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Problem 1 Consider an RLC circuit driven by a programmable current source i(t) as shown in the diagram. The voltage appearing across the capacitor is v(t). The governing equations for the system are given below: i(t) i.(t) i(t) = i1(t) + iz(t) dv(t) dt i2(t) = CS v(t) = i₁(t) R + L K(s) = V(s) I(s) i₂(t) Assume zero initial conditions, i.e., i₁(0) = 0 and v(0) = 0. ▪ Show that the input-output transfer function of the system, where the current source i(t) is the input, and the voltage across the capacitor v(t) is the output, is given by = di₁(t) dt + v(t) SL + R s2LC+SCR+1

Problem 1 Consider an RLC circuit driven by a programmable current source i(t) as shown in the diagram. The voltage appearing across the capacitor is v(t). The governing equations for the system are given below: i(t) i.(t) i(t) = i1(t) + iz(t) dv(t) dt i2(t) = CS v(t) = i₁(t) R + L K(s) = V(s) I(s) i₂(t) Assume zero initial conditions, i.e., i₁(0) = 0 and v(0) = 0. ▪ Show that the input-output transfer function of the system, where the current source i(t) is the input, and the voltage across the capacitor v(t) is the output, is given by = di₁(t) dt + v(t) SL + R s2LC+SCR+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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**Problem 1** Consider an RLC circuit driven by a programmable current source \( i(t) \) as shown in the diagram. The voltage appearing across the capacitor is \( v(t) \). The governing equations for the system are given below: - **Diagram Overview:** - The circuit consists of a current source \( i(t) \) connected in series with a resistor (R), inductor (L), and capacitor (C). - The current splits into \( i_1(t) \) through the resistor and inductor, and \( i_2(t) \) through the capacitor. - The voltage across the capacitor is labeled \( v(t) \). \[ i(t) = i_1(t) + i_2(t) \] \[ i_2(t) = C \frac{dv(t)}{dt} \] \[ v(t) = i_1(t)R + L \frac{di_1(t)}{dt} \] Assume zero initial conditions, i.e., \( i_1(0) = 0 \) and \( v(0) = 0 \). - Show that the input-output transfer function of the system, where the current source \( i(t) \) is the input, and the voltage across the capacitor \( v(t) \) is the output, is given by: \[ K(s) = \frac{V(s)}{I(s)} = \frac{sL + R}{s^2LC + sCR + 1} \]
Transcribed Image Text:**Problem 1** Consider an RLC circuit driven by a programmable current source \( i(t) \) as shown in the diagram. The voltage appearing across the capacitor is \( v(t) \). The governing equations for the system are given below: - **Diagram Overview:** - The circuit consists of a current source \( i(t) \) connected in series with a resistor (R), inductor (L), and capacitor (C). - The current splits into \( i_1(t) \) through the resistor and inductor, and \( i_2(t) \) through the capacitor. - The voltage across the capacitor is labeled \( v(t) \). \[ i(t) = i_1(t) + i_2(t) \] \[ i_2(t) = C \frac{dv(t)}{dt} \] \[ v(t) = i_1(t)R + L \frac{di_1(t)}{dt} \] Assume zero initial conditions, i.e., \( i_1(0) = 0 \) and \( v(0) = 0 \). - Show that the input-output transfer function of the system, where the current source \( i(t) \) is the input, and the voltage across the capacitor \( v(t) \) is the output, is given by: \[ K(s) = \frac{V(s)}{I(s)} = \frac{sL + R}{s^2LC + sCR + 1} \]
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