$LC$ Circuit

Consider an electrical circuit consisting of an inductor, of inductance $L$, connected in series with a capacitor, of capacitance $C$. See Figure 4. Such a circuit is known as an $LC$ circuit, for obvious reasons. Suppose that $I(t)$ is the instantaneous current flowing around the circuit. According to standard electrical circuit theory, the potential difference across the inductor is $L\,\dot{I}$. Again, from standard electrical circuit theory, the potential difference across the capacitor is $V=Q/C$, where $Q$ is the charge stored on the capacitor's positive plate. However, since electric charge is conserved, the current flowing around the circuit is equal to the rate at which charge accumulates on the capacitor's positive plate: i.e., $I = \dot{Q}$. Now, according to Kirchhoff's second circuital law, the sum of the potential differences across the various components of a closed circuit loop is equal to zero. In other words,

$$L\,\dot{I}+Q/C = 0.$$ (33)

Dividing by $L$, and differentiating with respect to $t$, we obtain

$$\ddot{I} + \omega^2\,I = 0,$$ (34)

where

$$\omega = \frac{1}{\sqrt{L\,C}}.$$ (35)

Comparison with Equation (17) reveals that (34) is a simple harmonic oscillator equation with the associated angular oscillation frequency $\omega$. We conclude that the current in an $LC$ circuit executes simple harmonic oscillations of the form

$$I(t) = I_0\,\cos(\omega\,t-\phi),$$ (36)

where $I_0>0$ and $\phi$ are constants. Now, according to Equation (33), the potential difference, $V=Q/C$, across the capacitor is minus that across the inductor, so that $V= -L\,\dot{I}$, giving

$$V(t) = \sqrt{\frac{L}{C}}\,I_0\,\sin (\omega\,t-\phi) = \sqrt{\frac{L}{C}}\,I_0\,\cos(\omega\,t-\phi-\pi/2).$$ (37)

Here, use has been made of the trigonometric identity $\sin\theta\equiv \cos(\theta-\pi/2)$. It follows that the voltage in an $LC$ circuit oscillates at the same frequency as the current, but with a phase shift of $\pi/2$. In other words, the voltage is maximal when the current is zero, and vice versa. The amplitude of the voltage oscillation is that of the current oscillation multiplied by $\sqrt{L/C}$. Thus, we can also write

$$V(t) = \sqrt{\frac{L}{C}}\,I(t-\omega^{-1}\,\pi/2).$$ (38)

Figure 4: An $LC$ circuit.

Comparing with Equation (22), it is clear that

$${\cal E} = \frac{1}{2}\,\dot{I}^{\,2} +\frac{1}{2}\, \omega^2\,I^2$$ (39)

is a conserved quantity. However, $\omega^2=1/L\,C$, and $\dot{I} = -V/L$. Thus, multiplying the above expression by $C\,L^2$, we obtain

$$E = \frac{1}{2}\,C\,V^2 + \frac{1}{2}\,L\,I^2.$$ (40)

The first and second terms on the right-hand side of the above expression can be recognized as the instantaneous energies stored in the capacitor and the inductor, respectively. The former energy is stored in the electric field generated when the capacitor is charged, whereas the latter is stored in the magnetic field induced when current flows through the inductor. It follows that (40) is the total energy of the circuit, and that this energy is a conserved quantity. Clearly, the oscillations of an $LC$ circuit can be understood as a cyclic interchange between electric energy stored in the capacitor and magnetic energy stored in the inductor, much as the oscillations of the mass-spring system studied in Section 2.1 can be understood as a cyclic interchange between kinetic energy stored by the mass and potential energy stored by the spring.

Suppose that at $t=0$ the capacitor is charged to a voltage $V_0$, and there is no current flowing through the inductor. In other words, the initial state is one in which all of the circuit energy resides in the capacitor. The initial conditions are $V(0)=-L\,\dot{I}(0)=V_0$ and $I(0) = 0$. It is easily demonstrated that the current evolves in time as

$$I(t) = -\frac{V_0}{\sqrt{L/C}}\,\sin(\omega\,t).$$ (41)

Suppose that at $t=0$ the capacitor is fully discharged, and there is a current $I_0$ flowing through the inductor. In other words, the initial state is one in which all of the circuit energy resides in the inductor. The initial conditions are $V(0)=-L\,\dot{I}(0)=0$ and $I(0)=I_0$. It is easily demonstrated that the current evolves in time as

$$I(t) = I_0\,\cos(\omega\,t).$$ (42)

Suppose, finally, that at $t=0$ the capacitor is charged to a voltage $V_0$, and the current flowing through the inductor is $I_0$. Since the solutions of the simple harmonic oscillator equation are superposable, it follows that the current evolves in time as

$$I(t) = -\frac{V_0}{\sqrt{L/C}}\,\sin(\omega\,t) + I_0\,\cos(\omega\,t).$$ (43)

Furthermore, according to Equation (38), the voltage evolves in time as

$$V(t)= - V_0\,\sin(\omega\,t-\pi/2) + \sqrt{\frac{L}{C}}\,I_0\,\cos(\omega\,t-\pi/2),$$ (44)

or

$$V(t) = V_0\,\cos(\omega\,t) + \sqrt{\frac{L}{C}}\,I_0\,\sin(\omega\,t).$$ (45)

Here, use has been made of the trigonometric identities $\sin (\theta-\pi/2)\equiv -\cos\theta$ and $\cos(\theta-\pi/2)\equiv \sin\theta$.

The instantaneous electrical power absorption by the capacitor, which can easily be shown to be minus the instantaneous power absorption by the inductor, is

$$P(t)=I(t)\,V(t) = I_0\,V_0\,\cos(2\,\omega\,t) + \frac{1}{2}... ...L}{C}}- \frac{V_0^{\,2}}{\sqrt{L/C}}\right)\sin(2\,\omega\,t),$$ (46)

where use has been made of Equations (43) and (45), as well as the trigonometric identities $\cos(2\,\theta)\equiv \cos^2\theta-\sin^2\theta$ and $\sin(2\,\theta)\equiv 2\,\sin\theta\,\cos\theta$. Hence, the average power absorption during a cycle of the oscillation,

$$\langle P\rangle \equiv \frac{1}{T}\int_0^T P(t)\,dt,$$ (47)

is zero, since it is easily demonstrated that $\langle\cos(2\,\omega\,t)\rangle=\langle \sin(2\,\omega\,t)\rangle=0$. In other words, any energy which the capacitor absorbs from the circuit during one part of the oscillation cycle is returned to the circuit without loss during another. The same goes for the inductor.

Figure 5: A simple pendulum.