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 **

Expressions Of Pj Problems

Second Order (RCL) Circuits - The Differential Equations

Math

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

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.

Single Variable Functions

Conics

Ordinary Differential Equations (ODEs)

Vector Spaces

Real Numbers

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.

Periodic Table

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