Question

Find the amplitude of the steady state current \(I_{p}\) in the \(R L C\) circuit shown in Figure 6.3 .1 if \(E(t)=U \cos \omega t+V \sin \omega t,\) where \(U\) and \(V\) are constants. Then find the value \(\omega_{0}\) of \(\omega\) maximizes the amplitude, and find the maximum amplitude.

Short answer

Answer: The amplitude of the steady-state current \(I_p\) in the RLC circuit can be found using the equation: $$I_p = \frac{\sqrt{(U^2 + V^2)}}{\sqrt{(R^2 + (\omega L - \frac{1}{\omega C})^2)}}.$$ The maximum amplitude is found when \(\omega\) is equal to the resonant frequency given by \(\omega_0 = \frac{1}{\sqrt{LC}}\). At this frequency, the maximum amplitude is given by: $$I_{p_{max}} = \frac{\sqrt{(U^2 + V^2)}}{R}.$$

Step-by-step solution

Write Driving Voltage in Phasor Domain

Let's first represent the driving voltage in the phasor domain. We have \(E(t) = U\cos\omega t + V\sin\omega t\). The corresponding phasor representation is: $$\tilde{E} = U + jV,$$ where \(\tilde{E}\) is the phasor of the driving voltage, and \(j\) denotes the imaginary unit.

Write the Impedance of the RLC Circuit

Now, we need to find the impedance of the RLC circuit. The impedance of a series RLC circuit is given by: $$Z = R + j(X_L-X_C) = R + j\biggl(\omega L - \frac{1}{\omega C}\biggr),$$ where \(R\), \(L\), and \(C\) are the resistance, inductance, and capacitance of the circuit, respectively.

Find the Phasor Current

Now, we will find the phasor of the current \(I_p\) in the circuit. Using Ohm's law, we have: $$\tilde{I} = \frac{\tilde{E}}{Z} = \frac{U + jV}{R + j\biggl(\omega L - \frac{1}{\omega C}\biggr)}.$$

Find the Amplitude of the Current

To find the amplitude of the current, we need to calculate the magnitude of the phasor current \(\tilde{I}\). We can use the formula for the magnitude of a complex number: $$I_p = |\tilde{I}| = \frac{\sqrt{(U^2 + V^2)}}{\sqrt{(R^2 + (\omega L - \frac{1}{\omega C})^2)}}.$$

Find the Value of \(\omega\) that Maximizes the Amplitude

To find the value of \(\omega\) that maximizes \(I_p\), we need to find the value of \(\omega\) that minimizes the denominator. We can use calculus for this purpose. Let's denote the denominator by \(g(\omega)\): $$g(\omega) = R^2 + (\omega L - \frac{1}{\omega C})^2.$$ To find the minimum value of \(g(\omega)\), we differentiate it with respect to \(\omega\) and set the derivative to zero: $$\frac{dg(\omega)}{d\omega} = 2(\omega L - \frac{1}{\omega C})\biggl(L + \frac{1}{\omega^2 C}\biggr) = 0.$$ Solving for \(\omega\), we get: $$\omega_0 = \frac{1}{\sqrt{LC}},$$ which is the value of \(\omega\) that maximizes the amplitude.

Find the Maximum Amplitude

Now, we can find the maximum amplitude by substituting this value of \(\omega_0\) into the above equation for \(I_p\): $$I_{p_{max}} = \frac{\sqrt{(U^2 + V^2)}}{\sqrt{(R^2 + (\omega_0 L - \frac{1}{\omega_0 C})^2)}} = \frac{\sqrt{(U^2 + V^2)}}{\sqrt{(R^2 + (\frac{L}{\sqrt{LC}} - \sqrt{\frac{L}{C}})^2)}} = \frac{\sqrt{(U^2 + V^2)}}{R}.$$ So, the amplitude of the steady-state current \(I_{p}\) in the RLC circuit is given by: $$I_p = \frac{\sqrt{(U^2 + V^2)}}{\sqrt{(R^2 + (\omega L - \frac{1}{\omega C})^2)}},$$ with \(\omega_0 = \frac{1}{\sqrt{LC}}\) maximizing the amplitude, and the maximum amplitude is given by: $$I_{p_{max}} = \frac{\sqrt{(U^2 + V^2)}}{R}.$$

Key concepts

Phasor Domain Representation

Understanding phasor domain representation is essential when analyzing RLC circuits. In the context of oscillating currents and voltages, a phasor is a complex number that represents the magnitude and phase of a sinusoidal function in a simplified form. Phasors convert differential equations into algebraic equations, thus making calculations involving circuits with alternating current (AC) much easier.

For a signal like \(E(t) = U\cos\omega t + V\sin\omega t\), the phasor representation removes the dependency on time and allows us to work with a static complex number \(\tilde{E} = U + jV\). This conversion to the frequency domain simplifies the process of circuit analysis, as it enables us to superimpose the effects of different circuit elements without having to solve differential equations.

RLC Circuit Impedance

In an RLC circuit, impedance takes the role of resistance for AC circuits. It's a complex number representing not only the opposition to the current by the circuit elements but also the phase difference between the voltage and the current. The impedance of a series RLC circuit is given by the formula \(Z = R + j(X_L - X_C)\), where \(R\) is the resistance, \(X_L = \omega L\) is the inductive reactance, and \(X_C = \frac{1}{\omega C}\) is the capacitive reactance.

The inductive reactance grows with frequency, while capacitive reactance decreases, indicating how inductors and capacitors react differently as the AC frequency changes. Impedance holds key information about the circuit's behavior at different frequencies and is crucial for determining the phasor current through the circuit.

Ohm's Law for Phasor Current

Ohm's Law for AC circuits extends the familiar formula \(V = IR\) to phasors, describing the relationship between phasor voltage, phasor current, and impedance as \(\tilde{V} = \tilde{I}Z\). This paves the way for analyzing how voltage and current relate in the frequency domain. For our RLC circuit with a driving voltage phasor \(\tilde{E}\), Ohm's Law in the phasor domain establishes that the phasor current \(\tilde{I}\) can be calculated as \(\tilde{I} = \frac{\tilde{E}}{Z}\).

In essence, the phasor form of Ohm's Law becomes a powerful tool in engineers' arsenals, effectively dealing with the complexities of AC circuit analysis by using the simple arithmetic of complex numbers. Understanding this law helps to predict current flow in RLC circuits under various conditions.

Maximizing Amplitude in RLC Circuit

Maximizing the amplitude of current in an RLC circuit involves adjusting the frequency of the input voltage. When the circuit reaches resonance, where the inductive and capacitive reactances equalize each other, the impedance is purely resistive, and the amplitude of current is at its peak.

The condition for resonance is found when \(\omega_0 L = \frac{1}{\omega_0 C}\), leading to \(\omega_0 = \frac{1}{\sqrt{LC}}\). At this resonant frequency, the circuit impedance becomes minimal, and thereby the current amplitude \(I_{p_{max}}\) reaches its maximum possible value, given by \(\frac{\sqrt{(U^2 + V^2)}}{R}\).

This topic is crucial for designing circuits that need to operate efficiently at certain frequencies, such as in radio transmitters and receivers, where the goal is often to maximize the signal amplitude without sacrificing the quality or introducing noise.