Question #105577
A series RLC circuit withR = 6 ohm, C = 0.02 Farad and L = 0.1 has no applied voltage. Find the subsequent current in the circuit if the initial charge, on the capacitor is q0 and the initial current is zero.
1
Expert's answer
2020-03-16T13:16:02-0400

Task:

R=6Ω;L=0.1H;C=0.02F;i(0)=0A;qC(0)=q0R = 6 \Omega; \\ L = 0.1 \text{H}; \\ C = 0.02 \text{F}; \\ i(0) = 0 \text{A}; \\ q_C(0) = q_0

Find: i(t)i(t)

Solution:



Current through the capacitor and voltage across the inductance can be described as:

iC=CdvCdt (1)vL=LdiLdt (2)i_{C}=C \cfrac{d v_{C}}{d t}\ (1)\\ v_{L}=L \cfrac{d i_{L}}{d t}\ (2)

In this case all the elements are connected in series, so iC=iL=ii_C = i_L = i . According to the second Kirchhoff’s circuit law:

vR+vL+vC=0 (3)v_{R}+v_{L}+v_{C}=0\ (3)

After inserting (1) and (2) into (3) the last equation can be rewritten as:

vR+vL+vC=Ri+Ldidt+1C0t0idt=0 (4)v_R + v_L + v_C = Ri + L \cfrac{di}{dt} + \cfrac{1}{C} \int\limits_0^{t_0} idt = 0\ (4)

After differentiation of the left and right parts of equation (4) it is possible to get:

ddt(Ri+Ldidt+1C0t0idt)=d0dtRdidt+Ld2idt+1Ci=0.1d2idt+6didt+50i=0\cfrac{d}{dt} (Ri + L \cfrac{di}{dt} + \cfrac{1}{C} \int\limits_0^{t_0} idt) = \cfrac{d0}{dt} \Leftrightarrow R \cfrac{di}{dt} + L \cfrac{d^2 i}{dt} + \cfrac{1}{C} i = 0.1\cfrac{d^2 i}{dt} + 6\cfrac{di}{dt} + 50i=0

It is a second order ordianary differential equation with constant coefficients. To solve this equation it is necessary to find the roots of the characteristic polynomial:

0.1λ2+6λ+50=00.1 \lambda^2 + 6 \lambda + 50 = 0

According to the Vieta's formuals:

{λ1+λ2=60.1=60λ1λ2=500.1=500{λ1=50λ2=10\begin{cases} \lambda_1+\lambda_2 = -\cfrac{6}{0.1} = -60 \\ \lambda_1 \cdot \lambda_2 = \cfrac{50}{0.1} = 500 \end{cases} \Bigg| \Rightarrow \begin{cases} \lambda_1 = -50 \\ \lambda_2 = -10 \end{cases}

So, the general solution for i(t)i(t) is equal to:

i(t)=C1eλ1t+C2eλ2t=C1e50t+C2e10t (5)i(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t} = C_1 e^{-50t} + C_2 e^{-10t}\ (5)

To get the partial solution it is necessary to solve the Cauchy's problem. First of all its is necessary to get initial condition for didt\cfrac{di}{dt} . According to (4):

Ldidt=RiuCdidt(0)=1L(Ri(0)uC(0)) (6)L \cfrac{di}{dt} = -Ri-u_C \Rightarrow \cfrac{di}{dt} (0) = \cfrac{1}{L} (-Ri(0)-u_C(0)) \ (6)

To get uC(0)u_C(0) it is convenient to use the defenition of capacity:

C=qCuCuC(0)=qC(0)C=q0C (7)C = \cfrac{q_C}{u_C} \Rightarrow u_C(0) = \cfrac{q_C(0)}{C} = \cfrac{q_0}{C}\ (7)

After inserting (7) into (6) the latter can be rewritten as:

didt(0)=1L(Ri(0)q0C) (8)\cfrac{di}{dt}(0) = \cfrac{1}{L} (-Ri(0) - \cfrac{q_0}{C})\ (8)

According to the task, i(0)=0i(0) = 0, so, according to (8):

didt(0)=1LCq0=500q0\cfrac{di}{dt}(0) = -\cfrac{1}{LC} q_0 = -500q_0

Now it is possible to solve the Cauchy's problem:

{i(0)=C1e500+C2e100=C1+C2=0didt(0)=ddt(C1e50t+C2e10t)(0)=50C110C2=500q0\begin{cases} i(0) = C_1 e^{-50\cdot 0} + C_2 e^{-10 \cdot 0} = C_1 + C_2 = 0 \\ \cfrac{di}{dt} (0) = \cfrac{d}{dt} (C_1 e^{-50t} + C_2 e^{-10t})(0) =-50C_1 -10C_2 =-500q_0 \end{cases}

So, the system for C1C_1 and C2C_2 is:

{C1+C2=050C110C2=500q0{C2=C140C1=500q0{C1=12.5q0C2=12.5q0 (9)\begin{cases} C_1+C_2=0\\ -50C_1-10C_2=-500 q_0 \end{cases} \Leftrightarrow \begin{cases} C_2 = -C_1\\ -40C_1=-500 q_0 \end{cases} \Leftrightarrow \begin{cases} C_1 = 12.5 q_0\\ C_2 = -12.5q_0 \end{cases} \ (9)

After inserting (9) in (5) it is possible to get partial solution:

i(t)=12.5q0e50t12.5q0e10ti(t) = 12.5q_0 e^{-50t} -12.5q_0 e^{-10t}

Answer: i(t)=12.5q0e50t12.5q0e10ti(t) = 12.5q_0 e^{-50t} -12.5q_0 e^{-10t} .



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