Second Order (RCL) Circuits - The Differential Equations
Strings (SiPjAjk) = S7P4A41 Base Sequence = 12735 String Sequence = 12735 - 4 - 41
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.
(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).
(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).
(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).
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.
Single Variable Functions
Ordinary Differential Equations (ODEs)
The Universe is composed of matter and radiant energy. Matter is any kind of mass-energy that moves with velocities less than the velocity of light. Radiant energy is any kind of mass-energy that moves with the velocity of light.
Composition And Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Molecular Structure Of Matter
Molecular Shapes: Bond Length, Bond Angle
Molecular Shapes: Valence Shell Electron Pair Repulsion
Molecular Shapes: Orbital Hybridization
Molecular Shapes: Sigma Bonds Pi Bonds
Molecular Shapes: Non ABn Molecules
Molecular Orbital Theory
More Pj Problem Strings