Reactance, impedance and resonance

The last lesson left the capacitor and inductor active in every cycle of an alternating supply, opposing the current in a way that a plain resistance cannot capture, because it depends on frequency and it shifts the timing between voltage and current. This closing lesson gives that opposition its name and its formula, combines it with resistance into a single quantity, and arrives at resonance, the effect that lets a circuit single out one frequency from many and the natural summit of the whole subject.

Reactance, opposition that depends on frequency

A capacitor passes an alternating current, but not without opposition, and the opposition it offers is called its reactance. It follows directly from the capacitor's nature. A capacitor charges and discharges each cycle, and the faster the cycling, the more readily charge sloshes in and out, so the higher the frequency, the less the capacitor opposes the current. Its reactance therefore falls as frequency rises:

X_{C} = \frac{1}{ω C}

An inductor is the mirror. It fights any change in current, and the faster the current alternates, the more violent the change it must fight, so the higher the frequency, the more it opposes. Its reactance rises with frequency:

X_{L} = ω L

Both reactances are measured in ohms, like resistance, because they too are a voltage divided by a current, but they differ from resistance in a crucial way: a reactance stores and returns energy rather than dissipating it, so a pure capacitor or inductor consumes no average power even while it opposes the current. It only borrows the energy for part of the cycle and hands it back in the next.

Example. A 10 microfarad capacitor is in a 50 hertz circuit. What is its reactance?

The angular frequency is $ω = 2 π \times 50 = 314$ radians per second, so $X_{C} = 1 / (ω C) = 1 / (314 \times 10 \times 10^{- 6}) = 1 / 0.00314 = 318$ ohms. At a higher frequency this would be smaller, since the capacitor passes fast signals more easily.

Now you. A 100 millihenry inductor is in the same 50 hertz circuit. What is its reactance?

Answer

$X_{L} = ω L = 314 \times 0.1 = 31.4$ ohms.

Phase, the shift in timing

Reactance opposes the current, but it also does something a resistor never does: it shifts the phase, the timing, between the voltage and the current. In a resistor, voltage and current rise and fall together, in step. In a capacitor, the current runs ahead of the voltage, because current must flow to build the voltage, so the flow leads the voltage it creates by a quarter of a cycle. In an inductor, the current lags the voltage by a quarter cycle, because the inductor's opposition delays the current from following the voltage. This quarter-cycle lead or lag is why reactance cannot simply be added to resistance as an ordinary number: resistance and reactance act at different moments in the cycle, and combining them has to respect that timing difference.

Impedance, resistance and reactance together

The quantity that combines resistance and reactance, honouring their timing difference, is impedance, written $Z$ , and it is the true measure of how a component or circuit opposes an alternating current. Because resistance acts in step and reactance acts a quarter cycle away, the two combine not by adding but the way the two sides of a right angle combine into a hypotenuse:

Z = \sqrt{R^{2} + X^{2}}

where $X$ is the net reactance. Impedance is the general form of which resistance is the special case at zero frequency, and Ohm's law survives into alternating current in the form $V = I Z$ , with impedance in place of resistance. Engineers track the timing as well as the size by treating these quantities as phasors, little arrows whose length is the size and whose angle is the phase, so that adding impedances becomes adding arrows, but the essential idea needs no more than the right-angle combination: resistance and reactance are perpendicular contributions to one total opposition.

Resonance, when the two reactances cancel

The most striking effect appears when a capacitor and an inductor are in a circuit together, because their reactances pull in opposite directions. As frequency rises the inductor's reactance grows and the capacitor's shrinks, so somewhere in between there is one frequency at which the two are exactly equal. At that frequency their opposite effects cancel, the net reactance falls to zero, and the circuit is left opposing the current only by whatever resistance it has. This is resonance, and the frequency at which it happens is found by setting $X_{L} = X_{C}$ :

ω L = \frac{1}{ω C} \Rightarrow ω = \frac{1}{\sqrt{L C}}

At resonance a series circuit passes current most easily, and the voltages across the inductor and capacitor can swing far larger than the source, as energy sloshes back and forth between the magnetic field of the coil and the electric field of the capacitor, each handing it to the other every cycle, the electrical echo of a pendulum swinging energy between motion and height. The circuit responds enormously at its resonant frequency and weakly at all others, which makes it a filter of exquisite selectivity.

Example. A circuit has a 100 microhenry inductor and a 100 picofarad capacitor. What is its resonant angular frequency, and roughly what ordinary frequency is that?

$ω = 1 / \sqrt{L C} = 1 / \sqrt{100 \times 10^{- 6} \times 100 \times 10^{- 12}} = 1 / \sqrt{10^{- 14}} = 10^{7}$ radians per second. Dividing by $2 π$ gives about 1.6 million hertz, 1.6 megahertz, squarely in the AM radio band, which is no accident.

Now you. A tuning circuit uses a 50 microhenry inductor. What capacitance sets its resonance at an angular frequency of $2 \times 10^{7}$ radians per second? Use $C = 1 / (ω^{2} L)$ .

Answer

$C = 1 / (ω^{2} L) = 1 / ((2 \times 10^{7})^{2} \times 50 \times 10^{- 6}) = 1 / (4 \times 10^{14} \times 5 \times 10^{- 5}) = 1 / (2 \times 10^{10}) = 5 \times 10^{- 11}$ F, or 50 picofarads.

Where the subject arrives

Resonance is where the whole course comes together, because it uses every idea before it. It needs the capacitor's electric-field store and the inductor's magnetic-field store, the frequency-dependent reactance of each, the phase difference that makes them cancel rather than add, and the alternating source that drives them. Its payoff is one of the quiet miracles of technology: a radio receiver is flooded with the signals of every station at once, and a resonant circuit, tuned by adjusting a capacitor or inductor until its resonant frequency matches one station, responds hugely to that one and ignores the rest, plucking a single voice out of the crowded air. From the bare fact that matter carries charge, the subject has reached the circuit that tunes a radio, and every step between, the field, the voltage, the current, the resistance, the two kinds of store and the alternating source, was a necessary rung. A reader who has climbed them all can now look at any linear circuit, direct or alternating, and know not just what it does but why, which is what it means to understand how electricity works.