We would like generalize what we have learned about DC circuits to AC circuits. First we will look at the concept of resistance and Ohm's Law. We have already learned that resistance represents a a type of loss. Electrical power flowing through a resistor is converted into heat. We also learned that capacitors and inductors possess a property called reactance, which represents their ability to temporarily store electrical power in electric or magnetic fields. Since both resistance and reactance affect the flow of current in an AC circuit, we will define a new quantity called impedance, defined as follows:
Impedance = Z = resistance + reactance = R+jX
Impedance has two parts. The resistive or real part represents power consumption. The reactive or imaginary part represents power storage. Ohm's Law can be used for AC analysis, providing that impedance is used in place of resistance:
V = IZ I = V/Z Z = V/I
In general both V and I will be complex numbers, since Z is complex. At this point it is important to realize that complex voltages and currents are quite common in AC analysis and that the complex notation is a convenient way to express differences in phase. Click here to see an illustration of these phase differences.
If we allow complex voltages and currents, Kirchoff's Laws can also be applied to AC circuits. Recall that Kirchoff's laws state that (a) the algebraic sum of the voltages around a loop is zero and (b) the algebraic sum of the currents entering/leaving a node is zero.
Before going farther, we need to sharpen our definition of reactance - in particular, how are frequency, capacitance, inductance and reactance related. For an inductor, inductance and inductive reactance are related through the following equation:
where: f = frequency (Hz)
L = inductance (H)
XL = inductive reactance
Inductive reactance is directly proportional to frequency. As the frequency of the AC voltage applied to an inductor increases, the opposition it offers to current flow increases. Inductive reactance is also directly proportional to inductance. If the inductance of an inductor is doubled, the inductive reactance is also doubled, if everything else is held constant. Inductive reactance is always imaginary and positive.
Here is an example:
What is the reactance of a 10 mH inductor at (a) 1 KHz, (b) 300 KHz, (c) 17 MHz?
(a)
(b)
(c)
The capacitive reactance of a capacitor is also dependent on frequency, and is given by the equation below:
where: f = frequency (Hz)
C = capacitance (F)
XC = capacitive reactance
Capacitive reactance is inversely proportional to frequency. As the frequency of the AC voltage applied to a capacitor increases, the opposition it offers to current flow decreases. Capacitive reactance is also inversely proportional to capacitance. If the capacitance of a capacitor is doubled, its reactance is halved, if everything else is held constant. Capacitive reactance is always imaginary and negative.
Here is an example:
What is the reactance of a 3300 pF capacitor at (a) 1 KHz, (b) 10 KHz, (c) 100 KHz?
(a)
(b)
(c)
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