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; Q0=0 coulombs; I0=2 amperes.
2. R=2 ohms; L=.05 henrys; C=.01 farads’; Q0=2 coulombs; I0=−2 amperes.
3. R=2 ohms; L=.1 henrys; C=.01 farads; Q0=2 coulombs; I0=0 amperes.
4. R=6 ohms; L=.1 henrys; C=.004 farads’; Q0=3 coulombs; I0=−10 amperes.
5. R=4 ohms; L=.05 henrys; C=.008 farads; Q0=−1 coulombs; I0=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. 110Q″+3Q′+100Q=5cos10t−5sin10t
7. 120Q″+2Q′+100Q=10cos25t−5sin25t
8. 110Q″+2Q′+100Q=3cos50t−6sin50t
9. 110Q″+6Q′+250Q=10cos100t+30sin100t
10. 120Q″+4Q′+125Q=15cos30t−30sin30t
Q6.3.3
11. Show that if E(t)=Ucosωt+Vsinωt where U and V are constants then the steady state current in the RLC circuit shown in Figure 6.3.1 is Ip=ω2RE(t)+(1/C−Lω2)E′(t)Δ, where Δ=(1/C−Lω2)2+R2ω2.
12. Find the amplitude of the steady state current Ip in the RLC circuit shown in Figure 6.3.1 if E(t)=Ucosωt+Vsinωt, where U and V are constants. Then find the value ω0 of ω 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. 110Q″+3Q′+100Q=Ucosωt+Vsinωt
14. 120Q″+2Q′+100Q=Ucosωt+Vsinωt
15. 110Q″+2Q′+100Q=Ucosωt+Vsinωt
16. 110Q″+6Q′+250Q=Ucosωt+Vsinωt
17. 120Q″+4Q′+125Q=Ucosωt+Vsinωt