For RL series circuit, R = 4? and L = 0.5 H. The source voltage V = 12 Vdc is ap

Question

For RL series circuit, R = 4? and L = 0.5 H. The source voltage V = 12 Vdc is applied. Find the current i(t) using the Laplace transform. Assume all initial conditions are zeros. a) First, write the voltage equation in terms of a differential equation. b) Second, apply the Laplace transform on the equation. For example, if you take the Laplace transform on V = 12 V, then the Laplace transform of 12 = 12/s, the Laplace transform of i(t) =I(s) etc. c) Rearrange the transformed equation for the current I(s). d) Take the inverse Laplace transform of I(s) to find the current i(t). e) The answer should be i(t) = 3(1-e^(-8t)). f) From the answer (e), what is the transient and the final (stable or steady-state) value? Describe i(t) from (e) more specifically. g) What is the time constant of the circuit? Calculate it.

Explanation / Answer

A resistor-inductor circuit (RL circuit), or RL filter or RL network, is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage source. Introduction The fundamental passive linear circuit elements are the resistor (R), capacitor (C) and inductor (L). These circuit elements can be combined to form an electrical circuit in four distinct ways: the RC circuit, the RL circuit, the LC circuit and the RLC circuit with the abbreviations indicating which components are used. These circuits exhibit important types of behaviour that are fundamental to analogue electronics. In particular, they are able to act as passive filters. This article considers the RL circuit in both series and parallel as shown in the diagrams. In practice, however, capacitors (and RC circuits) are usually preferred to inductors since they can be more easily manufactured and are generally physically smaller, particularly for higher values of components. This article relies on knowledge of the complex impedance representation of inductors and on knowledge of the frequency domain representation of signals. [edit]Complex impedance The complex impedance ZL (in ohms) of an inductor with inductance L (in henries) is The complex frequency s is a complex number, where j represents the imaginary unit: is the exponential decay constant (in radians per second), and is the angular frequency (in radians per second). [edit]Eigenfunctions The complex-valued eigenfunctions of any linear time-invariant (LTI) system are of the following forms: From Euler's formula, the real-part of these eigenfunctions are exponentially-decaying sinusoids: [edit]Sinusoidal steady state Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay). As a result, and the evaluation of s becomes [edit]Series circuit Series RL circuit By viewing the circuit as a voltage divider, we see that the voltage across the inductor is: and the voltage across the resistor is: [edit]Current The current in the circuit is the same everywhere since the circuit is in series: [edit]Transfer functions The transfer function for the inductor is Similarly, the transfer function for the resistor is [edit]Poles and zeros Both transfer functions have a single pole located at In addition, the transfer function for the inductor has a zero located at the origin. [edit]Gain and phase angle The gains across the two components are found by taking the magnitudes of the above expressions: and and the phase angles are: and [edit]Phasor notation These expressions together may be substituted into the usual expression for the phasor representing the output: [edit]Impulse response The impulse response for each voltage is the inverse Laplace transform of the corresponding transfer function. It represents the response of the circuit to an input voltage consisting of an impulse or Dirac delta function. The impulse response for the inductor voltage is where u(t) is the Heaviside step function and is the time constant. Similarly, the impulse response for the resistor voltage is

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