Basic Working Of Capacitance

Capacitance: The other type of reactance is capacitive reactance, whose effect is opposite that of inductive reactance. The basic capacitive device is a capacitor. 

A capacitor consists of two conducting surfaces or plates that face each other and are separated by a small gap. These plates can carry an electric charge; specifically, their charges will be opposite. By having an opposite charge on the opposing plate, very nearby but not touching, it is possible to collect a large amount of charge on each plate. 

We might say that the charge “sees” the opposite charge across the gap and is attracted to it, rather than only being repelled by its like charge on the same plate. In physical terms, there is an electric field across the gap that serves to hold the accumulation of charge on the plates. The gap could simply be air but is often filled with a better insulating (dielectric) material to prevent sparks from bridging the gap (though even such an insulating material can fail or leak charge; this is known as dielectric breakdown). For compactness, the plates are often made out of pliable sheets that are folded or rolled up inside a container. Like inductance, capacitance also occurs in devices that do not conform to an ideal shape; coaxial cables are an important example. Our discussion extends to all devices with capacitance. 

When presented with a d.c. voltage, a capacitor essentially behaves like a gap in the circuit. Initially, there is a brief period of building up the charge on both plates. But once the charge is built up and cannot go across, the capacitor acts as an open circuit, and no current will flow. 

An alternating current, however, can get across the capacitor. Recall from Section 1.1 that although a current represents a flow of charge, individual electrons do not actually travel a significant distance through a conductor; rather, each electron transmits an impulse or “push” to its neighbor. Because this impulse can be transmitted across the gap by means of the electric field, it is not necessary for electrons to physically travel across. This transmission only remains effective as long as the impulse (in other words, the voltage) keeps changing because once the charge has accumulated on the capacitor plate and a steady electric field is established, there is nothing more to transmit. Indeed, the current flow across a capacitor is proportional to the rate of change of the electric field, which corresponds to the rate of change of the voltage across the capacitor. As the voltage oscillates, the electric field continually waxes and wanes in alternating directions.

The greater the frequency, the more readily the current is transmitted since the rate of change of the voltage will be greater.8 The capacitive reactance, denoted by X or XC, is given by the inverse of the product of the angular frequency and the capacitance, which is denoted by C and measured in farads (F). The capacitance depends on the physical shape of the capacitor; it increases with the area of the plates and with decreasing separation between them (since the greater proximity of the charges will cause a stronger electric field), as long as there is no contact. In equation form,

                               X= - 1/wC

The equation shows that the magnitude of the capacitive reactance (neglecting the negative sign) increases with decreasing v, as it should. It also increases with decreasing capacitance. This is intuitive because a decrease in capacitance means that the plates are becoming less effective at supporting an electric field to transmit anything. The negative sign in the equation has to do with the effect of capacitance on a circuit being opposite that of an inductor, as we see shortly. Thus, when inductive and capacitive reactance are added together, they tend to cancel each other (see the example in Section 3.2.3). 

Like an inductor, a capacitor causes a phase shift between current and voltage in an a.c. circuit, but the crucial point is that the shift is in the opposite direction. A pure capacitance causes the current to lead the voltage by 908, as shown in Figure 3.7. We can see that the moment of greatest current flow coincides with the most rapid change in the voltage. A capacitor also stores and releases energy during different parts of the cycle. This energy resides in the electric field between the plates. The storage and release of energy by a capacitor occur at time intervals opposite to those of an inductor in the same circuit (this is consistent with the phase shift being in opposite directions). A capacitor and an inductor can, therefore, exchange energy between them in an alternating fashion. Like an ideal inductor, an ideal capacitor (without resistance) only exchanges but never dissipates energy. 

Also analogous to an inductor, there is an equation relating voltage and current for a capacitor. Here it is the voltage V that appears in terms of a rate of change with respect to time, and the capacitance C is the proportionality constant linking it to the current I:

                                                 

                                               I = C dV/ dt

This equation characterizes a capacitor as a thing that resists changes in voltage, which makes sense if we consider the capacitor plates as a large reservoir of charge: it takes a lot of current to effect a change in the potential. Again, the equation is consistent with the graph (Figure 3.7) that shows the alternating current reaches its maximum value at the instant that the voltage changes most rapidly.

Like inductance, the capacitance is geometry dependent and occurs to some extent with electrical devices of any shape. For example, there is a certain amount of capacitance between a transmission line and the ground. The parallel-plate capacitor is the strongest and simplest case, and it becomes much more difficult to derive a capacitance value for objects with different shapes. Such calculations are treated in standard electrical engineering texts.

COMPLEX REPRESENTATION

Complex numbers are a concise way to mathematically represent two aspects of a physical system at the same time. This will be necessary for describing impedance as a combination of resistance and reactance in the following section (readers familiar with complex notation may skip ahead).

 A complex number contains a real part, which is an ordinary number and directly corresponds to a measurable physical quantity, and an imaginary part, which is a sort of intangible quantity that, when projected onto physical reality, represents oscillatory behavior. 

An imaginary number is a multiple of the imaginary unit quantity ffiffiffiffiffiffiffi 1p . This quantity is denoted by i for imaginary in mathematics, and jin electrical engineering so as to avoid confusion with the label for current. The definition of j ¼ ffiffiffiffiffiffiffi 1p is another way of saying that j2 ¼21. It also implies that1 /j¼2j. 

This entity j embodies the notion of oscillation, or time-varying behavior, in its very nature. Consider the equation x2 ¼21. There is no real number that can work in this equation if substituted for x. Pick any positive number for x, and x2 is positive. Pick any negative number for x, and when you square it, the result is also positive. Pick zero, and x2 is zero. So we devise an abstract object we call j—not a real number, as we know it, but a thing which, it turns out, can also be manipulated just like a regular number. We define j as that thing that makes the equation x2 ¼21 true. 

You can think of j as the number that cannot decide whether it wants to be positive or negative. In fact, the equation x2 ¼21 can be translated into the logical statement, “This statement is false.” The statement cannot decide whether it is true or false; it flip-flops back and forth. The imaginary j is therefore in essence “flippety.”9 The number j makes the most sense when we represent it graphically, as is usually done in electrical engineering. Consider the number line with positive and negative real numbers, the positive numbers extending to the right and the negative numbers to the left of zero. Now think of multiplying a positive number by 21. What does this operation do? It takes the number from the right-hand side of the number line and drops it over to the left. For example,

3.21¼23. In other words, we can think of the multiplication by 21 as a rotation by 180 degrees about the origin (the zero point). 

Now we can take the result (say, 23) and multiply it again by 21. We get 23.21 ¼3. In other words, the number was again rotated by 1808. Successive applications of the “multiplication by 21” operation result in successive rotations.

 So far, we have restricted our imagination to numbers that lie on the real number line. Now suppose we “think outside the box” and ask the following question: What if there were an operation that rotates a number not by 1808, but by only 908? Rotating by 908 does not immediately seem meaningful, because it takes us off the number line into uncharted territory. But we do know this: If performed twice in succession, a rotation by 908 corresponds to a complete 1808 rotation. In other words, a rotation by 908, performed twice, gives the same result as multiplying by 21.

 But this leads us directly to the equation x2 ¼21, which asks for the number that, when multiplied by itself, becomes negative. If we define “multiplying by j” as “rotating by 908” and “multiplying by j2” is the same as “multiplying by j and then multiplying by j again,” then to multiply by j2 is to rotate by 1808, which is exactly the same thing as multiplying by 21. In this sense we can say that j2 and 21 are the same.

 Based on this rotation metaphor, j is conceptualized as the number that is measured in the new “imaginary” direction 908 off the real number line. We can now extend this concept of being “off the real number line” to an entire new, imaginary number line at right angles to the real one, intersecting at the origin or zero point. Along this new axis we can measure multiples of j upward and 2j downward.

With these two directions, we have in effect converted the one-dimensional number line into a two-dimensional plane of numbers, called the complex plane. This plane is defined by a real axis labeled Re (the old number line) and an imaginary axis labeled I'm (the new imaginary number line at right angles to the real one). We can now conceive of numbers that lie anywhere on this complex plane and represent a combination of real and imaginary numbers. These are the complex numbers. The complex number C ¼aþjb refers to the point a units to the right of the origin and b units above. We say that a is the real part and b is the imaginary part of C. In Figure 3.8, a ¼3 and b ¼4.

 Rather than specifying the real and imaginary components explicitly, another way of describing the same complex number is in reference to an arrow (vector) drawn from the origin to the point corresponding to the number. The length of the arrow is the magnitude of the complex number. This magnitude is a real number and is usually denoted by vertical lines, as in jCj. In addition, the angle between the arrow and the real axis is specified, which we denote by f. The representation in terms of magnitude and angle is called the polar form.

These alternative representations can be converted into one another by using any two of the relationships 

\

                                     jCj2 ¼ a2 þb2         sin ¼ b=C         cosf ¼ a=C      or        tanf ¼ b=a

 Adding or subtracting complex numbers is easily done in the rectangular component format: one simply combines the real parts and the imaginary parts, respectively. To multiply two complex numbers, both components of one are multiplied by both components of the other, and the results are added (like in a common binomial expression). However, multiplying, and especially dividing complex numbers is much easier in the polar format: the magnitudes are multiplied together (or divided) and the angles are added (for the division, the divisor angle is subtracted).

 The imaginary j and all the complex numbers that spring from it do not have the same utilitarian properties that real numbers do. You cannot eat j eggs for breakfast, and you cannot be the jth person in line at the post office. Nevertheless, complex numbers as operational devices do obey rules of manipulation that qualify them as “numbers” in the mathematical sense, and their special properties in these manipulations make them useful tools for representing certain real phenomena, especially phenomena that involve flippety, such as alternating current.10