 # Unit 4: Capacitors and Inductors in AC Circuits

In thi unit we’ll turn our attention to the behavior of capacitors and inductors in AC circuits. You saw in Unit 3 that resistors behave pretty much the same way in AC circuits as they do in DC circuits. But this is not true for capacitors and inductors. We use one set of rules (which you learned in EET 1150) for finding a capacitor’s voltage and current in DC circuits, and a different set of rules (which you’ll learn in this unit) for finding a capacitor’s voltage and current in AC circuits.

By the way, the rules that you learned in EET 1150 for finding the total capacitance and total inductance do hold for all circuits, DC or AC. But when you’re trying to figure out a capacitor’s current or voltage, you’ll use different techniques depending on whether the circuit is DC or AC.

AC circuit analysis relies heavily on trigonometry, which is the branch of mathematics dealing with angles. The last part of this e-Lesson covers some basic trigonometry, which you may have studied in a math class. But even if you’ve never studied trigonometry, you should be able to pick up this material without too much trouble.

##### Capacitors in AC Circuits
• In EET 1150 you learned some rules and equations that let you compute initial values, transient values, and steady-state values in circuits with capacitors. These rules and equations apply only to DC circuits; now we’ll learn entirely different rules and equations that you use if you have capacitors in an AC circuit.
##### Comparing Capacitors to Resistors
• Recall from Unit 2 that resistors in AC circuits can be analyzed using the same techniques (such as Ohm’s law) that you use for resistors in DC circuits. Recall also that in an AC circuit, a resistor’s voltage and current are in phase with each other, as shown here: • In AC circuits, there are two crucial differences between resistors and capacitors:
1. Current and voltage in a capacitor are not in phase with each other.
2. Instead of having a resistance, a capacitor in an AC circuit has something called a capacitive reactance, which is similar to a resistance.
##### Phase Relationships in a Capacitor
• Current and voltage in a capacitor are not in phase with each other.
• For sinusoidal waves, the current through a capacitor leads the voltage across it by 90º.  (In other words, the voltage lags the current by 90º.)
• For instance, in the diagram below, the short blue waveform represents the voltage across a capacitor, and the tall purple waveform represents the current through the capacitor. Do you agree that the current waveform leads the voltage waveform by 90º? • ##### A Memory Aid: ICE
• To remember that current leads voltage in a capacitor, remember the word ICE.
• Since I comes before E in the word "ICE," this reminds you that the current (I) in a capacitor (C) leads the voltage (E).
• Many textbooks use the letter E as the symbol for voltage, and that’s what E stands for in this memory aid. Our textbook uses V for voltage, but you don’t get a very easy-to-remember word if you write ICV. So if you want to use this memory aid you’ll have to think of E as the symbol for voltage.
##### Capacitive Reactance
• AC current through a capacitor depends not only on the size of the voltage across it, but also on the frequency of that voltage.
• A capacitor’s opposition to AC current is called capacitive reactance, abbreviated XC.  Like resistance, its unit is the ohm. But unlike resistance, its value changes as frequency changes.
• You can calculate a capacitor’s capacitive reactance if you know the capacitor’s size (in farads) and the frequency (or angular frequency) of the capacitor’s current or voltage. Capacitive reactance is given by the equation

XC = 1 ÷ (2pfC)

Since 2pf is equal to ω (the angular frequency), we could also write this equation as
XC = 1 ÷ (ωC)
• You can see from these equations that as frequency increases, capacitive reactance decreases. This means that a capacitor opposes low-frequency current more than it opposes high-frequency current.
• • ##### Ohm’s Law for Capacitors
• In an AC circuit, a capacitor’s current and voltage are related by an equation similar to Ohm’s law:

Ip = Vp ÷ XC
• In words, this says that a capacitor’s peak current is equal to the capacitor’s peak voltage drop divided by the capacitor’s reactance. Notice that this is the same as Ohm’s law for resistors, except that we’ve replaced the resistor’s resistance with the capacitor’s reactance.
• The same law also applies to peak-to-peak values and rms values:
Ipp = Vpp ÷ XC
and
Irms = Vrms ÷ XC
• Of course, you can also rearrange any of these equations algebraically if you wish to solve for voltage or capacitive reactance. For example, if you want to calculate peak voltage based on known values of capacitive reactance and peak current, you would rearrange the first equation to get Vp = Ip × XC.
• ##### Analyzing AC Capacitive Circuits
• Now that you know how to calculate a capacitor’s reactance, and now that you know how to use Ohm’s law with capacitors, you can analyze circuits that contain an AC voltage source connected to any combination of capacitors, such as the one shown below: • To analyze such a circuit, you use the same basic techniques that you used in EET 1150 to analyze DC resistive circuits.
• The only difference is that first you have to calculate each capacitor’s reactance; but from that point onward, you can treat reactances in the same way you’re used to treating resistances. For example, if you know a capacitor’s reactance and its current, then you can use Ohm’s law to find its voltage.
• One thing to be aware of is that reactances in series and parallel combine just like resistors in series and parallel. In other words:
• To find the total reactance of reactances connected in series, add the reactances.
• To find the total reactance of reactances connected in parallel, use the reciprocal formula.
• • ##### Not Ready Yet for AC Circuits with Capacitors and Resistors
• So now you should be able to handle AC circuits with capacitors.
• You’re still not ready, though, to analyze AC circuits that contain capacitors and resistors, such as the one shown below: • Circuits like this are trickier because of the complicated phase relationships between the various currents and voltages in the circuit. We’ll get to circuits like this one in a couple of weeks.

##### Power in a Capacitor
• Recall from Unit 2 that you can use any of the following equations to find the power dissipated by a resistor in an AC circuit:

P = Vrms  × Irms

P = Vrms2 ÷ R

P = Irms2 × R

• Recall also that power dissipation in a resistor means that the resistor is converting some of the circuit’s electrical energy to heat energy. This energy is lost from the circuit as the air around the resistor is heated.
• Ideally, a capacitor does not dissipate any power. What this means is that an ideal capacitor does not convert any of the circuit’s electrical energy to heat energy. Rather, an ideal capacitor stores electrical energy, which can be returned to the circuit at a later time.
• In reality, no capacitor is perfectly ideal, so a real capacitor will lose some electrical energy as heat.
• Because energy storage and loss is more complicated in capacitors than it is in resistors, we distinguish two kinds of power: true power and reactive power.
##### True Power
• From earlier courses you’re familiar with the concept of power. Up to now you’ve probably only studied one kind of power (which we call true power). But as we study capacitors and inductors, you’ll see that power is more complicated than you may have realized.
• True power is a measure of the rate at which a component or circuit loses energy. This energy loss is usually due to dissipation of heat (as in a resistor) or conversion to some other form of energy (as in a motor that converts electrical energy to motion). True power is the type of power that you’ve studied in earlier courses.
• We’ll use the symbol Ptrue for true power.
• Just like power in resistors, true power is measured in watts (W).
• ##### True Power in a Capacitor
• In an ideal capacitor, true power is zero, since ideal capacitors don’t lose any energy as heat.
• In a real capacitor, true power will not be zero, but generally it is small. There’s no easy way to compute true power in a real capacitor, since it depends on the capacitor’s physical construction. Because true power in capacitors is small, we’ll usually just assume that it’s zero.
##### Reactive Power
• Reactive power is a measure of the rate at which a component is storing energy or returning energy to the circuit.
• We use the symbol Pr for reactive power.
• You might think that reactive power would be measured in watts (W), just like true power. But it is not. Instead, we use a unit called the VAR, which stands for volt-ampere reactive. We do this to make it clear that we’re talking about energy that is stored and can be returned to the circuit, rather than energy that is lost from the circuit as heat.
• ##### Reactive Power in a Capacitor
• To compute a capacitor’s reactive power, we use equations that look very similar to the equations for a resistor’s power:

Pr = Vrms  × Irms

Pr = Vrms2 ÷ XC

Pr = Irms2 × XC

• Notice that these are the same as the equations for a resistor’s power, except in the second and third equations we replace resistance (R) with capacitive reactance (XC).
• ##### Comparing Inductors to Resistors and Capacitors
• You learned in Unit 3 that in an AC circuit, a resistor’s voltage and current are in phase with each other, as shown in this example: • You learned above that a capacitor has something called a capacitive reactance, which is similar to a resistance and can be substituted for resistance in Ohm’s law. You also learned that, for sinusoidal waves, the current through a capacitor leads the voltage across it by 90º, as shown in this example: • Like capacitors, inductors in AC circuits differ from resistors in two crucial ways:
1. Current and voltage in an inductor are not in phase with each other.
2. Instead of having a resistance, an inductor in an AC circuit has something called an inductive reactance.
##### Phase Relationships in an Inductor
• Current and voltage in an inductor are not in phase with each other.
• For sinusoidal waves, the voltage across an inductor leads the current through it by 90º.  (In other words, the current lags the voltage by 90º.) This is the opposite of what we saw above for capacitors.
• For instance, in the diagram below, the tall blue waveform represents the voltage across an inductor, and the short purple waveform represents the current through the inductor. Do you agree that the voltage waveform leads the current waveform by 90º? • ##### Ohm’s Law for Inductors
• An inductor’s current and voltage are related by an equation similar to Ohm’s law:

Ip = Vp ÷ XL

• In words, this says that an inductor’s peak current is equal to the inductor’s peak voltage drop divided by the inductor’s reactance.
• The same law also applies to peak-to-peak values and effective values:

Ipp = Vpp ÷ XL

and

Irms = Vrms ÷ XL

• You can also rearrange any of these equations algebraically if you wish to solve for voltage or inductive reactance. For example, if you want to calculate peak voltage based on known values of inductive reactance and peak current, you would rearrange the first equation to the form Vp = Ip × XL.
• ##### Analyzing AC Inductive Circuits
• Now that you know how to calculate an inductor’s reactance, and now that you know how to use Ohm’s law with inductors, you can analyze circuits that contain an AC voltage source connected to any combination of inductors, such as the one shown below: • To analyze such a circuit, you use the same basic techniques that you used in EET 1150 to analyze DC resistive circuits.
• The only difference is that first you have to calculate each inductor’s reactance; but from that point onward, you can treat reactances in the same way you’re used to treating resistances. For example, if you know an inductor’s reactance and its voltage drop, then you can use Ohm’s law to find its current.
• • ##### Not Ready Yet for AC Circuits with Inductors and Other Components
• So now you should be able to handle AC circuits with inductors.
• You’re still not ready, though, to analyze AC circuits that contain inductors and resistors or capacitors, such as the one shown below: • Circuits like this are trickier because of the complicated phase relationships between the various currents and voltages in the circuit. We’ll get to circuits like this one in a couple of weeks.

##### Power in an Inductor
• Recall from above that for capacitors in AC circuits we distinguish two kinds of power: true power and reactive power.
• We saw that true power is zero for an ideal capacitor.
• We also saw that reactive power is calculated using the same formulas used to calculate power in a resistor, except that we replace resistance (R) in these formulas with capacitive reactance (XC).
• As we’ll now see, similar comments apply to inductors in AC circuits.
##### True Power in an Inductor
• So far in this unit we’ve been treating inductors as ideal, which means we’ve been assuming that an inductor has zero winding resistance.
• True power is a measure of the rate at which an inductor loses energy as heat. In an ideal inductor, true power is zero.
• In a real inductor, true power is given by the equation

Ptrue = Irms2 × RW

• Notice that this is the usual formula for computing power in a resistor, except that we’ve replaced resistance (R) with winding resistance (RW).
• Since an inductor’s winding resistance is usually small compared to the circuit’s total resistance, an inductor’s true power is usually small compared to the circuit’s total power.
• True power is measured in watts (W).
##### Reactive Power in an Inductor
• Reactive power is a measure of the rate at which an inductor is storing energy or returning energy to the circuit.
• To compute an inductor’s reactive power, we use equations that look very similar to the equations for a resistor’s power:

Pr = Vrms  × Irms

Pr = Vrms2 ÷ XL

Pr = Irms2 × XL

• Notice that these are the same as the equations for a resistor’s power, except in the second and third equations we replace resistance (R) with inductive reactance (XL).
• Recall that reactive power is measured in volt-amperes reactive (VAR), not in watts (W).
• ##### Quality Factor of an Inductor
• An inductor’s quality factor is the ratio of the inductor’s reactive power to its true power.
• We use the symbol Q to stand for an inductor’s quality factor. So we can write:

Q = Pr ÷ Ptrue

• Since we know that Pr = Irms2 × XL and Ptrue = Irms2 × RW, we can rewrite this equation as

Q = (Irms2 × XL) ÷ (Irms2 × RW)

and then by canceling Irms2 from the numerator and the denominator, we can simplify this to

Q = XL ÷ RW

• So an inductor’s quality factor is equal to the ratio of its inductive reactance to its winding resistance.
• Since an ideal inductor has zero winding resistance, an ideal inductor’s quality factor is infinitely high. But a real inductor (with RW ≠ 0 Ω) has a finite quality factor.
• Quality factor is very important in the topic of resonance, which you’ll study later.
• ##### Review of Electrical Quantities
• It’s time to update our table of electrical quantities. The table shows the abbreviation for each quantity, along with the standard unit for measuring the quantity and the abbreviation for the unit.
• You studied the first eleven quantities (through inductance) in EET 1150. The others have been added in this course.
 Quantity Abbreviation Unit Abbreviation for the Unit charge Q coulomb C current I ampere A voltage (or emf) V (or E) volt V resistance R ohm Ω conductance G siemens S energy (or work) W joule J power P watt W efficiency η capacitance C farad F time constant τ second s inductance L henry H period T second s frequency f hertz Hz peak voltage Vp volts peak V p peak current Ip amps peak A p peak-to-peak voltage Vpp volts peak-to-peak V pp peak-to-peak current Ipp amps peak-to-peak A pp rms voltage Vrms volts rms V rms rms current Irms amps rms A rms time t second s varying voltage v volt V varying current i ampere A angle θ radian or degree rad or ° phase angle φ radian or degree rad or ° angular frequency ω radians per second rad/s capacitive reactance XC ohm Ω true power Ptrue watt W reactive power Pr volt-ampere reactive VAR inductive reactance XL ohm Ω quality factor Q
• If you haven’t already played them, be sure to check out the Electrical-Units Matching Game and the Electrical-Symbols Matching Game.

##### Trigonometry Review
• In the weeks ahead we’ll need the following concepts from trigonometry:
• Trigonometric functions: sin, cos, tan
• Inverse trigonometric functions: sin-1, cos-1, tan-1
• In the x-y plane, the positive x-axis is taken as the reference direction for all angles; in other words, we regard the postive x-axis as lying at an angle of 0°.
• Positive angles are measured counterclockwise from the positive x-axis. Therefore:
• The positive y-axis lies at an angle of +90°.
• The negative x-axis lies at an angle of +180°.
• The negative y-axis lies at an angle of +270°.
• The x-axis and y-axis divide the x-y plane into four quadrants, which we label quadrants I, II, III, and IV.
• All of this is summarized in the following diagram: ##### Negative Angles
• As mentioned above, positive angles are measured counterclockwise from the positive x-axis.
• Negative angles are measured clockwise from the positive x-axis. Therefore:
• The negative y-axis lies at an angle of −90°.
• The negative x-axis lies at an angle of −180°.
• The positive y-axis lies at an angle of −270°.
• This is summarized in the following diagram: • So we see that the same direction can be referred to in more than one way. For example, the negative y-axis lies at an angle of +270°, and it also lies at an angle of −90°. In other words, +270° and −90° are two different names for the same direction.
• Usually, we’ll express angles in quadrants I and II as positive angles between 0° and +180°, and we’ll express angles in quadrants III and IV as negative angles between 0° and -180°.
##### Vectors
• As you’ll recall from your math classes, a vector is a mathematical quantity that has both a magnitude (length) and a direction (angle).
• We’ll use the Greek letter θ (theta) to represent the vector’s angle.
• For example, the diagram below show a vector at an angle of 30°: • Often when we’re dealing with vectors, we’re interested in the finding the vector’s x-component and its y-component. (Or conversely, in some cases we start out knowing the vector’s x-component and its y-component, and we need to find the vector.) The following diagram shows a dotted line segment dropped vertically down from the vector’s tip to the x-axis. The length of this dotted segement is the vector’s y-component, and the length of the red line segment is the vector’s x-component. • Notice that in this diagram a right-angle triangle is formed by the vector, the dotted line segment, and the red line segment. Much of trigonometry is devoted to the study of right-angle triangles. We’ll now briefly review some of the main results of this branch of mathematics.
##### Right-Angle Triangle
• A right-angle triangle is a triangle in which two sides are at right angles (at 90°) to each other.
• In diagrams of right triangles, a small square is drawn in one angle to indicate that this is a 90° angle.
• In the diagram below, we’ve labeled the three sides so that we can easily refer to them by their labels. The labels "A", "B", and "C" have no special meaning–any other letters would serve just as well. • In the diagram below we’ve gone one step further by labeling one of the angles with the Greek letter theta, θ, so that we can easily refer to that angle. ##### Three Kinds of Question
• In later units, we’ll use right-angle triangles to represent electrical quantities such as voltage. Very often we’ll be in a situation where we know some things about a particular triangle, and we wish to figure out some other things.
• Here are the three sorts of question that we’ll need to be able to answer:
1. Given the lengths of two of the sides, what is the length of the third side? For example, in the triangle shown above, perhaps we know the lengths of sides A and C, and we wish to find the length of side B.
2. Given the length of one side and the size of one angle, what is the length of another side? For example, in the triangle shown above, perhaps we know the length of side C and the size of angle θ, and we wish to find the length of side B.
3. Given the lengths of two sides, what is the size of an angle? For example, in the triangle shown above, perhaps we know the lengths of sides A and B, and we wish to find how big the angle θ is (in degrees).
• Trigonometry is the key to being able to answer those questions.
##### Question #1: Finding Unknown Side From Two Known Sides
• The first question listed above was how to find the length of the third side in a right-angle triangle if we know the lengths of the other two sides.
• The solution to this problem is given by the Pythagorean theorem.
##### Pythagorean Theorem
• The Pythagorean theorem says that in any right-angle triangle, the length of the longest side squared is equal to the sum of the lengths of the two shorter sides squared.
• So for the triangle shown below, we can write this in symbols as

C2 = A+ B2. • Now a little algebra will let us solve for any one of the sides if we know the other two sides. In particular,
• To find side C if you know sides A and B, use C = √ (A2 + B2).
• To find side A if you know sides B and C, use A = √ (C2 – B2).
• To find side B if you know sides A and C, use B = √ (C2 – A2).
• ##### Fancy Names for the Sides of a Right-Angle Triangle
• Until now we’ve been using the labels A, B, and C to refer to the three sides of a right triangle. The three sides also have more formal names, which we’ll use below.
• The longest side of a right-angle triangle, which we’ve been calling side C, is called the hypotenuse.
• Once you’ve picked an angle of interest, the side farthest from that angle is called the opposite. For example, relative to angle θ, the side that we’ve been calling side A is the opposite, since it’s on the opposite side of the triangle from θ.
• The remaining side, which is next to (or "adjacent to") the angle of interest, is called the adjacent. For example, relative to angle θ, the side that we’ve been calling side B is the adjacent. • ##### Question #2: Finding Unknown Side From Known Side and Known Angle
• The second question listed above was how to find the length of a side in a right-angle triangle if we know the length of one other side and the size of an angle.
• The solution to this problem is given by the trigonometric functions.
##### Trigonometric Functions: sin, cos, tan
• The three basic trigonometric functions are the sine function, the cosine function, and the tangent function. These are abbreviated sin, cos, and tan.
• In a right triangle:
• sin θ = opposite ÷ hypotenuse;
• cos θ = adjacent ÷ hypotenuse;
• tan θ = opposite ÷ adjacent.
• A handy way to remember this is to remember the nonsense word "Sohcahtoa." Each letter in this word stands for one of the words in the three equations listed above. So if you can remember "Sohcahtoa," then you can remember those three equations.
• Your calculator has keys that give you the sine, cosine, or tangent of any angle.
• As you know, angles can be measured either in degrees or in radians. Scientific calculators can handle either unit of measurement, but you must be sure to put your calculator in the proper mode (degree mode or radian mode) whenever you’re doing a calculation that involves sines, cosines, or tangents.
• • Remember that the second type of question we wanted to be able to answer was to find the length of an unknown side in a right triangle if we know the length of any other side and the size of one angle. The equations above for sin θ, cos θ, and tan θ give us the key to answering this question.
• For example, suppose we know the length of the hypotenuse and the size of the angle θ, and we wish to find the length of the opposite. Then we can take the equation above for sin θ, which says

sin θ = opposite ÷ hypotenuse

and multiply both sides by the hypotenuse, giving us

sin θ × hypotenuse = opposite.

So to find the opposite (which was our goal) we’ll use our calculator to find the sine of θ and then multiply by the length of the hypotenuse.
• ##### Question #3: Finding Unknown Angle From Two Known Sides
• The third question listed above was how to find the size of an angle in a right-angle triangle if we know the lengths of two of the sides.
• The solution to this problem is given by the inverse trigonometric functions.
##### Inverse Trigonometric Functions
• The three inverse trigonometric functions are the inverse sine function, the inverse cosine function, and the inverse tangent function. These are abbreviated sin-1, cos-1, and tan-1.
• As mentioned above, angles can be measured either in degrees or in radians. You must be sure to put your calculator in the proper mode (degree mode or radian mode) whenever you’re doing a calculation that involves inverse sines, inverse cosines, or inverse tangents.
##### Inverse Sine: sin-1
• The inverse sine of a number is the angle whose sine is equal to that number:
θ = sin-1 k means that sin θ = k.
• This is not as confusing as it may sound. For example, since the sine of 60° is 0.866, it follows that 60° is the inverse sine of 0.866. Those are really just two different ways of saying the same thing:

the sine of 60° is 0.866 (or, in symbols, sin 60° = 0.866)

means the same thing as

60° is the inverse sine of 0.866 (in symbols, 60° = sin-1 0.866)
• ##### Inverse Cosine and Inverse Tangent: cos-1 and tan-1
• Similarly,
θ = cos-1 k means that cos θ = k.

and
θ = tan-1 k means that tan θ = k.
• Again, it’s not as confusing as it might look. For example, to say that 70° is the inverse cosine of 0.342 is just another way of saying that the cosine of 70° is 0.342.
Or, saying that 50° = tan-1 1.19 just means the same thing as saying that tan 50° = 1.19.
• If it just means the same thing, then what’s the point of these inverse trig functions? The point is, if we don’t know the size of an angle in a triangle but we do know the lengths of the triangle’s sides, then we can figure out the angle’s sine (or cosine or tangent). Then we can use the calculator’s sin-1 button (or cos-1 button or tan-1 button) to find the size of the angle.
• ##### !!!Caution!!!
• When you use a calculator to compute an inverse trigonometric function, the calculator’s answer may be in the wrong quadrant.
• Draw a sketch so that you can see which quadrant the answer should be in; then, if necessary, correct the calculator’s answer.
• Most often this will happen when you’re working with an angle in quadrants II or III.
• When you take the inverse tangent of an angle in quadrant II, the calculator will give an answer in quadrant IV, which you must adjust by adding 180°.
• Similarly, when you take the inverse tangent of an angle in quadrant III, the calculator will give an answer in quadrant I, which you must adjust by subtracting 180°.
• • This is the end of our trigonometry review.

##### Unit 4 Review
• This e-Lesson has covered several important topics, including:
• capacitive reactance in AC circuits
• phase relationships in capacitors in AC circuits
• true power and reactive power in a capacitor
• inductive reactance in AC circuits
• phase relationships in inductors in AC circuits
• true power and reactive power in an inductor
• trigonometric functions
• inverse trigonometric functions.
• To finish the e-Lesson, take this self-test to check your understanding of these topics.
• Congratulations! You’ve completed the e-Lesson for this unit.