Circuit Response Analysis

Abstract

The flow of current in the circuit branches and drop of voltage across circuit elements depend on their behavior and their ability to store energy. For instance, the voltage drop across a resistor is in phase with its current passing through. But that is not the same in a capacitor or an inductor. This makes the circuit KVL and KCL equations integrodifferential equations. The order of these equations depends on the number of energy-storing elements. In this chapter, the circuit elements are introduced and their equations are discussed. The order of a circuit is discussed, and responses of first- and second-order circuits to their initial condition and to external sources are analyzed.

Problems

• 4.1 Find the voltage across a 10 Ω resistor if the current flowing through is:

  • (a) i(t) = 10u(t)

  • (b) i(t) = 10tu(t)

  • (c) i(t) = 200 sin (60πt+10)

  • (d) i(t) = t ² u(t)

  • (e) i(t) = e ^−3t sin 10πt

• 4.2 Find the current of a 100 Ω resistor if the applied voltage is as follows:

  • (a) $v(t)=120\sqrt{2}\sin \left(100\pi t+10\right)$

  • (b) v(t) = u(t)+u(t − 2) − 2u(t − 3)

  • (c) v(t) = 10tu(t) − 20(t − 1)u(t − 1)+10(t − 3)u(t − 3)

  • (d) v(t) = e ^−3t sin 100πt

  • (e) v(t) = 100te ^−10t u(t)

• 4.3 Find the voltage induced across a L = 100 mH inductor when the current is:

  • (a) i(t) = 10u(t)

  • (b) i(t) = 10tu(t)

  • (c) i(t) = 200 sin (60πt+10)

  • (d) i(t) = t ² u(t)

  • (e) i(t) = e ^−3t sin 10πt

• 4.4 Find the current through a L = 100 mH inductor if the voltage applied across it is:

  • (a) $v(t)=120\sqrt{2}\sin \left(100\pi t+10\right)$

  • (b) v(t) = u(t)+u(t − 2) − 2u(t − 3)

  • (c) v(t) = 10tu(t) − 20(t − 1)u(t − 1)+10(t − 3)u(t − 3)

  • (d) v(t) = e ^−3t sin 100πt

  • (e) v(t) = 100te ^−10t u(t)

• 4.5 Find the voltage across a C = 100 μF capacitor when the current is:

  • (a) i(t) = 10u(t)

  • (b) i(t) = 10tu(t)

  • (c) i(t) = 200 sin (60πt+10)

  • (d) i(t) = t ² u(t)

  • (e) i(t) = e ^−3t sin 10πt

• 4.6 Find the current through a C = 100 μF capacitor if the voltage applied across it is:

  • (a) $v(t)=120\sqrt{2}\sin \left(100\pi t+10\right)$

  • (b) v(t) = u(t)+u(t − 2) − 2u(t − 3)

  • (c) v(t) = 10tu(t) − 20(t − 1)u(t − 1)+10(t − 3)u(t − 3)

  • (d) v(t) = e ^−3t sin 100πt

  • (e) v(t) = 100te ^−10t u(t)

• 4.7 The switch has been closed for long time. At time t = 0⁺, it is opened. Find v(t), i(t).

• 4.8 The switch has been closed for long time. At time t = 0⁺, it is opened. Find v(t), i(t).

• 4.9 The switch has been closed for long time. At time t = 0⁺, it is opened. Find v(t), i(t).

• 4.10 The switch has been closed for long time. At time t = 0⁺ it is opened. Find v(t), i(t).

• 4.11 The switch has been closed for long time. At time t = 0⁺, it is opened. Find v(t), i _c1(t), i _c2(t).

• 4.12 The switch has been closed for long time. At time t = 0⁺ it is opened. Find i(t), v ₁(t), v ₂(t).

• 4.13 Find the characteristics equations, characteristics roots, damping conditions, and the response in the following circuit.

• 4.14 Find the characteristics equations, characteristics roots, damping conditions, and the response in the following circuit.

• 4.15 Find the characteristics equations, characteristics roots, damping conditions, and the response in the following circuit.

• 4.16 Characteristics roots of a parallel RLC circuit are ${\lambda}_{1,2}=-1000\pm j5000\ \frac{\mathrm{rad}}{\mathrm{s}}$. Find the system’s natural voltage response, if the initial conditions are I ₀ =  − 25 A, V ₀ = 150 V.

• 4.17 Characteristics roots of a parallel RLC circuit are ${\lambda}_1=-1000\ \frac{\mathrm{rad}}{\mathrm{s}},{\lambda}_2=-5000\ \frac{\mathrm{rad}}{\mathrm{s}}$. Find the system voltage response if the circuit has initial conditions as ${V}_0=100\ \mathrm{V},\kern0.5em \frac{d}{dt}v\left({0}^{+}\right)=-12000\ \frac{\mathrm{V}}{\mathrm{s}}.$

• 4.18 Characteristics roots of a series RLC circuit are ${\lambda}_{1,2}=-1000\pm j15000\ \frac{\mathrm{rad}}{\mathrm{s}}.$ Find the system current response, if the initial conditions are I ₀ =  − 250 A, V ₀ =  − 200 V.

• 4.19 The response of a parallel RLC circuit recorded from the oscilloscope is as follows. Find the characteristics roots and characteristics equations.

• 4.20 Natural responses of some RLC circuits are as follows. Find the characteristics roots, characteristics equations, and initial conditions of the circuit.

  • (a) i(t) = 150e ^−200t cos 1000t

  • (b) v(t) = 100e ^−2000t sin (30000t+30)

  • (c) i(t) = 200e ^−300t+150e ^−120t

  • (d) v(t) = 200te ^−2000t+20e ^−2000t

• 4.21 Design an RLC series circuit such that the natural current response becomes:

$$ i(t)=100{e}^{-100t}\cos 750t\ \mathrm{A} $$

Select the values of R, L, and C and the initial conditions that result in the desired response.

Note that here might be multiple solutions for this design. Therefore, select the range to be no smaller than mH,μF in the inductor and capacitor.