Question

1. (a) Derive a differential equation to describe the input-output behaviour of the following circuit with input voltage $x(t)$ and output voltage $y(t)$: $i(t)$ $R = 2$ $L = 1$ $y(t)$ (b) Verify that the transfer function for the circuit in 1(a) in the Laplace domain is given by $H(s) = \frac{Y(s)}{X(s)} = \frac{s}{s + 2}$ and calculate the output $Y(s)$ for an input $X(s) \circ \bullet x(t) = e^{-ts}(t)$, i.e. a decaying exponential for positive t. (c) Derive the time domain signal $y(t) \circ \bullet Y(s)$ for the excitation in 1(b), state values for $y(0)$ and $y(t \to \infty)$, and sketch $y(t)$ approximately on the interval $t \in \{-1;5\}$.

          1. (a) Derive a differential equation to describe the input-output behaviour of the following
circuit with input voltage \(x(t)\) and output voltage \(y(t)\):

\(i(t)\)
\(R = 2\)
\(L = 1\)
\(y(t)\)

(b) Verify that the transfer function for the circuit in 1(a) in the Laplace domain is given
by

\(H(s) = \frac{Y(s)}{X(s)} = \frac{s}{s + 2}\)

and calculate the output \(Y(s)\) for an input \(X(s) \circ \bullet x(t) = e^{-ts}(t)\), i.e. a decaying
exponential for positive t.
(c) Derive the time domain signal \(y(t) \circ \bullet Y(s)\) for the excitation in 1(b), state values for
\(y(0)\) and \(y(t \to \infty)\), and sketch \(y(t)\) approximately on the interval \(t \in \{-1;5\}\).

$1. (a) Derive a differential equation to describe the input-output behaviour of the following circuit with input voltage x(t) and output voltage y(t): i(t) R = 2 L = 1 y(t) (b) Verify that the transfer function for the circuit in 1(a) in the Laplace domain is given by H(s) = (Y(s))/(X(s)) = (s)/(s + 2) and calculate the output Y(s) for an input X(s) ∘∙ x(t) = e^-ts(t), i.e. a decaying exponential for positive t. (c) Derive the time domain signal y(t) ∘∙ Y(s) for the excitation in 1(b), state values for y(0) and y(t →∞), and sketch y(t) approximately on the interval t ∈{-1;5}.$

Added by Steven M.

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