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A current source outputs a current that depends on time according to: I(t) = A cos @ịt + B sin @2t, where A = 0.6 A, B= 0.2 A, @1 = 3 rad/s, and @2 = 20 rad/s. Calculate the magnitude of the voltage across the inductor, which has inductance 5.2 H, at t = 12 s, in V. %3D

A current source outputs a current that depends on time according to: I(t) = A cos @ịt + B sin @2t, where A = 0.6 A, B= 0.2 A, @1 = 3 rad/s, and @2 = 20 rad/s. Calculate the magnitude of the voltage across the inductor, which has inductance 5.2 H, at t = 12 s, in V. %3D

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(Please answer to the fourth decimal place - i.e 14.3225)

**Current Source and Inductor Voltage Calculation** A current source outputs a current that depends on time according to: \[ I(t) = A \cos(\omega_1 t) + B \sin(\omega_2 t) \] where: - \( A = 0.6 \, \text{A} \) - \( B = 0.2 \, \text{A} \) - \( \omega_1 = 3 \, \text{rad/s} \) - \( \omega_2 = 20 \, \text{rad/s} \) Calculate the **magnitude** of the voltage across the inductor, which has an inductance of \( 5.2 \, \text{H} \), at \( t = 12 \, \text{s} \), in volts (V). ### Circuit Diagram The circuit consists of: - A current source denoted as \( I_1 \) at the left side - An inductor denoted as \( L_1 \) at the right side, connected in series with the current source The arrangement forms a simple series loop as shown in the diagram. ### Diagram Explanation The circuit diagram depicts a simple series circuit consisting of the following components: - **Current Source (\( I_1 \))**: Represented by a circle with an upward pointing arrow inside it, indicating the direction of current flow. - **Inductor (\( L_1 \))**: Represented by a coil, which is a common symbol for inductors in circuit diagrams. This setup requires understanding the relationship between the time-varying current provided by the source and the voltage induced across the inductor. The voltage across an inductor \( L \) is given by \( V_L(t) = L \frac{dI(t)}{dt} \). --- To solve for the voltage across the inductor, follow these steps: 1. Differentiate the current function \( I(t) \) with respect to time \( t \). 2. Multiply the result by the inductance \( L \). Given the complexity of the calculations, these steps can be detailed out in further educational content.
Transcribed Image Text:**Current Source and Inductor Voltage Calculation** A current source outputs a current that depends on time according to: \[ I(t) = A \cos(\omega_1 t) + B \sin(\omega_2 t) \] where: - \( A = 0.6 \, \text{A} \) - \( B = 0.2 \, \text{A} \) - \( \omega_1 = 3 \, \text{rad/s} \) - \( \omega_2 = 20 \, \text{rad/s} \) Calculate the **magnitude** of the voltage across the inductor, which has an inductance of \( 5.2 \, \text{H} \), at \( t = 12 \, \text{s} \), in volts (V). ### Circuit Diagram The circuit consists of: - A current source denoted as \( I_1 \) at the left side - An inductor denoted as \( L_1 \) at the right side, connected in series with the current source The arrangement forms a simple series loop as shown in the diagram. ### Diagram Explanation The circuit diagram depicts a simple series circuit consisting of the following components: - **Current Source (\( I_1 \))**: Represented by a circle with an upward pointing arrow inside it, indicating the direction of current flow. - **Inductor (\( L_1 \))**: Represented by a coil, which is a common symbol for inductors in circuit diagrams. This setup requires understanding the relationship between the time-varying current provided by the source and the voltage induced across the inductor. The voltage across an inductor \( L \) is given by \( V_L(t) = L \frac{dI(t)}{dt} \). --- To solve for the voltage across the inductor, follow these steps: 1. Differentiate the current function \( I(t) \) with respect to time \( t \). 2. Multiply the result by the inductance \( L \). Given the complexity of the calculations, these steps can be detailed out in further educational content.
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