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Mathematics LibreTexts

6.3E: The RLC Circuit (Exercises)

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    Q6.3.1

    In Exercises 6.3.1-6.3.5 find the current in the \(RLC\) circuit, assuming that \(E(t)=0\) for \(t>0\).

    1. \(R=3\) ohms;   \(L=.1\) henrys;   \(C=.01\) farads; \(Q_0=0\) coulombs;  \(I_0=2\) amperes.

    2. \(R=2\) ohms;   \(L=.05\) henrys;   \(C=.01\) farads’; \(Q_0=2\) coulombs;  \(I_0=-2\) amperes.

    3. \(R=2\) ohms;   \(L=.1\) henrys;   \(C=.01\) farads; \(Q_0=2\) coulombs;  \(I_0=0\) amperes.

    4. \(R=6\) ohms;   \(L=.1\) henrys;   \(C=.004\) farads’; \(Q_0=3\) coulombs;  \(I_0=-10\) amperes.

    5. \(R=4\) ohms;   \(L=.05\) henrys;   \(C=.008\) farads; \(Q_0=-1\) coulombs;  \(I_0=2\) amperes.

    Q6.3.2

    In Exercises 6.3.6-6.3.10 find the steady state current in the circuit described by the equation.

    6. \({1\over10}Q''+3Q'+100Q=5\cos10t-5\sin10t\)

    7. \({1\over20}Q''+2Q'+100Q=10\cos25t-5\sin25t\)

    8. \({1\over10}Q''+2Q'+100Q=3\cos50t-6\sin50t\)

    9. \({1\over10}Q''+6Q'+250Q=10\cos100t+30\sin100t\)

    10. \({1\over20}Q''+4Q'+125Q=15\cos30t-30\sin30t\)

    Q6.3.3

    11. Show that if \(E(t)=U\cos\omega t+V\sin\omega t\) where \(U\) and \(V\) are constants then the steady state current in the \(RLC\) circuit shown in Figure 6.3.1 is \[I_p={\omega^2RE(t)+(1/C-L\omega^2)E'(t)\over\Delta},\] where \[\Delta=(1/C-L\omega^2)^2+R^2\omega^2.\]

    12. Find the amplitude of the steady state current \(I_p\) in the \(RLC\) circuit shown in Figure 6.3.1 if \(E(t)=U\cos\omega t+V\sin\omega t\), where \(U\) and \(V\) are constants. Then find the value \(\omega_0\) of \(\omega\) maximizes the amplitude, and find the maximum amplitude.

    Q6.3.4

    In Exercises 6.3.13-6.3.17 plot the amplitude of the steady state current against \(ω\). Estimate the value of \(ω\) that maximizes the amplitude of the steady state current, and estimate this maximum amplitude. HINT: You can confirm your results by doing Exercise 6.3.12.

    13. \({1\over10}Q''+3Q'+100Q=U\cos\omega t+V\sin\omega t\)

    14. \({1\over20}Q''+2Q'+100Q=U\cos\omega t+V\sin\omega t\)

    15. \({1\over10}Q''+2Q'+100Q=U\cos\omega t+V\sin\omega t\)

    16. \({1\over10}Q''+6Q'+250Q=U\cos\omega t+V\sin\omega t\)

    17. \({1\over20}Q''+4Q'+125Q=U\cos\omega t+V\sin\omega t\)