The matlab code below is used to find the Laplace transform of a RLC circuit, th

Question

The matlab code below is used to find the Laplace transform of a RLC circuit, the Let R1=8?,R2=4?,R3= 6?,C=1/4F,L=1.6H,I1(0)=15A, and Q(0) = 2 C. The matlab output is is seen in the plots of the figures below.

QUESTION How does the current and charge vary with time? How does the magnitude and phase spectrum vary with time?

The matlab code below is used to find the Laplace transform of a RLC circuit, the Let R1=8?,R2=4?,R3= 6?,C=1/4F,L=1.6H,I1(0)=15A, and Q(0) = 2 C. The matlab output is is seen in the plots of the figures below.

QUESTION How does the current and charge vary with time? How does the magnitude and phase spectrum vary with time?

symsLCI1 (t) Q(t)s R- sym ( ' R%d", [1 3]); assume ([t LCR] >0) 4 E(t) = 1+sin(t); % Voltage = 1 V dll - diff(11,t); diff(Q, t ) ; 7 8 9 dQ= egn 1 eqn2 (R(2)/L)*dQ ( 1 / (R (2 ) +R(3) ) * (E-Q/C) ) (R (2)_R ( 1 ) )/L*11; d11 dQ - + == + R ( 2 ) / (R (2) +R (3) ) *11; - -- 11 cond! = 11(0)-0; 12 cond2 13 1 eqnlLTlaplace (eqni, t,s) 15 eqn2LTlaplace (eqn2, t, s) 16 7 syms 11 LT Q-LT s eqnlLT subs (eqniLT, [laplace (I1,t,s) laplace (Q,t,s), [11LT Q-LT) 19 20 eqn2LT = subs (eqn2LT, [laplace (11,t, s) laplace (Q,t, s)], [11-LTQ-LT)); 21 Q(0) 0; - -- vars = [11-LTQ-LT]; 23 24 [I1LT, Q-LTsolve (eqns, vars) 25 26 Ilsol-ilaplace (I1.LT, s,t) 27 Osol1-ilaplace (O-LT,s,t) 28 I1solsimplify (Ilsol) 9 Qsolsimplify (Qsol) 30 31 vars [RLCI1 (0) Q (0) 32 values-[8461.6 1/4 15 2]; 33 I1sol-subs (I1sol,vars, values) 34 Qsolsubs (Qsol, vars, values); 35 36 subplot (2,2,1) a7 fplot (I1sol, to 10]) 38 title ('wmcc 9104-Current' 3 ylabel 'I1(t)') 40 xlabel 't') 2 subplot (2,2,2) 43 fplot (Qsol, [0 10]) 44 title ('wmcc 9104-Charge) 45 ylabel ('Q(t) 46 xlabel 't 47 8 subplot (2,2, 3) 49 fplot (Ilsol, [5 25]) so title ('wmcc 9104-Current' 51 ylabel 'I1 (t)') 52 xlabel't') 53 text (7,0.25, 'Transient) 54 text (16,0.125, 'Steady State' 56 subplot (2,2, 4) 57 fplot (Qsol, [5 25])

Explanation / Answer

As we can see from current Vs time plot, for t=0 sec current is maximum i.e. 15A, but as time progresses due to inductor and capacitor present current decreases rapidly, which is a transient response and after 15 seconds it start oscilalting, not steady state is achieved but it is damped oscillatory.

In charge Vs time plot we can observe that at t=0 sec charge is 2C but as time progresses it is deacreasing, but not sharp as current. After that around t=12 sec onwards it is stabilizing in the range, which is oscillatory in nature, as time increases it starts stabilizing. Both charge and current shows decaying nature as time progresses due to capacitor and inductor present in the circuit. Current is decaying faster than charge.

Magnitude spectrum of current and charge is decaying in nature but it is stabilizing as time increases. Decay rate of current magnitude is very fast, sharp changes is observed at lower time but it is in steady state for higher time. Similarly charge magnitude is decaying nature but rate of decaying is slower as compared to current. This is because current is the rate of change of charge w.r.t. time hence current is faster decaying.

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