Simple Nature - Light and Matter

10.5.7 ImpedanceSo far we have been thinking in terms of the free oscillations of acircuit. This is like a mechanical oscillator that has been kicked butthen left to oscillate on its own without any external force to keepthe vibrations from dying out. Suppose an LRC circuit is drivenwith a sinusoidally varying voltage, such as will occur when a radiotuner is hooked up to a receiving antenna. We know that a currentwill flow in the circuit, and we know that there will be resonantbehavior, but it is not necessarily simple to relate current to voltagein the most general case. Let’s start instead with the special casesof LRC circuits consisting of only a resistance, only a capacitance,or only an inductance. We are interested only in the steady-stateresponse.The purely resistive case is easy. Ohm’s law givesI = V R.In the purely capacitive case, the relation V = q/C lets us calculateI = dqdt= C dVdt.This is partly analogous to Ohm’s law. For example, if we doublethe amplitude of a sinusoidally varying AC voltage, the derivativedV/ dt will also double, and the amplitude of the sinusoidally varyingcurrent will also double. However, it is not true that I = V/R, becausetaking the derivative of a sinusoidal function shifts its phase by90 degrees. If the voltage varies as, for example, V (t) = V o sin(ωt),then the current will be I(t) = ωCV o cos(ωt). The amplitude of thecurrent is ωCV o , which is proportional to V o , but it’s not true thatI(t) = V (t)/R for some constant R.A second problem that crops up is that our entire analysis ofDC resistive circuits was built on the foundation of the loop ruleand the junction rule, both of which are statements about sums. Toapply the junction rule to an AC circuit, for exampe, we would saythat the sum of the sine waves describing the currents coming intothe junction is equal (at every moment in time) to the sum of thesine waves going out. Now sinusoidal functions have a remarkableproperty, which is that if you add two different sinusoidal functionshaving the same frequency, the result is also a sinusoid with thatfrequency. For example, cos ωt + sin ωt = √ 2 sin(ωt + π/4), whichcan be proved using trig identities. The trig identities can get verycumbersome, however, and there is a much easier technique involvingcomplex numbers.z / In a capacitor, the currentis 90 ◦ ahead of the voltage inphase.Section 10.5 LRC Circuits 607