﻿ Second Order Resistor Capacitor Inductor (RCL) Circuits - The Differential Equations

Second Order (RCL) Circuits - The Differential Equations

Strings (SiPjAjk) = S7P4A41     Base Sequence = 12735     String Sequence = 12735 - 4 - 41

Expressions Of Pj Problems
Second Order (RCL) Circuits - The Differential Equations
Math

Figures 8.1(a) and 8.1(b) are second order (RCL) circuits. In figure 8.1(a), the capacitor and inductor are in parallel. In figure 8.1(b), the capacitor and inductor are in series.
Determine:
(a) The differential equation for the circuit in terms of the inductor current iL(t).
(b) The differential equation for the circuit in terms of the capacitor voltage vC(t).

The string:
(a) S7P4A41 (Linear Motion - Current Flowing Into Passive Elements). The motion of current flow in an electric circuit can be viewed as piece-wise-linear (current flow into a device) or looped (current flow in a circuit branch or the entire circuit). In a series connection, the piece-wise-linear current and looped current are the same. In a parallel connection, they are not. The looped current is stringed as S7P4A42 (Curved Motion).
(b) S7P3A32 (Force - Push).
The math:

(a) Pj Prblem of Interest is of type motion (Linear Motion).
Applying Kirchoff's Voltage Law (KVL) to loop containing i (fig.8.1(a)), we have:
v(t) - Ri(t) - vC(t) = 0 ----(1)
So, i(t) = [v(t) - vC(t)]/R ----(2)
Applying KVL to loop containing iL, we have:
vC(t) = vL(t) = L diL(t)/dt ----(3)
Applying Kirchoff's Current Law (KCL) to node a, we have:
i(t) - iC(t) - iL(t) = 0 ----(4)
So, [v(t) - vC(t)]/R -CdvC(t)/dt - iL(t) = 0 ----(5)
Substituting LdiL(t)/dt in place of vC(t) in equation (5) results in the following differential equation in terms of iL(t):
LC d2iL(t)/dt2 + (L/R)diL(t)/dt + iL(t) = (1/R)v(t) ----(6)
So, equation (6) is the differential equation in terms of inductor current, iL(t).
(b) The differential equation in terms of vC(t) is as follows:
LC d2vC(t)/dt2 + (L/R)dvC(t)/dt + vC(t) = (L/R)dv(t)/dt ----(7).
For the series circuit (fig.8.1(b) an integrodifferential equation arises as follows:
Applying KVL we have:
v(t) -Ri(t) - Ldi(t)/dt - (1/C)(∫-∞t d(i(t)/dt) = 0 -------(8)
Differentiating both sides of equation (8), and noting that i(t) = iC(t) we have:
RCdvC(t)/dt + LCd2vC(t)/dt2 + vC(t) = v(t) ----(9).

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