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If / = 8 u(-t) mA, obtain an expression for h as labeled in the circuit given which is valid for all t> O. 5Ω 500 mH 80 Ω 20i I mF vc The expression for ij valid for all t> O is Bt (Click to select) ♥ tµA, where B = s-1

If / = 8 u(-t) mA, obtain an expression for h as labeled in the circuit given which is valid for all t> O. 5Ω 500 mH 80 Ω 20i I mF vc The expression for ij valid for all t> O is Bt (Click to select) ♥ tµA, where B = s-1

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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**Circuit Analysis and Expression Derivation** **Problem Statement:** Given a current source \( I = 8 \, u(-t) \, \text{mA} \), find an expression for \( i_1 \) as labeled in the circuit diagram, which is applicable for all \( t > 0 \). **Circuit Description:** - The circuit includes: - A current source \( I \) supplying \( 8 \, u(-t) \, \text{mA} \). - An inductor with inductance of \( 500 \, \text{mH} \) carrying current \( i_L \). - A capacitor with capacitance of \( 1 \, \text{mF} \) with voltage \( v_C \). - A resistor of \( 80 \, \Omega \). - A resistor of \( 5 \, \Omega \). - A dependent voltage source \( 20i_1 \). **Objective:** To find the expression for \( i_1 \) that is valid for all \( t > 0 \). **Required Expression:** The expression for \( i_1 \) valid for all \( t > 0 \) is given by: \[ i_1 = \text{________} e^{Bt} (\text{Click to select}) \, \text{________} \, t \, \mu\text{A}, \quad \text{where } B = \text{________} \, \text{s}^{-1}. \] Fill in the blanks by solving the differential equations and applying boundary conditions relevant to the given circuit elements. **Graph/Diagram Explanation:** - The circuit shows interconnected elements comprising a source, reactive components (inductor & capacitor), resistors, and a dependent source. - Analysis typically involves applying Kirchhoff's laws and considering the impedance of the components. **Applications:** Understanding the response of such circuits is crucial in fields such as electrical engineering and physics, especially in the study of transient responses in RLC circuits.
Transcribed Image Text:**Circuit Analysis and Expression Derivation** **Problem Statement:** Given a current source \( I = 8 \, u(-t) \, \text{mA} \), find an expression for \( i_1 \) as labeled in the circuit diagram, which is applicable for all \( t > 0 \). **Circuit Description:** - The circuit includes: - A current source \( I \) supplying \( 8 \, u(-t) \, \text{mA} \). - An inductor with inductance of \( 500 \, \text{mH} \) carrying current \( i_L \). - A capacitor with capacitance of \( 1 \, \text{mF} \) with voltage \( v_C \). - A resistor of \( 80 \, \Omega \). - A resistor of \( 5 \, \Omega \). - A dependent voltage source \( 20i_1 \). **Objective:** To find the expression for \( i_1 \) that is valid for all \( t > 0 \). **Required Expression:** The expression for \( i_1 \) valid for all \( t > 0 \) is given by: \[ i_1 = \text{________} e^{Bt} (\text{Click to select}) \, \text{________} \, t \, \mu\text{A}, \quad \text{where } B = \text{________} \, \text{s}^{-1}. \] Fill in the blanks by solving the differential equations and applying boundary conditions relevant to the given circuit elements. **Graph/Diagram Explanation:** - The circuit shows interconnected elements comprising a source, reactive components (inductor & capacitor), resistors, and a dependent source. - Analysis typically involves applying Kirchhoff's laws and considering the impedance of the components. **Applications:** Understanding the response of such circuits is crucial in fields such as electrical engineering and physics, especially in the study of transient responses in RLC circuits.
The expression for \(i\) valid for all \(t > 0\) is: \[ -120 \, e^{B t} \sin \left(44.7213 \, t\right) \, \mu A, \text{ where } B = -0.077 \, s^{-1}. \] ### Explanation For \(t > 0\), the circuit is a series RLC connection, where \(R_{eq}\) consists of the two resistors and the dependent source. Find \(R_{eq}\) for \(t > 0\) by applying 1 V test source. \[ I_T = i_1 + \frac{1 V - (90\ \Omega)i_1}{5\Omega} = 162.5 \, mA \] \[ R_{eq} = V_T / I_T = 6.1538 \, \Omega \] \[ \alpha = \frac{R}{2L} = 6.1538 \, s^{-1} \] \[ \omega_0 = \frac{1}{\sqrt{LC}} = 44.72 \, rad/s \] \[ \omega_0 > \alpha \Rightarrow \text{underdamped} \] Find \(i\) through the series circuit first, and then find \(i_f\). Define \(i = i_L\): \[ i(t) = e^{-\alpha t} (B_1 \cos \omega_d t + B_2 \sin \omega_d t) \] \[ i_1(t) = \frac{i(t)R_{eq} - v_C(t)}{R_{eq}} + \frac{v_C(t)}{R_{eq}} \] \[ \omega_d^2 = \omega_0^2 - \alpha^2 = 44.3 \, rad/s \] \[ i(0) = 0 = A = B_1 \] ### Analyzing circuit at \(t < 0\), \[ v_C(0) = 36.9228 \, mV \] \[ \frac{di}{dt} \bigg|_{t=0} = \frac{\omega_0^2 v_C(0) - R_{eq}(0)}{L}
Transcribed Image Text:The expression for \(i\) valid for all \(t > 0\) is: \[ -120 \, e^{B t} \sin \left(44.7213 \, t\right) \, \mu A, \text{ where } B = -0.077 \, s^{-1}. \] ### Explanation For \(t > 0\), the circuit is a series RLC connection, where \(R_{eq}\) consists of the two resistors and the dependent source. Find \(R_{eq}\) for \(t > 0\) by applying 1 V test source. \[ I_T = i_1 + \frac{1 V - (90\ \Omega)i_1}{5\Omega} = 162.5 \, mA \] \[ R_{eq} = V_T / I_T = 6.1538 \, \Omega \] \[ \alpha = \frac{R}{2L} = 6.1538 \, s^{-1} \] \[ \omega_0 = \frac{1}{\sqrt{LC}} = 44.72 \, rad/s \] \[ \omega_0 > \alpha \Rightarrow \text{underdamped} \] Find \(i\) through the series circuit first, and then find \(i_f\). Define \(i = i_L\): \[ i(t) = e^{-\alpha t} (B_1 \cos \omega_d t + B_2 \sin \omega_d t) \] \[ i_1(t) = \frac{i(t)R_{eq} - v_C(t)}{R_{eq}} + \frac{v_C(t)}{R_{eq}} \] \[ \omega_d^2 = \omega_0^2 - \alpha^2 = 44.3 \, rad/s \] \[ i(0) = 0 = A = B_1 \] ### Analyzing circuit at \(t < 0\), \[ v_C(0) = 36.9228 \, mV \] \[ \frac{di}{dt} \bigg|_{t=0} = \frac{\omega_0^2 v_C(0) - R_{eq}(0)}{L}
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