# DC Circuits | First Order Circuits

## The Source-Free RC Circuit

A source-free RC circuit occurs when its dc source is suddenly disconnected. The energy already stored in the capacitor is released to the resistor(s).

Consider the circuit with an initially charged capacitor,

- The objective is to determine the circuit response, which we assume to be the voltage across the capacitor.
- Since the capacitor is initially charged, we can assume that at time , the initial voltage is , with the corresponding value of the energy stored as

- Applying KCL at the top node of the circuit yields . By definition, and . Thus,

This is a* first-order differential equation*, since only the first derivative is involved. Solving the equation,

- The voltage response of the circuit is an exponential decay of . This is the
*natural response*of the circuit.

The *natural response* refers to the behavior (in terms of voltages and currents) of the circuit, with no external excitation sources; graphically shown in Fig. 2.

- At t = 0, the initial condition is satisfied. As increases, the voltage approaches zero. The rapidity with which the voltage decreases is expressed in terms of the time constant .
- The time constant of a circuit is the time required for the response to decay to a factor of or 36.8% of its initial value. At , . Thus, . In terms of the time constant, .
- Since is less than 1% of after 5τ, it is customary to assume that the capacitor is fully discharged/charged after 5τ.
- The smaller the time constant, the faster the response/the faster the circuit reaches the steady state. Whether is small or large, the circuit reaches steady state in 5τ.
- With the voltage expressed in τ, we can find the current ,
- The power dissipated in the resistor is
- The energy absorbed by the resistor up to time is . As , which is the same as , the energy initially stored in the capacitor. This energy in the capacitor is eventually dissipated in the resistor.

In summary, the key to working with a source-free RC circuit is finding:

- The initial voltage across the capacitor.
- The time constant .

With these, we obtain the response as the capacitor voltage

- Once is obtained, other variables (capacitor current , resistor voltage , and resistor current ) can be determined.
- In finding , is often the Thevenin equivalent resistance at the terminals of the capacitor.

## The Source-Free RL Circuit

- Consider the series connection of a resistor and an inductor. The goal is to determine the circuit response, which is assumed to be the current through the inductor.

- We select the inductor current as the response in order to take advantage of the idea that it cannot change instantaneously.
- At
*t = 0*, we assume that the inductor has an initial current , or with the corresponding energy stored in the inductor as .

- Applying KVL around the loop in Fig.3, . But and . Thus,

Rearranging terms and integrating gives

The natural response of the RL circuit is an exponential decay of the initial current. The current response is shown in Fig. 4.

- It is evident from that the time constant for the RL circuit is . Thus, can be written as .
- With the current, we can find the voltage across the resistor as .
- The power dissipated in the resistor is .
- The energy absorbed by the resistor is .
- As the same as , the initial energy stored in the inductor. This energy in the inductor is eventually dissipated in the resistor.

The key to working with a source-free RL circuit is to find:

- The initial current through the inductor.
- The time constant of the circuit.

- With these, we obtain the response as the inductor current .
- Once we determine , other variables (inductor voltage , resistor voltage , and resistor current ) can be obtained.
- Note that in general, in is the Thevenin resistance at the terminals of the inductor.

## Singularity Functions

Singularity functions (*switching functions* in circuit analysis) are either discontinuous functions or functions with discontinuous derivatives.

*Unit Step Function u(t)*

- is 0 for negative values of and 1 for positive values of .

- undefined at , where it changes abruptly from 0 to 1

- dimensionless, like other mathematical functions such as sine and cosine
- The unit step function may be delayed or advanced. When is delayed by seconds

- When is advanced,

- Step function is used to represent an abrupt change in voltage or current. For example, the voltage

may be expressed in terms of the unit step function as

- If we let , then is simply the step voltage .
- A voltage source of is shown in Fig. 7(a); its equivalent circuit is shown in Fig. 7(b). It is evident in Fig. 7(b) that terminals a-b are short-circuited () for and that appears at the terminals for t>0.

- Similarly, a current source of is shown in Fig. 8(a), while its equivalent circuit is in Fig. 8(b).

- Notice that for t<0, there is an open circuit (), and flows for t>0.

*Unit Impulse Function*

- derivative of the unit step function

- is zero everywhere except at t=0, where it is undefined.

- The unit impulse may be regarded as an applied or resulting shock; a very short duration pulse of unit area, expressed mathematically as where denotes the time just before t=0 and is the time just after t=0.
- The unit area is the strength of the function. When an impulse function has a strength other than unity, the area of the impulse is equal to its strength; an impulse function has an area of 10.

- To illustrate how the impulse function affects other functions, evaluating the integral below where

This shows that we obtain the value of the function at the point where the impulse occurs; known as the *sampling* or *sifting* property.

- The special case is when , the integral becomes

*Unit Ramp Function **r(t)*

- Integrating the unit step function u(t) results in the unit ramp function
*r(t)*

- In general, a ramp is a function that changes at a constant rate.
- The unit ramp function is zero for negative values of t and has a unit slope for positive values of t.

- The unit ramp function may be delayed or advanced as shown:

For the delayed unit ramp function,

and for the advanced unit ramp function,

The three singularity functions are related by differentiation as

or by integration as

## Step Response of an RC Circuit

The *step response* of a circuit is its behavior when the excitation is the step function; the response due to a sudden application of a dc voltage/current.

Consider the RC circuit in Fig. 13(a) which can be replaced by the circuit in Fig. 13(b), where Vs is a constant dc voltage source.

We break the complete response into two components—one temporary and the other permanent, that is

- The
*t**ransient response*is the circuit’s temporary response that will decay to zero as time approaches infinity. - The
*steady-state response*is the response a long time after an external excitation is applied; it remains after the transient response has died out.

To find the step response of an RC circuit requires three things

- The initial capacitor voltage .
- The final capacitor voltage .
- The time constant .

- Let the response be the sum of the transient response and the steady-state response, .
- The transient response is always a decaying exponential, that is .
- After the transient response dies out, .
- Let be the initial capacitor voltage: . is the capacitor voltage just before switching and immediately after. From this, we obtain .
- Substituting to , we get

which can be written as

- where is the voltage at and is the final or steady-state value.
- If the switching takes place at time instead of , there is a time delay in the response so that

- where is the initial value at .

- If ,

The current through the capacitor is

## Step Response of an RL Circuit

To find the step response of an RL circuit requires three things:

- The initial inductor current at
- The final inductor current
- The time constant

Consider the RL circuit in Fig.14(a), which may be replaced by the circuit in Fig. 14(b). The inductor current i is the circuit response.

- Let the response be the sum of the transient response and the steady-state response, .
- The transient response is always a decaying exponential, that is
- After the transient response dies out, the inductor becomes a short circuit. The entire source voltage appears across . Thus, .
- Let be the initial current through the inductor: . From this, we obtain .
- Substituting and to , we get

which can be written as

- If the switching takes place at time instead of , .
- If ,

The voltage across the inductor is

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