Capacitive Reactance Help (page 2)
Capacitive Reactance—Capacitors and Current
Inductive reactance has its counterpart in the form of capacitive reactance, denoted X C . In many ways, inductive and capacitive reactance are alike. They’re both forms of “electrical inertia.” But in a capacitive reactance, the voltage has trouble keeping up with the current—the opposite situation from inductive reactance.
Capacitors And Current
Imagine two gigantic, flat, parallel metal plates, both of which are excellent electrical conductors. If a source of DC, such as that provided by a large battery, is connected to the plates (with the negative pole on one plate and the positive pole on the other), current begins to flow immediately as the plates begin to charge up. The voltage difference between the plates starts out at zero and builds up until it is equal to the DC source voltage. This voltage buildup always takes some time, because the plates need time to become fully charged. If the plates are small and far apart, the charging time is short. But if the plates are huge and close together, the charging time can be considerable. The plates form a capacitor, which stores energy in the form of an electric field.
Suppose the current source connected to the plates is changed from DC to AC. Imagine that you can adjust the frequency of this AC from a few hertz to many megahertz. At first, the voltage between the plates follows almost exactly along as the AC polarity reverses. As the frequency increases, the charge, or voltage between the plates, does not have time to get well established with each current cycle. When the frequency becomes extremely high, the set of plates behaves like a short circuit.
Capacitive reactance is a quantitative measure of the opposition that a capacitor offers to AC. It, like inductive reactance, varies with frequency and is measured in ohms. But X C is, by convention, assigned negative values rather than positive values. For any given capacitor, the value of X C increases negatively as the frequency goes down, and approaches zero from the negative side as the frequency goes up.
X C Vs Frequency
Capacitive reactance behaves, in some ways, like a mirror image of inductive reactance. In another sense, X C is an extension of X L into negative values. If the frequency of an AC source is given (in hertz) as f, and the value of a capacitor is given (in units called farads ) as C, then the capacitive reactance (in ohms), X C , can be calculated using this formula:
X C = –1/(2πfC )
Capacitive reactance varies inversely with the negative of the frequency. The function of X C versus f appears as a curve when graphed, and this curve “blows up negatively” (or, if you prefer, “blows down”) as the frequency nears zero. Capacitive reactance varies inversely with the negative of the capacitance, given a fixed frequency. Therefore, the function of X C versus C also appears as a curve that “blows up negatively” as the capacitance approaches zero. Relative graphs of these functions are shown in Fig. 9-10.
RC Phase Angle
When the resistance R in an electrical circuit is significant compared with the absolute value (or negative) of the capacitive reactance, the alternating voltage resulting from an alternating current lags that current by less than 90°. More often, it is said that the current leads the voltage (Fig. 9-11). If R is small compared with the absolute value of X C , the extent to which the current leads the voltage is almost 90°; as R gets relatively larger, the phase difference decreases. When R is many times greater than the absolute value of X C , the phase angle, ø RC , is nearly zero. If the capacitive reactance vanishes altogether, leaving just a pure resistance, then the current and voltage are in phase with each other.
The value of the phase angle ø RC , which represents the extent to which the current leads the voltage, can be found using a calculator. The angle is the arctangent of the ratio of the absolute value of the capacitive reactance to the resistance:
ø RC = arctan(|X C |/R )
Because capacitive reactance X C is always negative or zero, we can also say this:
ø RC = arctan(– X C / R )
Capacitive Reactance Practice Problems
Find the extent to which the current leads the voltage in an AC electronic circuit that has 96.5 ohms of resistance and -21.1 ohms of capacitive reactance. Express your answer in radians to three significant figures.
Use the above formula to find ø RC , setting X C = -21.1 and R = 96.5:
ø RC = arctan(| – 21.1|/96.5)
= arctan (21.1/96.5)
= arctan (0.21865)
= 0.215 rad
Practice problems for these concepts can be found at: Waves and Phase Practice Test
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