First Order Resistor Capacitor (RC) Parallel 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 Capacitor (RC) Parallel Circuits

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Figure 5.41 illustrates a first order RC parallel circuit comprising a voltage source = 100 V; two capacitors in parallel each of capacitance 4 F and four resistors.

The switch,*s _{1}* is always open. The switch

Determine:

(a) The capacitor's voltage

(b) The current through

**The string**:

(a) S_{7}P_{3}A_{32}

(b) 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 piecewise-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**:

(a) Pj Problem of Interest is of type *force*. Voltage problems are *force problems*.

(b) 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.5.23 we have:

Replace parallel capacitors with their equivalent:

Equivalent capacitance, C, of the two parallel capacitors
= *C _{1}* +

Determine the

(1) Remove the load.

(2) Zero all independent voltage and current sources.

(3) Calculate the total resistance, beginning from the resistances farthest from the load's original position.

In figure 5.42, the load (the capacitors in this case) have been removed. The voltage source is set to zero (a short circuit is in its place) and the current source is set to zero (an open circuit is in its place).

So, Thevenin equivalent resistance = [(3)(6)/(3 + 6)] + 4 = 6 Ω

The

(1) Remove the load, and open-circuit its terminals.

(2) Establish the open-circuit voltage

(3) Solve for

(4) The Thevenin voltage,

In figure 5.43, load is removed and the open-circuit voltage,

So, voltage across the 3 Ω resistor = v

Also, voltage across the 6 Ω resistor = v

Only the current source is in play. The voltage source is not in play since switch s

So, by the current division rule, current through the 3 Ω resistor = (6/9)4 = (2/3)4

A lSo, voltage across the 3 Ω resistor = (2/3)4 x 3 = 8 V.

So, v

So, Thevenin voltage, v

Time constant = RC = 6 x 8 = 48 secs; and A = 0 - 8 = -8; B = 8;

So, capacitor voltage v

Capacitor charge, q = CV = 4 x 8(1 - e

So, current in capacitor = dq/dt = (2/3)e

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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