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An electrical circuit has in series and electromotive force given by E(t) = 100 sin(40t) volts (V), a resistor of 10 ohms (Ω) and an inductor of 0.5 henrys (H). The current i(t) in the circuit is initially 0 amps (A)?
(a) Set up the initial value problem in terms of di/dt, i(t) and the initial current i(0). (b) Solve the initial value problem in (a) to find the current i(t) at time t > 0. (c) Find the steady-state current, that is, the current in the circuit after a long period of time.
To analyze the given electrical circuit, we'll use Kirchhoff's voltage law (KVL) and the relationship between voltage, current, resistance, and inductance.
Kirchhoff's voltage law states that the sum of the voltage drops across the components in a closed loop is equal to the electromotive force (EMF) in that loop.
Let's break down the given circuit:
Electromotive Force (EMF): The EMF is given by E(t) = 100 sin(40t) V, where t represents time in seconds.
Resistor: The resistor has a resistance of 10 Ω.
Inductor: The inductor has an inductance of 0.5 H.
Current: The current flowing through the circuit is denoted as i(t) and is initially 0 A.
To find the current i(t) in the circuit, we'll apply KVL. The sum of the voltage drops across the resistor and the inductor should be equal to the EMF.
Voltage drop across the resistor: V_R = i(t) * R = 10i(t) Ω
Voltage drop across the inductor: V_L = L * di(t)/dt = 0.5(di(t)/dt) H
Applying KVL: E(t) = V_R + V_L 100 sin(40t) = 10i(t) + 0.5(di(t)/dt)
To solve this second-order linear differential equation, we need to differentiate the equation with respect to time (t):
d/dt (100 sin(40t)) = d/dt (10i(t) + 0.5(di(t)/dt)) 4000 cos(40t) = 10(di(t)/dt) + 0.5(d^2i(t)/dt^2)
Now we have a second-order differential equation in terms of i(t). Rearranging the terms:
0.5(d^2i(t)/dt^2) + 10(di(t)/dt) - 4000 cos(40t) = 0
To solve this differential equation, we need to find the particular solution for i(t) that satisfies the initial condition i(0) = 0. The general solution will involve complementary and particular solutions.
Unfortunately, the given differential equation is nonlinear, and there is no simple analytical solution. To obtain the complete details of the current waveform, we'll need to solve this differential equation numerically using techniques like numerical integration or simulation software such as SPICE
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