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First Order Resistor Inductor (RL) Series Circuits


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

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First Order Resistor Inductor (RL) Series Circuits

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First Order Resistor Inductor Series Circuit
The circuit illustrated in figure 7.42 is a first order RL circuit. The switch is closed at t = 0 after having been open for an extended period of time.
Solve for the currents i1 and i2 using simultaneous differential equations for the circuit.

The string:
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 blow 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)
The math:
Any system that is able to store potential or kinetic energy, and to dissipate this energy, is a First Order System. In an electric circuit, a first order circuit is any circuit that consists of a single energy storage element such as a capacitor or an inductor; voltage or current sources and resistors.
Whenever a circuit is switched from one state to another, either by a change in the active elements or a change in the passive elements, there exists a transitional period during which the branch currents and element voltages change from their pre-switched values to their post-switched values. This transitional period is called the transient.
The circuit is said to be in steady state at the end of the transient.
First Order Circuits are described by First Order Ordinary differential equations (equations of first degree non-partial derivatives).
The solution to this first order ordinary differential equation consists of a homogeneous solution (or natural response) which is the solution of the transient; and the >particular solution (or forced response) which is the solution of the steady state. In other words, the complete solution (or complete response) is the sum of the homogeneous solution and the particular solution.
The complete response of a first order circuit is represented as follows:

f(t) = Ae-t/τ + B ------(1).

where A = initial value - final value; B = final value and τ = time constant .
τ = RC for a RC circuit. τ = L/R for a RL circuit.
In a RC circuit, A and B are voltage measurements.
In a RL circuit, A and B are current measurements.
The time constant is the time at which the function is 36.8% of its initial value. In other words, it is the time at which the function has undergone 63.2% of the change from f(0+) to f(∞)
Now for the problem of fig.7.42:
Pj Problem of Interest is of type motion. It could also be viewed as of type change because current is the rate of change of charge flow with respect to time. However, its representation as the flow of charge is more appropriate perspectively.
First Order Resistor Inductor Series Circuit
Initial current = I0 = 100/15 = 6.67 A.
Kirchoff's Voltage Law ( KVL) applied to the i1 loop gives:
5(i121 + 2di1/dt = 100 ------(1)
KVL applied to the i2 loop gives:
5(i122 = 100 -------(2)
Solving equations (1) and (2) simultaneously:
From equation (2), i2 = (100 - 5i1)/15.
Substituting this expression for i2 in equation (1), we have:
40i1 + 6di1/dt = 200 ----(3)
Homogeneous equation of (3) is:
40i1 + 6di1/dt = 0
From which i1h = Ae-(40/6)t
Particular equation of (3) is:
40i1 =200 (since 6di1/dt =0 at steady state)
From which i1p = 5
So, i1 = homogeneous solution + particular solution
So, i1 = Ae-(40/6)t + 5
At t = 0, I0 = 6.67 = A + 5; So A = 1.67 .
So, i1 = (1.67e-6.67t + 5) Amperes.
So,i2 = (100 - 5i1)/15 = (-0.556e-6.67t + 5) Amperes.

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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