Introduction
Inductive reactance has its counterpart in the form of capacitive reactance . This, too, can be represented as a ray starting at the same zero point as inductive reactance but running off in the opposite direction, having negative ohmic values (Fig. 15-8). When the ray for capacitive reactance is combined with the ray for inductive reactance, a complete real-number line is the result, with ohmic values that range from the huge negative numbers, through zero, to huge positive numbers.
Fig. 15-8 . Capacitive reactance can be represented on half-line or ray. There is no limit to how large it can get negatively, but it can never be positive.
Capacitors And DC
Imagine two large parallel metal plates, as described earlier. If you connect them to a source of dc, they draw a large amount of current at first as they become electrically charged. However, as the plates reach equilibrium, this current diminishes, and when the two plates attain the same potential difference throughout, the current is zero.
If the voltage of the battery or power supply is increased, a point is eventually reached at which sparks begin to jump between them. Ultimately, if the power supply can deliver the necessary voltage, this sparking, or arcing , becomes continuous. Then the pair of plates no longer acts like a capacitor. When the voltage across a capacitor is too great, the dielectric (whatever it is) no longer functions properly. This condition is known as dielectric breakdown .
In air-dielectric and vacuum-dielectric capacitors, dielectric breakdown is a temporary affair; it does not cause permanent damage. The device operates normally when the voltage is reduced, so the arcing stops. However, in capacitors with solid dielectric such as mica, paper, or tantalum, dielectric breakdown can burn or crack the dielectric, causing the component to conduct current even when the voltage is reduced below the arcing threshold. In such instances, the component is ruined.
Capacitors And AC
Suppose that the power source connected to a capacitor is changed from dc to ac. Imagine that you can adjust the frequency of this ac from a low initial value of a few hertz up to hundreds of hertz, then to many kilohertz, and finally to many megahertz or gigahertz.
At first, the voltage between the plates follows along with the voltage of the power source as the source polarity reverses over and over. However, the set of plates has a certain amount of capacitance. The plates can charge up fast if they are small and if the space between them is large, but they can’t charge instantaneously. As you increase the frequency of the ac source, there comes a point at which the plates do not get charged up very much before the source polarity reverses. The set of plates becomes sluggish. The charge does not have time to get established with each ac cycle.
At high ac frequencies, the voltage between the plates has trouble following the current that is charging and discharging them. Just as the plates begin to get a good charge, the ac current passes its peak and starts to discharge them, pulling electrons out of the negative plate and pumping electrons into the positive plate. As the frequency is raised, the set of plates starts to act more and more like a short circuit. Eventually, if you keep on increasing the frequency, the period of the wave is much shorter than the charging-discharging time, and current flows in and out of the plates in the same way as it would flow if the plates were shorted out.
Capacitive reactance is a quantitative measure of the opposition that the set of plates offers to ac. It is measured in ohms, just like inductive reactance and just like resistance. However, by convention, it is assigned negative values rather than positive ones. Capacitive reactance, denoted X C in mathematical formulas, can vary from near zero (when the plates are huge and close together and/or the frequency is very high) to a few negative ohms to many negative kilohms or megohms.
Capacitive reactance varies with frequency. It gets larger negatively as the frequency goes down and smaller negatively as the frequency goes up. This is the opposite of what happens with inductive reactance, which gets larger (positively) as the frequency goes up. Sometimes capacitive reactance is talked about in terms of its absolute value, with the minus sign removed. Then you might say that X C increases as the frequency decreases or that X C diminishes as the frequency is raised. However, it is best if you learn to work with negative X C values from the start.
Reactance And Frequency
Capacitive reactance behaves in many ways like a mirror image of inductive reactance. In another sense, however, X C is an extension of X L into negative values—below zero—with its own peculiar set of characteristics.
If the frequency of an ac source is given in hertz as f and the capacitance of a capacitor in farads is given as C , then the capacitive reactance is
X C = −1/(2π/ fC ) = −(2π fC ) −1 ≈ −(6.2832 fC ) −1
This same formula applies if the frequency is in megahertz (MHz) and the capacitance is in microfarads (μF). Remember that if the frequency is in millions, the capacitance must be in millionths. This formula also would apply for frequencies in kilohertz (kHz) and millifarads (mF), but for some reason, you’ll almost never see millifarads used in practice. Even millifarads are large units for capacitance; components of more than 1,000 μF (which would be 1 mF) are rarely found in real-world electrical systems.
Capacitive reactance varies inversely with the frequency. This means that the function X C versus f appears as a curve when graphed, and this curve “blows up negatively” as the frequency nears zero. Capacitive reactance also varies inversely with the actual value of capacitance given a fixed frequency. The function of X C versus C also appears as a curve that “blows up negatively” as the capacitance approaches zero. The negative of X C is inversely proportional to frequency, as well as to capacitance. Relative graphs of these functions are shown in Fig. 15-9.
Fig. 15-9 . Capacitive reactance is inversely proportional to the negative of the capacitance, as well as to the negative of the frequency.
Reactance And Frequency Practice Problem
Problem 1
A capacitor has a value of 0.00100 μF at a frequency of 1.00 MHz. What is the capacitive reactance?
Solution 1
Use the formula and plug in the numbers. You can do this directly because the data are specified in microfarads (millionths) and in megahertz (millions):
X C = −1/(6.2832 × 1.00 × 0.00100) = −1/(0.0062832) = −159 ohms
This is rounded to three significant figures because the data are specified only to this many digits.
Problem 2
What will be the capacitive reactance of the preceding capacitor if the frequency decreases to zero—that is, if the power source is dc?
Solution 2
In this case, if you plug the numbers into the formula, you’ll get zero in the denominator. Division by zero is not defined. In reality, however, there is nothing to prevent you from connecting a dc battery to a capacitor! You might say, “The reactance is extremely large negatively and, for all practical purposes, is negative infinity.” More appropriately, you should call the capacitor an open circuit for dc.
Problem 3
Suppose that a capacitor has a reactance of −100 ohms at a frequency of 10.0 MHz. What is its capacitance?
Solution 3
In this problem you need to put the numbers in the formula and solve for the unknown C . Begin with this equation:
−100 = −(6.2832 × 10.0 × C ) −1
Dividing through by −100:
1 = (628.32 × 10.0 × C ) −1
Multiply each side of this by C:
C = (628.32 × 10.0) −1
= 6283.2 −1
This can be solved easily enough. Divide out C = 1/6283.2 on your calculator, getting C = 0.00015915. Because the frequency is given in megahertz, this capacitance comes out in microfarads, so C = 0.00015915 μF. This must be rounded to C = 0.000159 μF in this scenario. You also can say C = 159 pF (remember that 1 pF = 0.000001 μF).
The arithmetic for dealing with capacitive reactance is a little messier than that for inductive reactance for two reasons. First, you have to work with reciprocals, and therefore, the numbers sometimes get awkward. Second, you have to watch those negative signs. It’s easy to leave them out or to put them in when they should not be there. They are important when looking at reactances in the coordinate plane because the minus sign tells you that the reactance is capacitive rather than inductive.
Points In The RC Quarter-plane
Capacitive reactance can be plotted along a half-line or ray just as can inductive reactance. Capacitive and inductive reactance, considered as one, form a real-number line. The point where they join is the zero-reactance point.
In a circuit containing resistance and capacitive reactance, the characteristics are two-dimensional in a way that is analogous to the situation with the RL quarter-plane. The resistance ray and the capacitive-reactance ray can be placed end to end at right angles to make the RC quarter-plane (Fig. 15-10). Resistance is plotted horizontally, with increasing values toward the right. Capacitive reactance is plotted downward, with increasingly negative values as you go down.
Fig. 15-10 . The RC impedance quarter-plane showing five points for specific complex-number impedances.
Complex-number impedances that contain resistance and capacitance can be denoted in the form R + jX C ; however, X C is never positive. Because of this, scientists often write R – jX C , dropping the minus sign from X C and replacing addition with subtraction in the complex-number rendition.
If the resistance is pure, say, R = 3 ohms, then the complex-number impedance is 3 − j 0, and this corresponds to the point (3, 0) on the RC quarter-plane. You might suspect that 3 − j 0 is the same as 3 + j 0 and that you need not even write the j 0 part at all. In theory, both these notions are correct. However, writing the j 0 part indicates that you are open to the possibility that there might be reactance in the circuit and that you’re working in two dimensions.
If you have a pure capacitive reactance, say, X C = −4 ohms, then the complex-number impedance is 0 – j 4, and this is at the point (0, −4) on the RC quarter-plane. Again, it’s important, for completeness, to write the 0 and not just the − j 4. The points for 3 − j 0 and 0 − j 4, and three others, are plotted on the RC quarter-plane in Fig. 15-10.
In practical circuits, all capacitors have some leakage conductance . If the frequency goes to zero, that is, if the source is dc, a tiny current will flow because no dielectric is a perfect electrical insulator. Some capacitors have almost no leakage conductance, but none are completely free of it. Conversely, all electrical conductors have a little capacitive reactance simply because they occupy physical space. Thus there is no such thing as a mathematically pure conductor of ac either. The points 3 − j 0 and 0 − j 4 are idealized.
Remember that the values for X C are reactances, not capacitances. Reactance varies with the frequency in an RC circuit. If you raise or lower the frequency, the value of X C changes. A higher frequency causes X C to get smaller negatively (closer to zero). A lower frequency causes X C to get larger negatively (farther from zero or lower down on the RC quarter-plane). If the frequency goes to zero, then the capacitive reactance drops off the bottom of the plane, out of sight. In this case you have two plates or sets of plates having opposite electric charges but no “action.”
Practice problems of these concepts can be found at: Alternating Current Practice Test
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