Question

The resistance, inductance, and capacitance in a Parallel \(R L C\) circuit are \(5000 \Omega, 1.25 \mathrm{H},\) and \(8 \mathrm{nF}\) respectively. a) Calculate the roots of the characteristic equation that describe the voltage response of the circuit. b) Will the response be over, under, or critically damped? c) What value of \(R\) will yield a damped frequency of \(6 \mathrm{krad} / \mathrm{s} ?\) What are the roots of the characteristic equation for the value of \(R\) found in \((\circ) ?\) e) What value of \(R\) will result in a critically damped response?

Short answer

a) Roots: \(-5000, -20000\). b) Overdamped. c) \(R = 1250\Omega\). d) Complex roots. e) \(R = 2500\Omega\).

Step-by-step solution

Write the Characteristic Equation

For a parallel \(RLC\) circuit, the characteristic equation is given by \(s^2 + \frac{1}{RC}s + \frac{1}{LC} = 0\). Given values:\(R = 5000 \Omega\), \(L = 1.25 \text{H}\), \(C = 8 \text{nF} = 8 \times 10^{-9} \text{F}\). Substitute these into the equation to get: \[s^2 + \frac{1}{5000 \times 8 \times 10^{-9}}s + \frac{1}{1.25 \times 8 \times 10^{-9}} = 0\].

Simplify the Characteristic Equation

Simplify the coefficients in the characteristic equation. For the term \(\frac{1}{RC}\), calculate \(RC = 5000 \times 8 \times 10^{-9} = 4 \times 10^{-5}\). Thus, \(\frac{1}{RC} = 25000\). For \(\frac{1}{LC}\), calculate \(LC = 1.25 \times 8 \times 10^{-9} = 10 \times 10^{-9}\), so \(\frac{1}{LC} = 10^8\). Thus, the equation becomes \[s^2 + 25000s + 10^8 = 0\].

Solve for Roots of the Quadratic Equation

Use the quadratic formula \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 25000\), and \(c = 10^8\) to find the roots. Calculate \(b^2 - 4ac = 25000^2 - 4\cdot10^8 = 6.25\times10^8 - 4\times10^8 = 2.25\times10^8\). So, the roots are \[s = \frac{-25000 \pm \sqrt{2.25\times10^8}}{2}\].

Evaluate the Roots

Calculate \(\sqrt{2.25\times10^8} = 15000\). Hence, the roots are \[s = \frac{-25000 \pm 15000}{2}\], giving solutions \(-5000\) and \(-20000\).

Determine the Type of Damping

Since the roots are real and distinct, the system is **overdamped**.

Find R for a Damped Frequency

The damped frequency \(\omega_d\) is given as \(6000 \text{rad/s}\). The damped frequency relates to the system's roots: \(\omega_d = \sqrt{\omega_0^2 - \left(\frac{1}{2RC}\right)^2}\), where \(\omega_0 = \sqrt{\frac{1}{LC}} = 10^4 \text{rad/s}\). Set \(6000 = \sqrt{\left(10^4\right)^2 - \left(\frac{1}{2R \times 8 \times 10^{-9}}\right)^2}\) and solve for \(R\).

Solve for R in Damped Frequency

Squaring both sides and simplifying: \[\left(6000\right)^2 = \left(10^4\right)^2 - \left(\frac{1}{2R \times 8 \times 10^{-9}}\right)^2\] \(\Rightarrow 3.6\times10^7 = 10^8 - \frac{1}{\left(16\times10^{-9}R\right)^2}\). Solve for \(R\): \[\frac{1}{R^2} = \frac{10^8 - 3.6\times10^7}{16\times10^{-9}} = 6.4\times10^{6}\]. Thus, \(R = 1250\Omega\).

Roots of the New Characteristic Equation

Substitute \(R = 1250\) into the characteristic equation. The new equation is \[s^2 + \frac{1}{1250 \times 8 \times 10^{-9}}s + 10^8 = 0\]. Solve this using the quadratic formula.

Calculate Roots for New R

With \(R = 1250\), \(RC = 1\times10^{-5}\) so \(\frac{1}{1250 \times 8 \times 10^{-9}} = 10^5\). The roots are \[s = \frac{-10^5 \pm \sqrt{b^2 - 4ac}}{2}\] with \(b^2 - 4ac = 10^{10} - 4\times10^8 = 9.6\times10^9\), giving complex roots.

Find R for Critical Damping

For critical damping, \(b^2 = 4ac\). Compute \(\left(\frac{1}{RC}\right)^2 = 4\times10^8\), so \(\frac{1}{RC} = 20000\), \(RC = 5 \times 10^{-5}\). Solving \(5000C = 5 \times 10^{-5}\) gives \(R = 2500\) for critical damping.

Key concepts

Characteristic Equation

In a parallel RLC circuit, the key to understanding the behavior of the circuit lies in the characteristic equation. This equation represents the relationship between resistance \(R\), inductance \(L\), and capacitance \(C\) in terms of the variable \(s\), which corresponds to the complex frequency domain.

The general form of the characteristic equation for a parallel RLC circuit is given by:

• \(s^2 + \frac{1}{RC}s + \frac{1}{LC} = 0\)

For a given circuit, once the values for \(R\), \(L\), and \(C\) are known, these can be plugged into the characteristic equation to solve for the values of \(s\), which are crucial in analyzing the circuit's voltage response. Understanding these roots is vital as they determine the damping behavior of the system, explaining how quickly and in what manner the circuit will respond to changes.

Damping in RLC Circuits

The concept of damping in RLC circuits defines how oscillations decay over time. In a parallel RLC circuit, damping is a result of the resistance that dissipates energy as the circuit releases stored energy in the form of oscillations.

Damping can be categorized into three types based on the roots of the characteristic equation:

• Overdamping: Occurs when the roots are real and distinct, as in the case where resistance is high relative to other factors, leading to a slow return to equilibrium without oscillating.
• Critical damping: This is the threshold between overdamping and underdamping, where the system returns to equilibrium as quickly as possible without oscillating. This happens when the roots are real and repeated.
• Underdamping: When the roots are complex, the system oscillates before settling down due to lower resistance, allowing some energy to convert back and forth between the inductor and capacitor.

Understanding these damping types helps predict how changes in resistance affect circuit behavior.

Quadratic Formula in Circuits

Solving the characteristic equation often involves using the quadratic formula to find its roots. The formula is a mathematical tool to solve equations of the form \(ax^2 + bx + c = 0\), and it's particularly useful in circuit analysis.

For a quadratic equation \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we identify:

• \(a = 1\)
• \(b = \frac{1}{RC}\)
• \(c = \frac{1}{LC}\)

By substituting these into the formula, we determine the values of \(s\), which are critical in analyzing circuit responses. The discriminant \(b^2 - 4ac\) tells us about the nature of the roots:

• If positive, the roots are real and distinct indicating overdamping.
• If zero, the roots are real and repeated, indicating critical damping.
• If negative, the roots are complex, indicating underdamping.

Critical and Overdamping

Critical and overdamping are particular types of damping responses in an RLC circuit. They each describe how quickly or sluggishly a circuit can respond to changes in an electrical signal.

- **Overdamping** occurs when the resistance is high enough to prevent oscillations. The roots of the characteristic equation are real and separate.
- **Critical damping** is a delicate balance where the system returns to equilibrium without oscillating, but as quickly as possible. Here, the roots are real and repeated.

Each of these states can be achieved by adjusting the circuit parameters, mainly the resistance, to match certain conditions. Engineers often aim for critical damping when a quick yet stable response is necessary, such as in certain automatic control systems.

Damped Frequency in Circuits

The damped frequency in circuits describes the frequency at which a damped oscillator like an RLC circuit oscillates as it returns to steady-state. This is typically less than the natural frequency of the system due to energy loss through damping.

The formula for the damped frequency \(\omega_d\) is:

• \(\omega_d = \sqrt{\omega_0^2 - \left(\frac{1}{2RC}\right)^2}\)

Where \(\omega_0 = \sqrt{\frac{1}{LC}}\) is the natural angular frequency. As the damping effect increases (i.e., higher resistance), the damped frequency decreases. Thus, controlling the damped frequency is important in designing circuits for specific applications, ensuring they perform optimally without excessive oscillation.