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,
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
. Thus,
. By definition,
and
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.
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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
initial value. At
,
. Thus,
or 36.8% of its
. 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:
1. The initial voltage
2. The time constant .
across the capacitor.
With these, we obtain the response as the capacitor voltage
Once
is obtained, other variables (capacitor current , resistor voltage
, and resistor current
determined.
In finding
, is often the Thevenin equivalent resistance at the terminals of the capacitor.
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) can be
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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
stored in the inductor as
Applying KVL around the loop in Fig.3,
, or
with the corresponding energy
.
. 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.
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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
in the inductor is eventually dissipated in the resistor.
.
, the initial energy stored in the inductor. This energy
The key to working with a source-free RL circuit is to find:
1. The initial current
through the inductor.
2. 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
obtained.
Note that in general,
in
.
, and resistor current
) can be
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 .
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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
When
is delayed by
seconds
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
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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.
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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)
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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,
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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
1. The initial capacitor voltage
2. The final capacitor voltage
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.
.
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3. 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:
switching and
Substituting
.
.
immediately after. From this, we obtain
to , we get
.
is the capacitor voltage just before
.
which can be written as
where
is the voltage at
If the switching takes place at time
where
If
and
is the initial value at
is the final or steady-state value.
instead of
, there is a time delay in the response so that
.
,
The current through the capacitor is
Step Response of an RL Circuit
To find the step response of an RL circuit requires three things:
1. The initial inductor current
2. The final inductor current
3. The time constant
at
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
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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
across
Let
. Thus,
.
be the initial current through the inductor:
Substituting
appears
.
and
. From this, we obtain
to , we get
which can be written as
If the switching takes place at time
If
,
instead of
,
.
The voltage across the inductor is
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