In this lesson, we’re going to learn how to calculate some of the important quantities associated with current, voltage, and power in alternating-current circuits. We’ll also learn about the different possible relationships between alternating current and alternating voltage, depending on whether the circuit is resistive, capacitive, or inductive.
Let’s start by defining alternating current. Current is the rate at which charge flows along a path. In an electrical circuit such as these resistors connected in series, the path is formed by the wires and the components of the circuit. As we can see from this example, there’re two possible directions for the current in this circuit. The charge can either flow clockwise through the circuit or in the opposite direction, counterclockwise through the circuit. If we introduce a source of electromotive force or emf to the circuit, like a cell of voltage 𝑉, the emf provided by the cell will drive a current in the circuit. In this case, the conventional current will have a clockwise direction because conventional current is defined as going from the positive terminal to the negative terminal of a cell.
If we physically re-move and return around the cell, we reverse the positive and negative terminals relative to the rest of the circuit, and the direction of the current is now counterclockwise. In both of these examples with the cell driving current in the circuit, the current is a direct current because it only has one direction as determined by the orientation of the cell. There’s a different kind of source called an alternating voltage source that can smoothly change between a positive terminal on top and a negative terminal on bottom and a negative terminal on top and a positive terminal on bottom, just like our two orientations of the cell.
Therefore, the emf from the alternating voltage source will switch back and forth between driving the current in the clockwise direction and the counterclockwise direction around the circuit. We therefore call such a current an alternating current because it switches back and forth between the two possible directions in the circuit. Similarly, we call the emf produced by our source an alternating emf because it drives a current that changes directions. Before we learn about a simple source of alternating emf, it’s worth understanding how we would represent a direct current and an alternating current on a graph.
To make a graph of current with respect to time, we need to represent the current at a particular time with a number. We’ll let the magnitude of our number be the magnitude of the current. And then since current has one of two possible directions, we’ll let positive numbers represent current in one direction and negative numbers represent current in the opposite direction. It doesn’t matter which direction we choose to be positive and which one we choose to be negative as long as we’re consistent for a particular circuit. So let’s draw direct current. This is a current that only has one direction. This blue line represents a direct current.
Note that this current is not constant in time; it does change sizes. But the value is always greater than zero, which means it always has the direction associated with positive current. So it’s a direct current because it only has one direction. This green line also represents a direct current. The size of this current is constant in time, but its value is always less than zero. So it has the opposite direction to the current represented by the blue line. On the other hand, the current represented by this orange curve is not a direct current, but an alternating current. The value of this current switches from greater than zero at some times to less than zero at other times. In other words, this current changes direction.
All right, let’s now move on to the alternating emf produced by a simple AC generator. The simplest AC generator consists of a wire loop in a uniform magnetic field. As the loop rotates in the field, the magnetic flux through the loop changes, which causes an alternating emf to be induced across the terminals of the loop. Let’s derive a quantitative formula for this emf as a function of time. We can then use this formula to understand some quantitative and qualitative properties of the emf, as well as alternating quantities in general. We’ll start with Faraday’s law of induction, which states that the induced emf is equal to the negative of the changed magnetic flux through a loop with respect to time.
All right, so let’s calculate the flux through our loop. We can think about magnetic flux as the amount of magnetic field passing through the loop. So if the area of the loop is 𝐴 and the strength of the magnetic field is 𝐵, then the total magnetic field passing through the loop would just be 𝐴 times 𝐵. This is true when the loop is totally perpendicular to the field. But remember, our loop is rotating. Here are two pictures of the loop in the magnetic field looking from the side. In the first picture, the loop is perpendicular to the magnetic field, and we can clearly see that it cuts across five magnetic field lines.
However, in the second picture, as the loop has rotated through an angle 𝜃, the loop now only cuts across three magnetic field lines, even though it’s the same length. So even though the area of the loop hasn’t changed, the total magnetic field passing through the loop certainly has, and so the flux is different as well. To figure out this new flux, let’s notice that the original loop represented by the solid line is cutting across the same number of magnetic field lines as this other smaller vertical loop represented by this vertical dashed line. And we know what the flux through this other loop is because it’s perpendicular to the magnetic field. It’s just the area of this loop times the strength of the magnetic field.
Now the flux, through both loops is the same. So we can say that the loop represented by the solid line has an effective area equal to the area of the loop represented by the dashed line. To figure out this effective area, let’s first call the length of the solid line 𝑙. Second, since the loop is rotating at some fixed speed, we can replace the angle 𝜃 with the quantity 𝜔 times 𝑡. 𝜔 is the angular speed of rotation, so 𝜔 times 𝑡 is the total angle that the loop has rotated. Next, we notice that since the two dashed lines in this diagram are parallel, the angle between the two orange lines is also 𝜔𝑡. This lets us calculate the length of the orange dashed line as 𝑙 times cos of 𝜔𝑡.
We don’t need to correct the width of the loop because it’s always perpendicular to the magnetic field throughout the rotation. So this lets us write the effective area of the loop as the width of the loop times the length of the loop times cos 𝜔𝑡. But the width times the length is just the original area of the loop. So the effective area for calculating the magnetic flux as the loop rotates in the field is the area of the loop times cos of 𝜔𝑡, where 𝜔 is the angular speed of rotation and 𝑡 is the time elapsed. Adjusting our previous expression for flux to get the flux at any particular time, we only need to include the factor of cos of 𝜔𝑡.
This turns our original factor of area into the effective area. The flux that we’ve calculated is the flux for one single loop. But many generators actually contain multiple loops to increase the output of emf. If there are 𝑛 identical loops and each loop has a flux of 𝐴𝐵 cos 𝜔𝑡, then the total flux through all of the loops will be 𝑛 times 𝐴𝐵 cos 𝜔𝑡. Let’s now find the negative of the change in this expression with respect to time to get the emf.
The only part of our expression for flux that depends on time is cos 𝜔𝑡. This means that we can write the emf as 𝑛𝐴𝐵, all of which are constant in time, times negative the change in cos 𝜔𝑡 with respect to time. It turns out that this change is negative 𝜔 sin 𝜔𝑡. And when we take the negative of that, we get negative negative 𝜔 sin 𝜔𝑡. And we’re just left with positive 𝜔 sin 𝜔𝑡. This gives us our final expression for the alternating electromotive force produced by an AC generator. We have that emf is equal to 𝑛, the number of wire loops, times 𝐴, the area of each loop, times 𝐵, the strength of the magnetic field, times 𝜔, the speed of rotation of the loops, times sin of 𝜔𝑡, the time since the loop was vertical.
Now that we’ve finished our quantitative derivation, let’s use this formula to understand some of the qualitative properties of this alternating emf. To get a qualitative picture of this emf, the first thing we’ll do is draw a graph. The graph of emf with respect to time has a sinusoidal shape due to the sin 𝜔𝑡 part of the expression for emf. Since the sine function varies between positive one and negative one, the extreme values for the emf are just given by the constant in front of the sin 𝑛𝐴𝐵𝜔. In a similar vein, we can clearly see from this graph that the emf takes on both positive values and negative values. In a physical sense, this corresponds to the emf’s ability to drive current in both the positive and negative directions of a particular circuit.
Furthermore, as we can see, the value of the emf changes with time. So it’s useful to try to come up with some value that would represent the average behavior of the emf. One natural value we might consider is the arithmetic mean of the emf over one cycle. We’ll call this 𝐸 sub avg for average emf. And we’ll find it by finding the total emf in one cycle divided by the length of the cycle. However, if we look at our graph carefully, we see that every time the emf attain some positive value, there’s a corresponding time where the emf attains a negative value with the same magnitude. But that means that the total emf from these two times is zero since the values cancel.
Applying this to the whole cycle, we see that the total emf over the entire cycle is zero because all the positive emfs are canceled by corresponding negative emfs. If the total emf is zero, then the average emf is also zero. This is a true fact, but it doesn’t carry any information about the peak values for the emf. And in fact, no matter how many wire loops there are, what their area is, what the strength of the magnetic field is, or how fast the rotating, the average emf from an AC generator will always be zero. So we want a different kind of average that will include information about the peak values for the emf. One value that will turn out to be particularly useful for electronics applications is the root mean square, or rms.
Each word, root, mean, and square, refers to one of the steps for calculating an rms value. The first step is to square each number. Since the square of the negative number is a positive number, this makes sure that all of the numbers we’re dealing with are now positive. The next step is to find the mean of these squared numbers. This will always be positive or zero, since a square number is either positive or zero. The final step is to take the square root of the mean that we found in step two. We do this step to balance the squaring that we did in step one. It turns out that the rms value of a sinusoid has a very nice and simple form.
The rms value of a sinusoid is exactly one over the square root of two of the maximum value. This means that the root mean square emf produced by our generator is one over the square root of two 𝑛𝐴𝐵𝜔. Note that this is indeed a value that has information about the maximum size of the emf. Furthermore, as a very useful approximation, one over the square root of two was approximately equal to 0.7. So the rms value for the emf is approximately 0.7 times 𝑛𝐴𝐵𝜔. Now that we understand some of the average properties of alternating emf, we can apply similar principles to the current and power in alternating-current circuits.
In an alternating-current circuit, if the emf follows a sinusoid in time, so does the current. The graph of current versus time will have a very similar shape to the graph of the emf versus time. It will be sinusoidal in shape with a maximum size of 𝐼 peak that could be either positive or negative. Therefore, just like the emf, we’ll have that the arithmetic mean of the current over a single cycle is zero and the root mean square current is one over the square root of two times the maximum value for the current, or approximately 0.7 times the maximum current. In a normal resistive direct-current circuit, the power dissipated is the square of the current times the resistance. For alternating-current circuits that are purely resistive, where the current varies with time, this relationship still holds as long as we also specify that the power is now varying with time.
The maximum power would then be dissipated when the current reaches its positive or negative maximum value. However, unlike current and emf, the power dissipated is always greater than or equal to zero because 𝑅 is a positive number or zero. And even though 𝐼 can be negative, 𝐼 squared is always positive or zero. Therefore, if we try to take the arithmetic mean of the power over one cycle of the current, we won’t just get zero. What we actually get is one-half of the maximum power or one-half 𝐼 peak squared times 𝑅. But one-half is just one over the square root of two squared, so we can rewrite 𝑃 average as one over the square root of two times the maximum current as a quantity squared times 𝑅.
But this is useful because we have a definition for one over the square root of two times the maximum current. It’s just the root mean square current. Then we can see that the average power dissipated in an alternating-current circuit is the same as the power that would be dissipated in a direct-current circuit if the current in the direct-current circuit had the same value as the rms value for the current in the alternating-current circuit. So using the rms value for the current not only gives us information about the peak value for the current, but also gives us a connection between the average power dissipated in an alternating-current circuit and the power dissipated in a direct-current circuit.
All right, let’s now see how alternating voltage and alternating current relate to each other in resistive, capacitive, and inductive circuits. To illustrate these relationships, we’ve drawn three circuits. Each circuit has an alternating voltage source connected to a single component. In the first circuit, the component is a resistor of resistance 𝑅. In the second circuit, it’s a capacitor of capacitance 𝐶. And in the third circuit, it’s an inductor of inductance 𝐿. The letter 𝐿 is used in honor of the physicist Heinrich Lenz, because Lenz’s law plays an important role in inductors. To help with the comparison, we’ll treat each alternating voltage source as identical, so each circuit has the same alternating emf driving current.
For each circuit, we’ve drawn a sinusoidal emf on a graph as a blue dashed curve. Let’s now see what we expect for the current for each circuit. For the resistive or 𝑅 circuit, we have Ohm’s law, which tells us that voltage and current are directly proportional. This is distinctly stated by Ohm’s law 𝑉 equals 𝐼𝑅 and is true even when the current is alternating. In this case, the current, represented by the green solid line, will have exactly the same shape as the emf but with a different magnitude. In a capacitive or 𝐶 circuit, the emf and current are not directly proportional. Rather, the current follows this solid green curve, which is zero at times when the emf is maximum and is maximum at times when the emf is zero.
If we look carefully, we’ll see that the current follows a sinusoid with the same shape as the emf but reaches its maximum value one-quarter of a period before the emf does. In the language of phase, we see that the current leads the emf by 90 degrees. To understand this relationship intuitively, recall that the voltage across a capacitor is determined by how much charge is on the plates. And current is the flow of charge with time. So the emf causes a current in the circuit to deposit charge on the capacitor.
If we recall that conventional current is defined as the flow of positive charge from the positive terminal of the voltage source to the negative terminal of the voltage source, we can see that the resulting charge on the capacitor will be positive on the same side as the positive terminal of the voltage source and negative on the same side as the negative side of the voltage source, since positive charge is flowing into the top plate and flowing away from the bottom plate. If the emf from the alternating voltage source is larger than the voltage across the capacitor, the current will be driven through the capacitor depositing charge as pictured.
Thus, as long as the emf is getting larger, we expect current to flow as shown. On the other hand, if the emf is getting more negative relative to the voltage across the capacitor, we expect the current to flow in the opposite direction. Looking at our graph, we see that, indeed, when the emf goes from its most positive to its most negative value — in other words, the emf is decreasing — the current is negative. And when the emf goes from its most negative to its most positive value — in other words, increasing — the current is positive. In an inductive or 𝐿 circuit, the current would have a similar shift to the capacitive circuit, but in the other direction.
Thus, in this case, the maximum electromotive force occurs one-quarter of a period before the maximum current. In other words, the emf leads the current by 90 degrees. To understand this intuitively, recall that as the emf drives a current through the inductor, it produces a magnetic field pointing in a particular direction that oscillates with the current. As the emf changes, the current in the inductor changes, which changes the size of the magnetic field. By Lenz’s law, this changing magnetic field induces a current to oppose that change. This causes the current to be delayed, and the result is that the emf changes effectively before the current does, which results in a phase lead of 90 degrees.
It’s worth noting before we review what we’ve learned that the emf in each of these circuits is sinusoidal. And so the current is also sinusoidal, even though it sometimes lags or leads the emf.
All right, let’s review the key points we learned in this lesson. Our focus was alternating-current circuits. We defined alternating current as a current that changes direction periodically with time. We saw that if the graph of the alternating current is a sinusoid, and indeed for any sinusoid, the average value defined is the arithmetic mean is zero. However, we can define a different sort of average called the root mean square, which is not zero. For the special case of a sinusoid, the root mean square is equal to one over the square root of two times the maximum value, which is approximately 0.7 times the maximum value.
We also saw that a sinusoidal alternating current is often driven by a sinusoidal electromotive force produced from an AC generator. We derived the formula number of loops times the area of each loop times the strength of the magnetic field times the angular frequency of rotation times the sin of 𝜔 times the time since the loop was perpendicular to the magnetic field to give the emf at any given time. We saw that in a resistive circuit, the emf and current are in phase. In a capacitive circuit, the current leads the emf by 90 degrees. And in an inductive circuit, the emf leads the current by 90 degrees. Finally, we saw that the average power dissipated in a resistive AC circuit is the rms current squared times the resistance.