Question
The series $R L C$ circuit shown in Figure P3.52 has the following parameters: $C=0.04 \mathrm{~F}$, $L=1 \mathrm{H}, R=6 \Omega, i_L(0)=4 A$, and $v_c(0)=-4 \mathrm{~V}$. Find the equation for the current $i(t)$.Figure P3.52 can't copy
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The given circuit is a series RLC circuit. The governing differential equation for the current \( i(t) \) in a series RLC circuit is given by: \[ L \frac{d^2i(t)}{dt^2} + R \frac{di(t)}{dt} + \frac{1}{C} i(t) = 0 \] Substituting the values \( L = 1 \, \text{H} Show more…
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The electrical circuit in the accompanying figure is called a parallel LRC circuit; it contains a resistor with resistance $R$ ohms $(\Omega),$ an inductor with inductance $L$ henries (H), and a capacitor with capacitance $C$ farads (F). It is shown in electrical circuit analysis that at time $t$ the current $i_{L}$ through the inductor and the voltage $v_{C}$ across the capacitor are solutions of the system$$\left[\begin{array}{l} i_{L}^{\prime}(t) \\ v_{C}^{\prime}(t) \end{array}\right]=\left[\begin{array}{cc} 0 & 1 / L \\ -1 / C & -1 /(R C) \end{array}\right]\left[\begin{array}{l} i_{L}(t) \\ v_{C}(t) \end{array}\right]$$ (a) Find the general solution of this system in the case where $R=1 \mathrm{ohm}, L=1$ henry, and $C=0.5$ farad. (b) Find $i_{L}(t)$ and $v_{C}(t)$ subject to the initial conditions $i_{L}(0)=2$ amperes and $v_{C}(0)=1$ volt. (c) What can you say about the current and voltage in part (b) over the "long term" (that is, as $t \rightarrow \infty$ )? (FIGURE CAN'T COPY)
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Consider a series circuit (Figure 4$)$ consisting of a resistor of $R$ ohms, an inductor of $L$ henries, and a variable voltage source of $V(t)$ volts (time t in seconds). The current through the circuit $I(t)$ (in amperes) satisfes the differential equation $$ \frac{d I}{d t}+\frac{R}{L} I=\frac{1}{L} V(t) $$ Find the solution to Eq. ( 10 ) with initial condition $I(0)=0,$ assuming that $R=100 \Omega, L=5 \mathrm{H},$ and $V(t)$ is constant with $V(t)=10 \mathrm{V}$ .
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34. RLC Series Circuit. In the study of an electrical circuit consisting of a resistor, capacitor, inductor, and an electromotive force (see Figure 4.9), we are led to an initial value problem of the form (20)$$\begin{array}{l}{L \frac{d I}{d t}+R I+\frac{q}{C}=E(t)} \\ {q(0)=q_{0}} \\{I(0)=I_{0}}\end{array}$$ where L is the inductance in henrys, R is the resistance in ohms, C is the capacitance in farads,$$ E(t)$$ is the electromotive force in volts,$$q(t)$$ is the charge in coulombs on the capacitor at time $$t, \text { and } I=d q / d t$$ is the current in amperes. Find the current at time t if the charge on the capacitor is initially zero, the initial current is zero,$$L=10 \mathrm{H}, R=20 \Omega, C=(6260)^{-1} \mathrm{F}$$ and $$E(t)=100 \mathrm{V}$$ [Hint: Differentiate both sides of the differential equation in (20) to obtain a homogeneous linear second-order equation for $$I(t)$$ Then use (20) to determine $$d I / d t \text { at } t=0.1$$
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