LC Circuit

Imagine yourself standing on a beach, staring at the waves coming in and out of the shore in perfect balance. The waves appear to go out and recede in perfect and equal time, exhibiting wave-like behavior like those we see in trigonometry functions in mathematics. The phenomenon of sinusoidal waves is something we see across all reaches of physics, one examples we will be looking at today are inductor and capacitor circuits, also referred to as LC Circuit. You may be familiar with inductors from previous courses; they are made up of conducting materials that allow currents to be induced via a changing magnetic field. On the other hand, we have capacitors, which store electrical energy. these two electrical components complement each other and allow for LC circuits to exhibit wave-like behavior in the frequency of their current. Keep reading to find out more!

Fig. 1 - Wave-like behavior is observed across all fields of physics, including circuits.

LC Circuit Equations

Before we go into detail about the equations and frequency in LC circuits, let's take a more detailed look at the setup of the circuit. Referring to the figure below, we have a charged capacitor of capacitance \(C\) connected to a solenoid with an inductance \(L\).

Fig. 2 - The structure of an LC circuit with a charged capacitor connected to a solenoid

If you are familiar with capacitors, you would know that these electrical components store electrical energy through the separation of charge between their parallel plates. You can check out our other articles on capacitors to learn more! These parallel plate capacitors are charged by connecting them to a power source in a circuit, allowing for electrons to build up on one side of the plate. This separation of charge generates an electric field between the plates, storing energy within it.

Now that we have a charged capacitor, we then connect this to an inductor, which in this case is a solenoid. As there is no longer a power source connected to the capacitor, there is no electrical potential keeping the electrons at one of the parallel plates. Thus, the capacitor discharges and generates a current which flows through the circuit, as well as through the solenoid. By running a current through the solenoid, we now have a magnetic field surrounding the coil. However, as the capacitor runs out of electrical energy to discharge, the magnetic field becomes weaker and weaker. On the other hand, this weakening magnetic field results in a changing magnetic flux, which then induces a current in the solenoid through the phenomenon of electromagnetic induction. This current then charges the capacitor up until the magnetic field reaches zero, and we go back to the beginning!

We can see now that the inductor and the capacitor work together in perfect balance with one another. The electrical energy stored in each component oscillates between one another, similar to the oscillations of a sinusoidal wave. Now that we have understood LC circuits qualitatively. Let's express it quantitatively through mathematical expression.

Firstly, we need to establish the total voltage within the circuit. As the components are connected in series, we can put together their individual voltages to give

\[ V_{\text{T}} = V_{\text{L}} + V_{\text{C}} ,\]

where \(V_{\text{T}}\) is the total voltage, \(V_{\text{L}}\) is the voltage of the inductor, and \(V_{\text{C}}\) is the voltage of the capacitor. They are all measured in units of volts \(\mathrm{V}\). Now we think back to our knowledge of capacitors and inductors and recall their equations relating them to voltage. For an inductor, this is given by

\[ V_{\text{L}} = L \frac{\mathrm{d} I}{\mathrm{d} t},\]

where \(L\) is the inductance measured in units of henrys \(\mathrm{H}\), \(I\) is the current measured in amperes \(\mathrm{A}\), and \(t\) is time measured in seconds \(\mathrm{s}\). Similarly, for a capacitor, the equation is given by

\[ V_{\text{C}} = \frac{Q}{C} ,\]

where \(Q\) is the charge on the parallel plates of the capacitor measured in coulombs \(\mathrm{C}\) and \(C\) is the capacitance of the capacitor measured in farads \(\mathrm{F}\). Now we can substitute these into our expression for the total voltage in the circuit,

\[ V_{\text{T}} = L \frac{\mathrm{d} I}{\mathrm{d} t} + \frac{Q}{C} .\]

Resonant Frequency of LC Circuit

Now that we have our equation of an LC circuit, we can use it to derive the resonant frequency of the system. We can define the resonant frequency as the following.

As our LC circuit is not connected to an external power source, we can say that the oscillations it exhibits are at its natural or resonant frequency. Now using the LC circuit equation, we can take its time derivative as

\[ \begin{align} \frac{\mathrm{d} V_{\text{T}}}{\mathrm{d}t} &= L \frac{\mathrm{d^2} I }{\mathrm{d} t^2} + \frac{1}{C} \frac{\mathrm{d} Q}{ \mathrm{d} t } \\ \frac{\mathrm{d} V_{\text{T}}}{\mathrm{d}t} &= L \frac{\mathrm{d^2} I }{\mathrm{d} t^2} + \frac{1}{C} I \\ 0 &= L \frac{\mathrm{d^2} I }{\mathrm{d} t^2} + \frac{1}{C} I. \end{align} \]

Now to unpack what we have derived above. In the first line, we have taken the time derivative of both components on the right-hand side. This only includes the current \(I\) and the charge \(Q\) as they are the only time-dependent components, while the inductance \(L\) and the capacitance \(C\) are constants.

Subsequently, we can rewrite the time derivative of the charge \( \frac{\mathrm{d} Q}{\mathrm{d} t} \) as the current \(I\), due to the fact that the current is the rate of flow of charge. Finally, in the last line, we replace the time derivative of the total voltage with zero. This is because we know that the total voltage \(V_{\text{T}}\) in the circuit remains constant, as there is no external power source.

Now you may recognize this equation as a second-order, homogenous differential equation, which we can solve to determine the frequency of the system. We will go through the full derivation of the frequency, but we can first define it as

\[ \omega_0 = \frac{1}{\sqrt{LC}},\]

where \(\omega_0\) is the resonant frequency measured in units of \(\mathrm{\frac{rad}{s}}\).

LC Circuit Frequency

When solving second-order differential equations, we first use a general solution that we can substitute into the equation. This is given by

\[ I(t) = A e ^{Bt} ,\]

where \(A\) and \(B\) are constants to be determined. We can take the derivative of this to give

\[ \frac{\mathrm{d} I}{\mathrm{d} t} = AB e^{Bt} ,\]

and take the second derivative again as

\[ \frac{\mathrm{d^2}I}{\mathrm{d} t^2} = AB^2 e^{Bt} .\]

Substituting this into our second-order differential equation, we get

\[ AB^2 e^{Bt} + \frac{1}{LC} A e^{Bt} = 0 .\]

Now to use this to solve our equation, we need to find the roots of the characteristic equation, which we can find by

\[ Ae^{Bt} \left( B^2 + \frac{1}{LC} \right) = 0 .\]

To solve this, we have a trivial solution given by \(A = 0\) and we know that the exponential function never reaches zero, so we are left with

\[ \begin{align} B^2 + \frac{1}{LC} &= 0 \\ B^2 &= -\frac{1}{LC} \\ B &= \pm i \sqrt{\frac{1}{LC}} \\ B &= \pm i \omega_0 . \end{align} \]

Thus we are left with a characteristic equation with complex roots, giving a general solution in terms of the exponentials

\[ I(t) = c_1 e^{i\omega_0 t} + c_2 e^{-i \omega_0 t} ,\]

where \(c_1\) and \(c_2\) are constants to be determined. However, we know that the current \(I\) is a real, observable quantity, thus we can discard the imaginary part of the function if we can separate it from the real part. This is done by substituting in Euler's equation to result in

\[ \begin{align} I(t) &= c_1 \left( \cos(\omega_0 t) + i \sin(\omega_0 t) \right) + c_2 \left( \cos(\omega_0 t) - i \sin(\omega_0 t) \right) \\ I(t) &= (c_1 + c_2) \cos(\omega_0 t) + i(c_1 - c_2) \sin(\omega_0 t) . \end{align} \]

Discarding the imaginary part, we are left with

\[ I(t) = A\cos(\omega_0 t ) .\]

Where we have replaced \(c_1 + c_2\) with \(A\) since they are both just constants that can be rewritten as another constant. Now to solve for \(A\), we need to consider an initial condition, at time \(t = 0 \, \mathrm{s}\), when the inductor is first connected to the capacitor, we know that the current in the circuit is \( I_0\), the initial current. Thus

\[ I( t = 0\, \mathrm{s} ) = I_0 .\]

Finally, our expression for the current in an LC circuit is

\[ I(t) = I_0 \cos(\omega_0 t ) .\]

As a result, we see that \(\omega_0\) is indeed the resonant frequency of an LC circuit. We can convert this angular frequency into the frequency in hertz using the equation

\[ \begin{align} \omega_0 &= 2 \pi f \\ \frac{1}{\sqrt{LC}} &= 2 \pi f \\ f &= \frac{1}{\sqrt{LC} \, 2 \pi } , \end{align} \]

where \(f\) is the frequency of the current in an LC circuit measured in hertz \(\mathrm{Hz}. \)

Time Constant of LC Circuit

Firstly, let's define what a time constant is.

What types of systems have time constants? These are systems that evolve over time and eventually reach a steady state, resulting in no variation in the system over time. An example of this type of system in circuits would be a resistor-capacitor circuit, also referred to as an RC circuit. When the charged capacitor is connected to the resistor, it discharges over time, allowing a current to run through the resistor. However, once the charge on the capacitor is depleted, there is no current left in the circuit to charge the capacitor again, unlike an LC circuit, thus reaching a steady state.

Fig. 3 - An RC circuit reaches a steady state when the capacitor has been fully discharged.

If we compare this to the LC circuit we have looked at before, it should be clear that an LC circuit does not reach a steady state, as the current is constantly evolving due to the exchange of electrical energy between the inductor and the capacitor. Therefore, an LC circuit does not have a time constant.

LC Circuit Maximum Current

Finally, we want to determine the value of the maximum current flowing through the circuit, and what values of time it occurs at. Since we have the equation for the current, we can differentiate it with respect to time to find the maximum current. However, from our knowledge of sinusoidal functions, we are also able to read the amplitude from the function as \(I_0\). This tells us that the maximum current within an LC circuit is the initial current at time \(t = 0 \, \mathrm{s} \).

Now let's differentiate the function to find the values of time at which the maximum current occurs. This gives us

\[ \frac{\mathrm{d} I}{\mathrm{d} t} = -I_0 \omega_0 \sin(\omega_0 t) .\]

Setting this function to zero, we find

\[ \begin{align} -I_0 \omega_0 \sin(\omega_0 t) &= 0 \\ \sin(\omega_0 t) &= 0 \\ \omega_0 t &= 0, \pi, 2\pi \\ t &= 0, \frac{\pi}{\omega_0} , \frac{2\pi}{\omega_0} . \end{align} \]

Thus we have our maximum current and the time it occurs at.