Inductors and Capacitors, Part II

This article describes the basics of capacitance and inductance. An understanding of calculus is helpful, but not necessary. A basic understanding of voltage and current is assumed. This article uses a higher level of mathematics than most of the tutorials. A knowledge of complex arithmatic is also helpful.

Complex Numbers

Complex numbers are those that involve the square root of -1. Since a square root can never be negative (it doesn't matter if the number is positive or negative, if you square it you end up with a positive number), this is called an imaginary number. At first you might think that dealing with an imaginary number is just another excuse for those mathematical type persons to go off and fill a page with useless equations. However, it turns out that these imaginary numbers have properties that make them very useful in electronics.

In mathematics, the square root of -1 is usually written using a lower case letter i. In electronics, i stands for current (intensity), so we use j for the square root of -1 instead.

Complex numbers have two parts. The first is the real term, which is the type of number we are used to dealing with on a daily basis. The second part is the imaginary term, which is a multiple of j. Either term may be zero.

Reactance and Complex Numbers

The reactance of a capacitor (Xc) is given by the formula Xc=1/(jwC), where C is the capacitance and w=2(pi)f (and f is the frequency). The w is usually written as a greek omega, but hey, this is only an ASCII text, so we have to make do with a w. Note that the reactance changes with the frequency, just as the current vs. voltage changed with the frequency in part I of this article.

The inductive reactance (Xl) equals jwL, where L is the inductance, and w=2(pi)f. Note that the reactance of capacitors and inductors both contain no real terms.

You can think of the reactance as being somewhat equivalent to the resistance.

Phasors

Sorry, we aren't talking about Star Trek here. A Phasor is simply a complex number. If you use the real and imaginary numbers as x and y values on a graph, you can plot complex voltages, currents, impedences, etc. as vectors. This is a little complicated, but hopefully using impedence as an example will make it more clear.

Impedence is simply how much the electricity is impeded, or restrained. For a simple resistor, the impedence was the resistance. For a capacitor or inductor, the impedence is the reactance. Impedence is, in general, a complex number. It contains a real part and an imaginary part. The real part comes from real resistance. The imaginary part comes from reactance. This is easy to remember, since inductors and capacitors don't absorb energy like a resistor (which converts the energy into heat). Instead, inductors and capacitors store and release energy.

We usually use a capital Z to represent impedence. Since it has a real and an imaginary component, Z=R+X. Now, here's the fun part. Remember Ohm's Law? Well, we can still use it, even with capacitors and inductors (this is the beauty of phasors, we can use simple algebra and phasors to solve problems instead of being forced to use calculus).

We usually use a capital V and I to represent voltage and current as phasors. The formula is simply V=IZ (very similar to V=IR for resistors), only now we are using phasors instead of simple real numbers.

As an example, let's assume we have a 150 ohm resistor, a 2 mH inductor, and a 473 uF capacitor all in series, connected to a voltage source of 120 volts AC, 60 Hz. The total impedence is Z=R+Xl+Xc, or Z=150+jw(.002)+1/(jw(.000473)), and w=2(pi)(60), or 120(pi). One important thing to note here is that 1/j is equal to -j. If this doesn't make sense, consider what happens if you multply the top and bottom both by j. You end up with j/(j squared), and (j squared) is negative one, and j/(-1) equals -j. If we continue solving for Z, we get Z=150+0.75j-5.61j (approximately), or Z=150-4.86j. Note that we just added up all of the impedences, the same way we did for resistors in series.

To solve for the current, it is easiest to use the polar notation for the phasor. If you graph the real and imaginary terms as vectors, you end up with a resultant vector. This vector has a magnitude and a direction. The magnitude is the square root of R squared plus X squared (where R and X are the real and imaginary parts), and the angle, usually written as a greek theta, is the inverse tangent of (X/R). We write this as (magnitude)<(angle), where the < is really an angle sign. If you are writing it by hand, you make it look more like an angle than a less than sign. Some people continue the bottom of the line under the angle value, so that it cannot be mistaken for a less than sign.

In our example, the magnitude is 150.07 and the angle is -1.86. If we write Z in polar form, it is then 150.07<-1.86.

Ok, now we are ready to find the current. The formula is V=IZ. We know Z and V, so we need to solve for I. If we rearrange the formula, we get I=V/Z, or I = (120<0)/(150.07<-1.86). To divide the phasors, we simply divide the magnitude and subtract the angle. So, I=(120/150.07)<(0-(-1.86)), or 0.800<1.86.

We can convert this back to rectangular form (using the real and imaginary terms) very easily. The real part is (magnitude)cos(angle), and the imaginary part is (magnitude)sin(angle). I is therefore 0.8cos(1.86)+0.8sin(1.86)j, or 0.800+0.026j.

Note that the angle (called the phase angle) of the current is greater than the angle of the voltage (which was zero). This means that the current is leading the voltage. Current in capacitive circuits leads the voltage, and lags behind the voltage in inductive circuits.

Summary of Phasors

The use of phasors can be a little confusing at first, but it is a valuable technique for solving circuits, especially in power line circuits where the frequency is a constant. In other circuits, plotting a few values may give you a good idea of how the circuit is operating at different frequencies. Remember that all signals can be broken down into a sum of various sine waves, so that looking at the response of individual frequencies can let you see how the circuit will perform for any arbitrary waveform.

The use of phasors requires conversions between polar and rectangular forms, but otherwise lets you use simple algebra to solve the equations. This allows us to use the simple V=IZ and P=VI formulas, instead of being forced to deal with things like i=C(dv/dt). To multiply two phasors together (in polar form), multiply the magnetudes and add the angles. To divide, divide the magnitude and subtract the angles. Addition and subtraction are more easily done in rectangular form. Just add/subtract all of the real terms as neccessary, and then add/subtract all of the imaginary terms.

Watts and Vars

If you deal with computer uninterruptable power supplies, you know that they are rated in volt-amps instead of watts. What is a volt-amp, you ask? Well, quite simply it is the volts multiplied by the amps. From the above discussion on phasors, you should now realize that this may result in a complex number (for a computer UPS, all they give you is the magnitude in the specs). If you convert this into a real and an imaginary component, the real part is called the watts, and the imaginary part is called the vars (var = volt amp reactive).

You may also encounter a term called the power factor. The power factor is the cosine of the current angle subtracted from the voltage angle. A power factor of 1 means that the voltage and current are perfectly in phase. The power company likes to have a power factor of 1 on their lines, since this makes the transmission of power more efficient. Since most household loads are slightly inductive (due to motors in hair dryers, vacuum cleaners, dishwashers, etc), the power company adds capacitors to the line to compensate for the inductance.

Frequency Effects and Practical Applications

Capacitors and inductors are usually used in circuits which take advantage of the fact that their response varies with frequency. One use is power supply filters. If you put a capacitor in parallel with your load across the terminals of the power supply, as the frequency gets higher, the reactance of the capacitor gets smaller. At higher frequencies, the capacitor effectively provides a short circuit across the power supply, but at DC, the reactance is extremely large, so large that the capacitor does not seem to even exist in the circuit. When capacitors are used in this way, they are called bypass capacitors, since they bypass the power supply.

Remember from the formula Xc=1/(jwC) that the reactance gets smaller as C gets larger. Therefore, larger capacitors work better as power supply bypass capacitors. You will often see large electrolytics used as the main filter capacitors in a power supply. However, since electrolytics do not function properly at higher frequencies, you will also often see smaller ceramic disc capacitors in parallel with the electrolytics. These capacitors work well at higher frequencies, and function to get rid of the noise that the electrolytics would not filter off. In many circuits, especially higher frequency digital circuits, individual chips will have a small ceramic capacitor (typically 0.1 uF) bypassing the power leads. This capacitor is installed physically very close to the chip and care is taken to keep the signal path very short from the capacitor to the chip. This allows the capacitor to filter off any high frequency noise that might be present, and helps to keep the chip operating more reliably. Op amps will often oscillate or do weird things if they are lacking a small bypass capacitor. Higher speed digital circuits will also often misbehave without small capacitors on each chip.

Inductors can also be used to filter off power supply noise. Inductors are placed in series in the power supply line. At low frequencies, they are essentially a short circuit, allowing all of the DC power to go through the line. At higher frequencies, the reactance starts to climb (remember Xl=jwL, so Xl is bigger when w is bigger). This presents a higher impedence to the noise source, and less noise current flows. Inductors are often called chokes when they are used this way.

Capacitors and inductors are very often used in filter circuits. Circuits can be made which pass or reject only high or only low frequencies, or perhaps let only a particular frequency band through. Because of the complexity of this subject, filters are discussed in more detail in a seperate article.

As was previously mentioned, capacitors and inductors may be used to balance out reactance on a line, as is often done with power lines.

One neat trick is to use a capacitor to drop voltage from an AC power line. This is sometimes done in places where a heavy transformer is undesirable. However, it has all of the pitfalls of using a resistor to drop voltage in a circuit. First, the actual voltage dropped varies, depending on the current that is flowing. This means that this technique is only good when a constant amount of current is flowing. Second, because a transformer is not used, there is no isolation from the main AC line. This technique also has a benefit that no heat is generated, unlike a resistor.

Note that a diode and a capacitor can be used as an AM demodulating circuit. The diode rectifies the signal, and the capacitor filters off the higher frequency carrier. This is important, because diodes and capacitors used for other purposes will sometimes make a perfect AM receiver in the middle of your circuit.

A little bit of variation

Sometimes it is valuable to have a variable capacitor or a variable inductor. In a radio receiver, for example, the receiving frequency might depend on the resonnant frequency of a particular RLC circuit (a circuit with a resistor, capacitor, and inductor - see the article on filters for more details). By varying the capacitance, you can vary the frequency that the radio receives. Older stereos and televisions used this principle. The stereo would have a dial, which was usually connected to a variable capacitor (usually by a long string and pulley system). The capacitor was constructed by simply using alternating plates, one set being stationary and the other movable. By varying the spacing of the plates, the capacitance was varied. Older television dials did the same thing, but had fixed plate spacing depending on the channel selected, instead of being completely tunable like stereos.

Variable inductors are also common. These are sometimes made by using a conductor wiper, similar to how variable resistors are constructed, and sometimes are made by using a variable core inside the inductor.

Variable capacitors and inductors will be specified by their range, in addition to the normal capacitor and inductor specs (max voltage, max current, etc).