First-Order Circuit
First-Order Circuit
A first-order circuit can only contain one energy
storage element (a capacitor or an inductor).
The circuit will also contain resistance.
A first-order circuit is characterized by a first
order differential equation.
Two types of First-Order Circuit
RC Circuits
R
C
RL Circuits
R
L
Ways to excite the circuits
By initial conditions of the storage elements in
the circuits
By independent sources
Source-Free Circuit
A source-free circuit is one where all
independent sources have been disconnected
from the circuit after some switch action. The
voltages and currents in the circuit typically will
have some transient response due to initial
conditions (initial capacitor voltages and initial
inductor currents).
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
resistors.
Source-Free RC Circuit
Applying KCL,
assuming the initial voltage is V0
Natural Response
The natural response of a circuit refers to the
behaviour (in terms of voltages and currents) of
the circuit itself, with no external sources of
excitation.
Natural Response of RC Circuit
The natural response depends on the nature of the
circuit alone, with no external sources. In fact,
the circuit has a response only because of the
energy initially stored in the capacitor.
Time Constant
The rapidity with which the voltage decreases is
expressed in terms of the time constant,
denoted by the lower case Greek letter tau, .
Time Constant of RC Circuit
The time constant of a circuit is the time required
for the response to decay by a factor of 1/e or
36.8 percent of its initial value.
= RC
Source-Free RC Circuit
Key to working with Source-free RC
Circuit
The initial voltage v(0) = Vo across the capacitor.
The time constant .
Problems:
1. In the figure, let vC(0) = 15 V. Find vC, vx , and ix
for t > 0.
Problems:
2. From the circuit, let vC(0) = 30 V. Determine vC,
vx , and io for t 0.
Problems:
3. If the switch in the figure opens at t = 0, find
v(t) for t 0 and wC(0).
Source-Free RL Circuit
Source-Free RL Circuit
Applying KVL,
assuming the initial current is I0
Natural Response of RL Circuit
This shows that the natural response of the RL
circuit is an exponential decay of the initial
current.
Time Constant of RL Circuit
Source-Free RL Circuit
Key to working with Source-free RL
Circuit
The initial current i(0) = Io across the inductor.
The time constant .
Problems:
1. For the circuit, find i(t) for t > 0.
Problems:
2. Determine i, io, and vo for all t in the circuit.
Assume that the switch was closed for a long time.
Singularity Functions
Also called switching functions.
They serve as good approximations to the
switching signals that arise in circuits with
switching operations.
Singularity functions are functions that either
are discontinuous or have discontinuous
derivatives.
Singularity Functions
Unit Step Functions
Unit Impulse Functions
Unit Ramp Functions
Unit Step Function
The unit step function u(t) is 0 for negative
values of t and 1 for positive values of t.
Unit Impulse Function
The derivative of the unit step function u(t) is
the unit impulse function (t).
The unit impulse function (t) is zero
everywhere except at t = 0, where it is
undefined.
Unit Ramp Function
Integrating the unit step function u(t) results in
the unit ramp function r(t).
The unit ramp function is zero for negative
values of t and has a unit slope for positive
values of t.
Step Response
The step response of a circuit is its behavior
when the excitation is the step function, which
may be a voltage or a current source.
The step response is the response of the circuit
due to a sudden application of a dc voltage or
current source.
Step Response RC Circuit
Applying KCL,
Natural Response
The natural response or transient response is the
circuits temporary response that will die out
with time.
Forced Response
The forced response or steady-state response is
the behavior of the circuit a long time after an
external excitation is applied.
Step Response of the RC Circuit
To find the Step Response of a RC
Circuit
1. The initial capacitor voltage v(0).
2. The final capacitor voltage v().
3. The time constant .
Problems:
1. Find v(t) for t > 0 in the circuit Assume the
switch has been open for a long time and is
closed at t = 0. Calculate v(t) at t = 0.5.
2. The switch the circuit is closed at t = 0. Find i(t)
and v(t) for all time.
Note that u(t) = 1 for t < 0 and 0 for t > 0. Also,
u(t) = 1 u(t).
Step Response of the RL Circuit
To find the Step Response of a RL
Circuit
1. The initial inductor current i(0) at t = 0+.
2. The final inductor current i().
3. The time constant .
Problems:
1. Find i(t) in the circuit for t > 0. Assume that the
switch has been closed for a long time.
Problems:
2. The switch in the circuit has been closed for a
long time. It opens at t = 0. Find i(t) for t > 0.
