Circuit Theory/TF Examples/Example33

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Series RLC circuit with two initial conditions Example 33 for wikibook circuit theory

Find io(t) if Vs(t) = 1 + cos(3t).

Choose Starting Point[edit]

Because of the initial conditions, going to start with Vc(t) and then work our way through the initial conditions to io.

Transfer Function[edit]

H(s) = \frac{V_c}{V_s} = \frac{\frac{1}{sC}}{\frac{1}{sC} + \frac{1}{\frac{1}{R_1} + \frac{1}{sL}} + R_2}

The MuPad commands are going to be:

L :=1; R1:=.5; R2:=1.5; C:=.5;
simplify((1/(s*C))/(1/(s*C) + 1/(1/R1 + 1/(s*L)) + R2))

Which results in:

 H(s) = \frac{8s + 4}{8s^2 + 11s + 4}

Homogeneous Solution[edit]

Set the denominator of the transfer function to 0 and solve for s:

solve(8*s^2 + 11*s + 4)

Imaginary roots:

s_{1,2} = \frac{-11 \pm \sqrt{7}i}{16}

So the solution has the form:

V_{c_h} = e^{-\frac{11t}{16}}(A\cos \frac{7t}{16} + B\sin \frac{7t}{16}) + C

Particular Solution[edit]

After a very long time the capacitor opens, no current flows, so all the source drop is across the capacitor. The source is a unit step function thus:

V_{c_p} = 1

Initial Conditions[edit]

mupad screen shot leading up to computing constant B

Adding the particular and homogenous solutions, get:

V_c(t) = 1 + e^{-\frac{11t}{16}}(A\cos \frac{7t}{16} + B\sin \frac{7t}{16}) + C

Doing the final condition again, get:

V_c(\infty) = 1 = 1 + C \Rightarrow C = 0

Which implies that C is zero.

From the given initial conditions, know that Vc(0+) = 0.5 so can find A:

V_c(0_+) = 0.5 = 1 + A \Rightarrow A = -0.5

Finding B is more difficult. From capacitor terminal relation:

VC := 1 + exp(-11*t/16)*(-.5*cos(7*t/16) + B*sin(7*t/16))
IT := diff(VC,t)

The total current is:

i_T(t) = C{d V_c \over dt} = \frac{11e^{-\frac{11t}{16}}}{16}(0.5\cos \frac{7t}{16} - B \sin \frac{7t}{16}) + \frac{7e^{-\frac{11t}{16}}}{16}(0.5\sin \frac{7t}{16} + B\cos \frac{7t}{16})

The loop equation can be solved for the voltage across the LR parallel combination:

V_C + V_{LR} + R_2 C{d V_c \over dt} - V_s = 0
V_{LR} = V_s - V_C - R_2 i_t = 1 - V_c - 1.5*i_t
VLR := 1 - VC - 1.5*IT

We know from the inductor terminal relation that:

i_L = \frac{1}{L} \int V_{LR} dt + C_1
IL := 1/.5 * int(VLR,t)

At this point mupad gave up and went numeric. In any case, it is clear from t = ∞ where the inductor current has to be zero that the integration constant is zero. This enables us to compute B from the inductor initial condition.

t :=0

Set the time to zero, set IL equal to the initial condition of .2 amps and solve for B:

solve(IL=0.2, B)

And get that B is -0.2008928571 ...

mupad screen shot finding the desired output io

The desired answer is io which is just VLR/R_1. To calculate need to start new mupad session because t is zero now. Start with:

B := -0.2008928571;
R1 :=0.5;

Repeat the above commands up to VLR and then add:

io = VLR/R1

i_o = 3e^{-\frac{11t}{16}}(0.0879\cos\frac{7t}{16} - 0.219\sin\frac{7t}{16}) - 0.0625e^{-\frac{11t}{16}}(0.5\cos\frac{7t}{16} + 0.201\sin\frac{7t}{16})