30.9 LC Circuits

Telecommunication—radios, televisions, cell phones—is based on electromagnetic signals that oscillate at a well-defined frequency. These oscillations are generated and detected by a simple circuit consisting of an inductor and a capacitor. This is called an LC circuit. In this section we will learn why an LC circuit oscillates and determine the oscillation frequency.

FIGURE 30.42 shows a capacitor with initial charge $Q_{0}$ , an inductor, and a switch. The switch has been open for a long time, so there is no current in the circuit. Then, at $t = 0$ , the switch is closed. How does the circuit respond? Let’s think it through qualitatively before getting into the mathematics.

The figure illustrates a circuit diagram of a capacitor and an inductor connected by a switch.

FIGURE 30.42 Full Alternative Text

As FIGURE 30.43 shows, the inductor provides a conducting path for discharging the capacitor. However, the discharge current has to pass through the inductor, and, as we’ve seen, an inductor resists changes in current. Consequently, the current doesn’t stop when the capacitor charge reaches zero.

The figure consists of four sequential parts labeled “A” to D that illustrate the oscillation in an L C circuit using a spring block system.

FIGURE 30.43 Full Alternative Text

A block attached to a stretched spring is a useful mechanical analogy. Closing the switch to discharge the capacitor is like releasing the block. The block doesn’t stop when it reaches the origin; its inertia keeps it going until the spring is fully compressed. Likewise, the current continues until it has recharged the capacitor with the opposite polarization. This process repeats over and over, charging the capacitor first one way, then the other. That is, the charge and current oscillate.

The goal of our circuit analysis will be to find expressions showing how the capacitor charge Q and the inductor current I change with time. As always, our starting point for circuit analysis is Kirchhoff’s voltage law, which says that all the potential differences around a closed loop must sum to zero. Choosing a cw direction for I, Kirchhoff’s law is

Δ V_{C} + Δ V_{L} = 0

(30.43)

The potential difference across a capacitor is $Δ V_{C} = Q / C$ , and we found the potential difference across an inductor in Equation 30.36. Using these, Kirchhoff’s law becomes

\frac{Q}{C} - L \frac{d I}{d t} = 0

(30.44)

Equation 30.44 has two unknowns, Q and I. We can eliminate one of the unknowns by finding another relation between Q and I. Current is the rate at which charge moves, $I = d q / d t$ , but the charge flowing through the inductor is charge that was removed from the capacitor. That is, an infinitesimal charge dq flows through the inductor when the capacitor charge changes by $d Q = - d q$ . Thus the current through the inductor is related to the charge on the capacitor by

I = - \frac{d Q}{d t}

(30.45)

Now I is positive when Q is decreasing, as we would expect. This is a subtle but important step in the reasoning.

A photograph of a lady using a cell phone. — A cell phone is actually a very sophisticated two-way radio that communicates with the nearest base station via high-frequency radio waves—roughly 1000 MHz. As in any radio or communications device, the transmission frequency is established by the oscillating current in an LC circuit.

Equations 30.44 and 30.45 are two equations in two unknowns. To solve them, we’ll first take the time derivative of Equation 30.45:

\frac{d I}{d t} = \frac{d}{d t} (- \frac{d Q}{d t}) = - \frac{d^{2} Q}{d t^{2}}

(30.46)

We can substitute this result into Equation 30.44:

\frac{Q}{C} + L \frac{d^{2} Q}{d t^{2}} = 0

(30.47)

Now we have an equation for the capacitor charge Q.

Equation 30.47 is a second-order differential equation for Q. Fortunately, it is an equation we’ve seen before and already know how to solve. To see this, we rewrite Equation 30.47 as

\frac{d^{2} Q}{d t^{2}} = - \frac{1}{L C} Q

(30.48)

Recall, from Chapter 15, that the equation of motion for an undamped mass on a spring is

\frac{d^{2} x}{d t^{2}} = - \frac{k}{m} x

(30.49)

Equation 30.48 is exactly the same equation, with x replaced by Q and $k / m$ replaced by $1 / L C$ . This should be no surprise because we’ve already seen that a mass on a spring is a mechanical analog of the LC circuit.

We know the solution to Equation 30.49. It is simple harmonic motion $x (t) = x_{0} cos ω t$ with angular frequency $ω = \sqrt{k / m .}$ Thus the solution to Equation 30.48 must be

Q (t) = Q_{0} cos ω t

(30.50)

where $Q_{0}$ is the initial charge, at $t = 0$ , and the angular frequency is

ω = \sqrt{\frac{1}{L C}}

(30.51)

The charge on the upper plate of the capacitor oscillates back and forth between $+ Q_{0}$ and $- Q_{0}$ (the opposite polarization) with period $T = 2 π / ω$ .

As the capacitor charge oscillates, so does the current through the inductor. Using Equation 30.45 gives the current through the inductor:

I = - \frac{d Q}{d t} = ω Q_{0} sin ω t = I_{max} sin ω t

(30.52)

where $I_{max} = ω Q_{0}$ is the maximum current.

An LC circuit is an electric oscillator, oscillating at frequency $f = ω / 2 π$ . FIGURE 30.44 shows graphs of the capacitor charge Q and the inductor current I as functions of time. Notice that Q and I are $90 °$ out of phase. The current is zero when the capacitor is fully charged, as expected, and the charge is zero when the current is maximum.

The figure illustrates a graphical representation of the oscillations of an L C circuit.

FIGURE 30.44 Full Alternative Text

EXAMPLE 30.15 An AM radio oscillator

You have a 1.0 mH inductor. What capacitor should you choose to make an oscillator with a frequency of 920 kHz? (This frequency is near the center of the AM radio band.)

SOLVE The angular frequency is $ω = 2 π f = 5.78 \times 10^{6} rad/s$ . Using Equation 30.51 for $ω$ gives the required capacitor:

C = \frac{1}{ω^{2} L} = 3.0 \times 10^{- 11} F = 30 pF

An LC circuit, like a mass on a spring, wants to respond only at its natural oscillation frequency $ω = 1 / \sqrt{L C}$ . In Chapter 15 we defined a strong response at the natural frequency as a resonance, and resonance is the basis for all telecommunications. The input circuit in radios, televisions, and cell phones is an LC circuit driven by the signal picked up by the antenna. This signal is the superposition of hundreds of sinusoidal waves at different frequencies, one from each transmitter in the area, but the circuit responds only to the one signal that matches the circuit’s natural frequency. That particular signal generates a large-amplitude current that can be further amplified and decoded to become the output that you hear.

30.9 LC Circuits

FIGURE 30.42 An LC circuit.

FIGURE 30.43 The capacitor charge oscillates much like a block attached to a spring.

FIGURE 30.44 The oscillations of an LC circuit.

EXAMPLE 30.15 An AM radio oscillator