[latex]\boxed{\text{emf}_0=206\ \text{V}}[/latex]

This value is larger than the average emf of 131 V calculated in Example 50.1 because the induced voltage varies continuously during the rotation. The emf reaches its maximum value only when the rate of change of magnetic flux is greatest, while at other times it is smaller and even passes through zero. The average over one-quarter of a revolution is therefore less than the peak value.
Healthcare Connection: Medical facilities rely on generators that produce a stable peak voltage and frequency to ensure the reliable operation of sensitive equipment. Although the instantaneous voltage continually oscillates in an AC generator, electrical systems are designed so that the effective (RMS) voltage delivered to medical devices remains constant and within strict safety standards.

Commercial electric generators look quite different from the simplified models shown in this chapter, but they operate according to exactly the same physical principles. The mechanical energy needed to rotate the generator may come from many different sources, including falling water in hydroelectric plants, high-pressure steam in fossil-fuel or nuclear power plants, wind turbines, or internal combustion engines.
Figure 50.5 shows a steam turbine connected to an electrical generator. High-pressure steam flows over the turbine blades, causing the shaft to rotate. This rotational motion drives the generator, where electromagnetic induction converts mechanical energy into electrical energy.

Figure 50.5: Steam turbine coupled to an electric generator. Steam produced by heating water drives the turbine blades, converting thermal energy into mechanical energy, which the generator then converts into electrical energy through electromagnetic induction. (Credit: Nabonaco, Wikimedia Commons)

Although generators and electric motors appear very similar, they perform opposite energy conversions. A generator converts mechanical energy into electrical energy, whereas a motor converts electrical energy into mechanical energy. In fact, the same machine can often operate as either a motor or a generator simply by reversing the direction of energy transfer.

This dual nature has practical applications. Some early automobiles used their starter motor as a generator once the engine was running. Today, hybrid and electric vehicles routinely recover energy during braking by allowing their electric motors to operate as generators, converting some of the vehicle's kinetic energy back into electrical energy to recharge the battery. In the next section on back emf, we will explore this close relationship between motors and generators in greater detail.