Question

A circuit has in series a resistor \(\mathrm{R} \Omega\), and inductor of \(L \mathrm{H}\), and a capacitor of \(C\) farads. The initial current is zero and the initial charge on the capacitor is \(Q_{0}\) coulombs. (a) Show that the charge and the current are damped oscillatory functions of time if and only if \(R<2 \sqrt{L / C}\), and find the expressions for the charge and the current in this case. (b) If \(R \geq 2 \sqrt{L / C}\), discuss the nature of the charge and the current as functions of time.

Short answer

In an RLC circuit with inductor \(L\), resistor \(\mathrm{R} \Omega\), and capacitor of \(C\) farads, if \(R<2 \sqrt{L / C}\), the charge and current are damped oscillatory functions of time: \[Q(t)=Q_1e^{r_1t}+Q_2e^{r_2t}\] \[I(t)=Q'(t)=r_1Q_1e^{r_1t}+r_2Q_2e^{r_2t}\] where \(r_{1,2}=-\frac{R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}}\). If \(R \geq 2 \sqrt{L / C}\), the functions will be either critically damped or overdamped, and the charge and current decay monotonically.

Step-by-step solution

Set up the differential equation for the circuit

To begin with, we recall Kirchoff's voltage law: the sum of voltages around any closed loop of a circuit must be zero. In this case, the voltage across the resistor is given by \(V_R = IR\), the voltage across the inductor is given by \(V_L = L \frac{dI}{dt}\), and the voltage across the capacitor is given by \(V_C = \frac{Q}{C}\), where \(I\) is the current and \(Q\) is the charge on the capacitor. Therefore, we have: \[IR+L\frac{dI}{dt}+\frac{Q}{C} = 0\] Since \(Q = \int Idt\), we can rewrite the equation as: \[R\frac{dQ}{dt}+L\frac{d^2Q}{dt^2}+\frac{Q}{C}=0\]

Solve the differential equation for different cases

Now, we consider two cases: (1) \(R < 2\sqrt{\frac{L}{C}}\) and (2) \(R \geq 2\sqrt{\frac{L}{C}}\). For case (1), we have a damped oscillatory solution for the charge and current. We can write the homogeneous second-order differential equation as: \[\frac{d^2Q}{dt^2}+\frac{R}{L}\frac{dQ}{dt}+\frac{1}{LC}Q=0\] To solve this, we first assume \(Q(t)=e^{rt}\), where \(r\) is a constant. Substituting this into the equation, we get: \[r^2Q+RrQ+Q=0\] Dividing through by \(Q\) and \(L\), we obtain the characteristic equation: \[r^2+\frac{R}{L}r+\frac{1}{LC}=0\] We solve the characteristic equation for the roots \(r_{1,2}\) using the quadratic formula, which results in: \[r_{1,2}=-\frac{R}{2L}\pm\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}}\] Since \(R<2\sqrt{\frac{L}{C}}\), the discriminant is positive and the roots are distinct. Thus, we have damped oscillatory functions for the charge and the current: \[Q(t)=Q_1e^{r_1t}+Q_2e^{r_2t}\] \[I(t)=Q'(t)=r_1Q_1e^{r_1t}+r_2Q_2e^{r_2t}\]

Discuss the case when \(R \geq 2\sqrt{\frac{L}{C}}\)

For case (2), we have \(R \geq 2\sqrt{\frac{L}{C}}\), which means the discriminant of the characteristic equation is either equal to zero or negative. This results in either critically damped (\(\Delta = 0\)) or overdamped (\(\Delta < 0\)) solutions. In both cases, the charge and current decay monotonically. For critically damped, we have a single root \(r_{1}=r_2=-\frac{R}{2L}\) and for overdamped, we have two negative roots. In conclusion, this exercise required setting up and solving the differential equation governing an RLC circuit. Specifically, we showed that the charge and current are damped oscillatory functions of time if and only if \(R<2\sqrt{\frac{L}{C}}\), and we found the expressions for the charge and current in this case. We also discussed the nature of the charge and current when \(R \geq 2\sqrt{\frac{L}{C}}\).

Key concepts

RLC Circuit

An RLC circuit is an electrical circuit consisting of three components: a resistor (R), an inductor (L), and a capacitor (C), all connected in series or parallel. In a series configuration, the resistor resists the flow of current, the inductor stores energy in the magnetic field, and the capacitor stores energy in the electric field. Understanding how these components interact is crucial for analyzing the circuit's behavior over time.
RLC circuits are fundamental in many electronic applications, as they can filter signals, tune frequencies, and manage power flow. The combination of these components results in differential equations that describe the circuit's dynamics. Such circuits can exhibit complex behavior, including oscillations and damping, which depend heavily on the component values and their configuration in the circuit.

Damped Oscillations

Damped oscillations occur in an RLC circuit when energy is gradually lost, usually in the form of heat from the resistor. These oscillations decrease over time until the system eventually comes to rest. In the context of the circuit, damped oscillations are observed when the resistance is less than a critical value defined by the inequality: \(R < 2\sqrt{\frac{L}{C}}\).

• If this condition is met, the circuit will experience underdamped oscillations, characterized by periodic fluctuations with decreasing amplitude.
• The reduction in amplitude continues with each cycle, reflecting the energy loss in the circuit.

This behavior has practical applications, such as in radio tuners and audio equipment, where controlled oscillations are needed for proper functioning.

Kirchoff's Voltage Law

Kirchoff's Voltage Law (KVL) is a fundamental principle used to analyze electrical circuits. It states that the sum of all voltages around a closed loop in a circuit must equal zero. This law helps in setting up the differential equations needed to describe an RLC circuit's behavior over time.
In an RLC circuit:

• The voltage drop across the resistor is given by \(V_R = IR\).
• The voltage across the inductor is \(V_L = L \frac{dI}{dt}\).
• The voltage across the capacitor is \(V_C = \frac{Q}{C}\).

Applying KVL to these components results in the equation:
\[IR + L \frac{dI}{dt} + \frac{Q}{C} = 0\] This equation forms the basis for deriving further relationships in the study of the circuit's oscillatory behavior.

Characteristic Equation

The characteristic equation is a crucial element in analyzing second-order differential equations for circuits, such as RLC circuits. By assuming a solution of the form \(Q(t) = e^{rt}\), we can substitute back into the differential equation and derive a polynomial equation in terms of \(r\), called the characteristic equation.
For an RLC circuit, the characteristic equation is:
\[r^2 + \frac{R}{L}r + \frac{1}{LC} = 0\]
Solving this equation using the quadratic formula provides the roots, \(r_1\) and \(r_2\), which determine the nature of the solution:

• If the roots are complex, the system exhibits damped oscillations.
• If the roots are real and equal, the system is critically damped.
• If the roots are distinct and real, it is overdamped.

This equation is vital for predicting how the circuit will behave over time.

Second-order Differential Equation

A second-order differential equation involves the second derivative of a function and is a key tool in modeling the behavior of RLC circuits. When applied to an RLC circuit, it takes into account the rate of change of current or charge through the system, leading to equations that describe how these quantities evolve over time.
The general form used in an RLC circuit is:

• \(L \frac{d^2Q}{dt^2} + R \frac{dQ}{dt} + \frac{Q}{C} = 0\)

Solving this equation provides insight into how the charge \(Q(t)\) and current \(I(t)\) change with time. The solutions are crucial for understanding whether the circuit exhibits oscillatory behavior, and if so, whether the oscillations are damped. This form of differential equation is common in a wide variety of physical systems beyond just electrical circuits.