Notes on First Order Circuits
Tyler Shewbert
November 5, 2015
Part I
Source Free RC Circuit
-A first-order circuit is characterized by a first-order differential equation.
-There are two ways to excite RC and RL circuits, the first way is call source-free. We assume that the energy is
initially stored in the capacitive or inductive element. They may have dependent sources. The second way is by using
independent sources.
-A source-free RC circuit occurs when a DC source is suddenly disconnected from a circuit. The energy that the
capacitor has already store will now be released to the power dissipating resistors.
The initial voltage:
V (0) = V0
(1)
The stored energy at this point is simply
w(0) =
1
CV 2
2
(2)
1
dt
RC
(3)
Using KCL a first order differential equation is found:
dv
=
v
Solving this differential equation produces
v(t) = V0 e
t/RC
(4)
Which says that the voltage response of the RC circuit is a decay exponentially of V0 . This is also called the natural
response, which is the behavior a circuit exhibits with no external sources.
For a capacitor, when the voltage source steps to zero, the capacitor will now
The time constant is written ⌧ = RC. This will be the same regardless of what the output is. It is based only on the
capacitance and resistance of the circuit.
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The power dissipated by the resistor is
V02
e
R
p(t) = viR =
2t/⌧
(5)
and the energy is of course
wR =
ˆ
p(t)dt
(6)
The keys to working a source free RC circuit are to first find the initial voltage across the capacitor, and then find the
time constant. If the circuit has one capacitor and several resistors, then Thevenin circuits can be used to equivocate the
circuit.
Part II
Source Free RL Circuit
We will now consider the case of an inductor in series with a resistor, the instant after a source has been removed. Since
the inductor current cannot change instantaneously, we assume the current to be a function of time i(t) as we did with
voltage in the capacitor. When the source is removed, by the principles of Lenz’s law, the inductor will now become a
source, sourcing current into the resistor. So the inductor has an initial current I0 , an energy value that corresponds to
the definition of energy stored in an inductor. Using KVL around the loop, similar to KCL in the capacitor, we find a
differential equation that can be written as
L
di R
+ i=0
dt
L
(7)
Solving this by rearranging terms and integrating gives us the solution of the differential equation:
i(t) = I0 e
Rt/L
(8)
Similar to the capacitor, the initial energy stored in the inductor is dissipated by the resistor. This process begins as
soon as the independent source is removed from the circuit. Also, a Thevenin equivalent circuit can be used to model this
circuit.
Part III
Singularity or Switching Functions
A singularity or switching function is a function that is either discontinuous or their derivative is discontinuous. There
are three widely used singularity functions, the unit step, the unit impulse, and the unit ramp.
The unit step function u(t) is zero for negative values and one for positive values of t. The unit step function is
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undefined at t = 0. It is a scalar, dimensionless value.
The unit impulse function is also know as the delta function. It states that the value (t) is zero everywhere except
at t = 0, where it is not defined. This is often used as a resulting or applied shock. When a function is integrated with
the impulse function, we obtain the value of the function at the point where the impulse occurs.
Integrating the unit step function creates the unit ramp function r(t). This function is zero for negative values of
t and has a unit slope for positive values of t.
All three functions are related by differentiation and integration.
Part IV
Step Responses of an RC Circuit
The step response of a circuit refers to the circuits behavior when it is excited by either a step voltage or current source.
First we will consider an RC circuit with a voltage source. We know that the voltage of a capacitor cannot change
instantaneously. If we apply KCL to such a circuit, the resulting differential equation is found:
C
dv v
+
dt
Vs u(t)
=0
R
(9)
where v is the voltage across the circuit and Vs is the source voltage. Applying boundary conditions and solving this
equation using rearranging we get two answers, one for positive t:
v(t) = Vs + (V0
Vs )e
t/⌧
(10)
and one for negative t:
(11)
v(t) = V0
Conceptually what is happening is that current is being stepped, and by Lenz’s Law sinks current into the circuit after
the step current is applied.
There are two parts to the equation defining v(t). First, the natural response, if there is stored energy in the capacitor
is defined by:
v n = V0 e
t/⌧
(12)
and the forced response from the step:
vf = Vs (1
e
t/⌧
)
(13)
Which becomes the above equation for v(t).
This is also broken up into transient response and steady-state response. The transient response is temporary. It is
the response that will die out with time. The steady-state response is the behavior of the circuit after a long period of
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excitation. This is written mathematically as:
v(t) = v(1) + [v(0)
v(1)]e
t/⌧
(14)
This equation only applies to step responses, meaning when the input is constant.
Part V
Step Response of an RL Circuit
For an RL circuit, a similar path can be taken that was used for an RC circuit. This time, we know that the current i(t)
must be continuous, so we can say that the RL circuit equation should be made of two parts, a transient and steady-state
response as was the RC circuit. The transient part has already been defined as:
it = Ae
t/⌧
(15)
where A will be definedI0 is the initial current through the circuit. At t = 0, I0 = A +
Vs
R,
since I0 = i(0+ ) = i(0 ) .
Combining all of this, i(t) can be rewritten:
i(t) =
Vs
+ (I0
R
Vs
)e
R
t/⌧
(16)
or
i(t) = i(1) + [i(0)
i(1)]e
t/⌧
(17)
Part VI
First-Order Op Amp Circuits
An op amp circuit that has a reactive device will exhibit first order behavior also.
These notes were developed from: [1, 2]
References
[1] Charles Alexander and Matthew Sadiku. Fundamentals of Electric Circuits. McGraw-Hill Education, 5 edition, 1 2012.
[2] SC Petersen. Class notes on first order circuits, November 2015.
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