Second Order (RCL) Circuits - The Differential Equations

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{4}A_{41} Base Sequence = 12735 String Sequence = 12735 - 4 - 41 **

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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 *i _{L}(t)*.

(b) The differential equation for the circuit in terms of the capacitor voltage

**The string**:

(a) S_{7}P_{4}A_{41} (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 S_{7}P_{4}A_{42} (Curved Motion).

(b) S_{7}P_{3}A_{32} (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) - v_{C}(t) = 0 ----(1)

So, i(t) = [v(t) - v_{C}(t)]/R ----(2)

Applying
KVL to loop containing i_{L}, we have:

v_{C}(t) = v_{L}(t) = L di_{L}(t)/dt ----(3)

Applying Kirchoff's Current Law (KCL) to node a, we have:

i(t) - i_{C}(t) - i_{L}(t) = 0 ----(4)

So, [v(t) - v_{C}(t)]/R -Cdv_{C}(t)/dt - i_{L}(t) = 0 ----(5)

Substituting Ldi_{L}(t)/dt in place of v_{C}(t) in equation (5)
results in the following differential equation in terms of i_{L}(t):

LC d^{2}i_{L}(t)/dt^{2} + (L/R)di_{L}(t)/dt + i_{L}(t) = (1/R)v(t) ----(6)

So, equation (6) is the differential equation in terms of inductor current, i_{L}(t).

(b) The differential equation in terms of v_{C}(t) is as follows:

LC d^{2}v_{C}(t)/dt^{2} + (L/R)dv_{C}(t)/dt + v_{C}(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) = i_{C}(t) we have:

RCdv_{C}(t)/dt + LCd^{2}v_{C}(t)/dt^{2} + v_{C}(t) = v(t) ----(9).

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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