First Order Resistor Inductor (RL) Series Circuits

**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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First Order Resistor Inductor (RL) Series Circuits

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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 *i _{1}* and

**The string**:

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 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 S_{7}P_{4}A_{42} (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.

Initial current = I_{0} = 100/15 = 6.67 A.

Kirchoff's Voltage Law (
KVL) applied to the i_{1} loop gives:

5(i_{1}21 + 2di_{1}/dt = 100 ------(1)

KVL applied to the i_{2} loop gives:

5(i_{1}22 = 100 -------(2)

Solving equations (1) and (2) simultaneously:

From equation (2), i_{2} = (100 - 5i_{1})/15.

Substituting this expression for i_{2} in equation (1), we have:

40i_{1} + 6di_{1}/dt = 200 ----(3)

Homogeneous equation of (3) is:

40i_{1} + 6di_{1}/dt = 0

From which i_{1h} = Ae^{-(40/6)t}

Particular equation of (3) is:

40i_{1} =200 (since 6di_{1}/dt =0 at steady state)

From which i_{1p} = 5

So, i_{1} = homogeneous solution + particular solution

So, i_{1} = Ae^{-(40/6)t} + 5

At t = 0, I_{0} =
6.67 = A + 5; So A = 1.67
.

So, i_{1} = (1.67e^{-6.67t} + 5) Amperes.

So,i_{2} = (100 - 5i_{1})/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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