First Order Resistor Capacitor (RC) 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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The circuit illustrated in figure 7.25 is a first order circuit. The capacitor has an initial charge Q_{0} = 800 μC; and capacitance = 4μF. Calculate the current flowing through the capacitor and its charge for t > 0, if the switch S, is closed at t = 0.

**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**:

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.

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.

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.7.25 we have:

Time constant, τ = RC = 10 x 4 x 10^{-6} = 4 x 10^{-5}

Initial voltage across capacitor,
v_{0} = initial charge/capacitance

So,
v_{0} = (800 x 10^{-6})/(4 x 10^{-6}) = 200

Final voltage across capacitor, v = voltage of voltage source = 100 V.

So, A = 200 - 100 = 100 and B = 100

So, from equation (1), voltage across capacitor at time t, v_{c}(t) is:

v_{c}(t) = 100e^{-t/(4 x 10-5)} + 100 = 100(1 + e^{-25000t}))

So, charge of capacitor for t >: 0,
q is as follows:

q = CV = (4 x 10^{-6}) x 100(1 + e^{-25000t}) = (4 x 10^{-4})(1 + e^{-25000t}) Coulombs

Current in capacitor, i = dq/dt (the rate of change of charge)

So, i = d(4 x 10^{-4})(1 + e^{-25000t})/dt = -10e^{-25000t} 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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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.

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