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,
Figure 1: A source-free RC circuit. R and C may be the equivalent resistance and capacitance of combinations of resistors and capacitors.
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.
Figure 2: The voltage response of the RC circuit
At , 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.
Figure 3: A source-free RL circuit
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.
Figure 4: The current response of the RL circuit
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
Figure 5: The unit step function
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,
Figure 6: (a) The unit step function delayed by to, (b) the unit step advanced by to
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.
Figure 7: (a) Voltage source of Vo u(t), (b) its equivalent circuit
Similarly, a current source of is shown in Fig. 8(a), while its equivalent circuit is in Fig. 8(b).
Figure 8: (a) Current source of Io u(t), (b) its equivalent circuit.
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.
Figure 9: The unit impulse function
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.
Figure 10: Three impulse functions
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.
Figure 11: The unit ramp function
The unit ramp function may be delayed or advanced as shown:
Figure 12: The unit ramp function (a) delayed by to, (b) advanced by to.
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.
Figure 13: An RC circuit with voltage step input
We break the complete response into two components—one temporary and the other permanent, that is
The transient 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.
Figure 14: An RL circuit with a step input voltage
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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, the initial voltage is 
, with the corresponding value of the energy stored as 
. By definition, 
and 
. Thus,
circuit is an exponential decay of 
. This is the natural response of the circuit.
, 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 
.
or 36.8% of its initial value. At 
, 
. Thus, 
. In terms of the time constant, 
.
is less than 1% of 
after 5τ, it is customary to assume that the capacitor is fully discharged/charged after 5τ.
,
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.
across the capacitor.
.
is obtained, other variables (capacitor current 
, resistor voltage 
, and resistor current 
) can be determined.
, 
is often the Thevenin equivalent resistance at the terminals of the capacitor.
through the inductor.
, or 
with the corresponding energy stored in the inductor as 
.
. But 
and 
. Thus,
that the time constant for the RL circuit is 
. Thus, 
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.
.
.
.
the same as 
, the initial energy stored in the inductor. This energy in the inductor is eventually dissipated in the resistor.
through the inductor.
of the circuit.
.
, other variables (inductor voltage 
, resistor voltage 
, and resistor current 
) can be obtained.
in 
is the Thevenin resistance at the terminals of the inductor. 
is 0 for negative values of 
and 1 for positive values of 
, where it changes abruptly from 0 to 1
is delayed by 
seconds
is advanced,
, then 
is simply the step voltage 
.
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.
is shown in Fig. 8(a), while its equivalent circuit is in Fig. 8(b).
), and 
flows for t>0.
is zero everywhere except at t=0, where it is undefined.
where 
denotes the time just before t=0 and 
is the time just after t=0.
has an area of 10.
, the integral becomes
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.
.
.
.
.
be the initial capacitor voltage: 
. 
is the capacitor voltage just before switching and 
immediately after. From this, we obtain 
.
to , 
we get
is the voltage at 
and 
is the final or steady-state value.
instead of 
, there is a time delay in the response so that
is the initial value at 
.
,
at 
.
appears across 
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. From this, we obtain 
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