# Chapter 1: Circuit Variables by 81a2A3IL

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```									EECE 311 Lecture Notes #3
CHAPTER 7: RESPONSE OF FIRST-
ORDER RL AND RC CIRCUITS
ASSIGNMENTS
   Read the textbook, Electric Circuits, 9th Ed.,
Chapter 7, pp.228-259.
Do HW#4: Chapter 7: 4, 5, 7, 12, 14, 16, 23, 26,
30, 35, 36, 37, 51, 53, 55, 56, 85.
OBJECTIVES
   Be able to determine the natural response of both RL and RC circuits.
   Be able to determine the step response of both RL and RC circuits.
   Know how to analyze circuits with sequential switching.
   Be able to analyze op amp circuits containing resistors and a single
capacitor
CONTENTS

   7.1 The Natural Response of an RL Circuit
   7.2 The Natural Response of an RC Circuit
   7.3 The Step Response of RL and RC Circuits
   7.4 A General Solution for Step and Natural Responses
   7.5 Sequential Switching
   7.6 Unbounded Response
   7.7 The Integrating Amplifier
TYPES OF FIRST ORDER CIRCUITS
7.1 THE NATURAL RESPONSE OF AN RL
CIRCUIT

   The natural response of an RL circuit can best be described in
terms of the circuit shown in Fig. 7.3.
   Finding the natural response requires finding the voltage and
current at the terminals of the resistor after the switch has
been opened, that is, after the source has been disconnected
and the inductor begins releasing energy
   If we let t = 0 denote the instant when the switch is opened,
the problem becomes one of finding v(t) and i(t) for t ≥0.
   For t ≥ 0, the circuit shown in Fig. 7.3 reduces to the one
shown in Fig. 7.4
EXPRESSIONS FOR THE CURRENT FOR RC
CIRCUIT
   Kirchhoff's voltage law to obtain an expression involving i, R,
and L. Summing the voltages around the closed loop gives

Where use the passive sign convention.
   The highest order derivative appearing in the equation is 1;
hence the term first-order.

   Initial inductor current:
The first instant after the switch has been opened, the current
in the inductor remains unchanged. If we use 0- to denote the
time just prior to switching, and 0+ for the time immediately
following switching, then

   Natural response of an RL circuit:
The initial current in the inductor is oriented in the same
direction as the reference direction of i.

fig 7.5 shows this response
SIGNIFICANCE OF TIME CONSTANT FOR RC
CIRCUIT
   The expressions for i(t) (Eq. 7.7) and v(t) (Eq. 7.8) include a
term of the form e-(R/L)t The coefficient of t—namely, R/L—
determines the rate at which the current or voltage
approaches zero. The reciprocal of this ratio is the time
constant of the circuit, denoted
SIGNIFICANCE OF TIME CONSTANT FOR RC
CIRCUIT

   Table 7.1 gives the value of e- t/ for integral multiples of
from 1 to 10.
   The existence of current in the RL circuit shown in Fig. 7.1(a)
is a momentary event and is referred to as the transient
response of the circuit.
   The response that exists a long time after the switching has
taken place is called the steady-state response.
   The phrase a long time then also means the time it takes the
circuit to reach its steady-state value.
TYPICAL STEPS INVOLVED IN CALCULATION
FOR NATURAL RESPONSE OF AN RL CIRCUIT

   Calculating the natural response of an RL circuit can be summarized as follows:

1. Find the initial current, I0, through the inductor.
2. Find the time constant of the circuit, = L/R.
3. Use Eq. 7.15, Ioe- t/ , to generate i(t) from Io and .

   All other calculations of interest follow from knowing i(t).
   Text Examples 7.1 and 7.2 illustrate the numerical calculations associated with the natural response of
an RL circuit.
7.2 THE NATURAL RESPONSE OF AN RC
CIRCUIT
   We begin by assuming that the switch has been in position a
for a long time, allowing the loop made up of the dc voltage
source Vg, the resistor R1, and the capacitor C to reach a
steady-state condition. Thus the voltage source cannot
sustain a current, and so the source voltage appears across
the capacitor terminals.
   When the switch is moved from position a to position b (at t =
0), the voltage on the capacitor is Vg. Because there can be no
instan-taneous change in the voltage at the terminals of a
capacitor, the problem reduces to solving the circuit shown in
Fig. 7.11
EXPRESSION FOR VOLTAGE OF AN RC CIRCUIT
   Using the lower junction between R and C as the reference
node and summing the currents away from the upper junction
between R and C gives

   Initial capacitor voltage:
The initial voltage on the capacitor equals the voltage source
voltage Vg, or

   Time constant for RC circuit:
The time constant for the RC circuit equals the product of the
resistance and capacitance, namely,

   Natural Response of an RC circuit:
Fig 7.12 shows the graphical representation of the response
and the equation is given as
TYPICAL STEPS INVOLVED IN CALCULATION
FOR NATURAL RESPONSE OF AN RC CIRCUIT

   Calculating the natural response of an RC circuit can be summarized as follows:

1. Find the initial voltage, v0, through the capacitor.
2. Find the time constant of the circuit, = RC.
3. Use Eq. 7.25, v(t)= Voe- t/ , to generate v(t) from vo and .

   All other calculations of interest follow from knowing v(t).
   Text Examples 7.3 and 7.4 illustrate the numerical calculations associated with the natural response of
an RL circuit.
7.3 THE STEP RESPONSE OF RL CIRCUIT
   After the switch in Fig. 7.16 has been closed, Kirchhoff's
voltage law requires that

   After deriving equation for natural response of RL circuit
from above equation we get,

   Fig 7.17 shows response for the RL circuit when Io = 0
   Also Fig 7.18 shows Inductor voltage vs. time
7.3 THE STEP RESPONSE OF RC CIRCUIT

   We can find the step response of a first-order RC circuit by
analyzing the circuit shown in Fig. 7.21. For mathematical
convenience, we choose the Norton equivalent of the network
connected to the equivalent capacitor.

   From the above equation we derive equation for step response
of RC circuit.
7.4 A GENERAL SOLUTION FOR STEP RESPONSE
AND NATURAL RESPONSES
   To generalize the solution of these four possible circuits, we
let x(t) represent the unknown quantity, giving x(t) four
possible values. It can represent the current or voltage at the
terminals of an inductor or the current or voltage at the
terminals of a capacitor.
   We know that the differential equation describing any one of
the four circuits in Fig. 7.24 takes the form

   Where the value of K can be zero, the final value of x will be
constant; that is, the final value must satisfy Eq. 7.54, and,
when x reaches its final value, the derivative dx/dt must be
zero.
   After deriving general solution for natural and step responses
of RL and RC circuits. Time t0 = time of change or switching
DETAILED STEPS INVOLVED TO COMPUTE STEP
AND NATURAL RESPONSES
   When computing the step and natural
responses of circuits, it may help to follow
these steps:….
   Identify the variable of interest for the circuit. For RC
circuits, it is most convenient to choose the capacitive
voltage; for RL circuits, it is best to choose the inductive
current.
   Determine the initial value of the variable, which is its
value at to. Note that if you choose capacitive voltage or
inductive current as your variable of interest, it is not
necessary to distinguish between t = t0- and t = t0.+ This is
because they both are continuous variables abler. If you
choose another variable, you need to remember that its
initial value is defined at t = to+.
   Calculate the final value of the variable, which is its value
as t —> ∞.
   Calculate the time constant for the circuit.

   Note: The expressions t0- and t0+ are analogous to 0- and 0+.
Thus x(t0-) is the limit of x(t) as t —> t0 to from the left, and
x(t0+) is the limit of x(t) as t —> to from the right.
7.5 SEQUENTIAL SWITCHING
   Whenever switching occurs more than once in a circuit, we have sequential switching.
    For example, a single, two-position switch may be switched back and forth, or multiple switches may
be opened or closed in sequence. The time reference for all switching's cannot be t = 0. We determine
the voltages and currents generated by a switching sequence by using the tech-niques described
previously in this chapter. We derive the expressions for v(t) and i(t) for a given position of the switch
or switches and then use these solutions to determine the initial conditions for the next position of the
switch or switches.
   With sequential switching problems, a premium is placed on obtaining the initial value x(to). Recall
that anything but inductive currents and capacitive voltages can change instantaneously at the time
of switching. Thus solving first for inductive currents and capacitive voltages is even more pertinent
in sequential switching problems. Drawing the circuit that pertains to each time interval in such a
problem is often helpful in the solution process.
   Refer Text examples 7.11, 7.12
7.6 UNBOUNDED RESPONSE
   A circuit response may grow, rather than decay, exponentially with time. This type of response, called
an unbounded response, is possible if the circuit contains dependent sources. In that case, the
Thévenin equivalent resistance with respect to the terminals of either an inductor or a capacitor may
be negative. This negative resistance generates a negative time con-stant, and the resulting currents
and voltages increase without limit.
   In an actual circuit, the response eventually reaches a limiting value when a component breaks down
or goes into a saturation state, prohibiting fur-ther increases in voltage or current.
   When we consider unbounded responses, the concept of a final value is confusing. Hence, rather than
using the step response solution given in Eq. 7.59, we derive the differential equation that describes
the circuit containing the negative resistance and then solve it using the separation of variables
technique.
   Refer Text example 7.13
7.7 INTEGRATING AMPLIFIER
   We assume that the operational amplifier is ideal. Thus we
7.7 INTEGRATING AMPLIFIER
   Vo(to) is the initial o/p voltage which is 0 , Further deriving
we get,
7.7 INTEGRATING AMPLIFIER
   The output voltage is proportional to the integral of the
input voltage only if the op amp operates within its linear
range, that is, if it doesn't saturate.
   Refer Text Examples 7.14 and 7.15 further illustrate the
analysis of the integrating amplifier
SUMMARY
   A first-order circuit may be reduced to a Thévenin (or Norton) equivalent connected to either a single
equiva-lent inductor or capacitor.
   The natural response is the currents and voltages that exist when stored energy is released to a
circuit that contains no independent sources.
   The time constant of an RL circuit equals the equiva-lent inductance divided by the Thévenin
resistance as viewed from the terminals of the equivalent inductor.
   The time constant of an RC circuit equals the equiva-lent capacitance times the Thévenin resistance
as viewed from the terminals of the equivalent capacitor.
   The step response is the currents and voltages that result from abrupt changes in dc sources
connected to a circuit. Stored energy may or may not be present at the time the abrupt changes take
place.
   The solution for either the natural or step response of I both RL and RC circuits involves finding the
initial and final value of the current or voltage of interest and the time constant of the circuit.
Equations 7.59 and 7.60 summarize this approach.
   Sequential switching in first-order circuits is analyzed by dividing the analysis into time intervals
correspon-ding to specific switch positions. Initial values for a particular interval are determined from
the solution corresponding to the immediately preceding interval.
   An unbounded response occurs when the Thévenin resistance is negative, which is possible when
the first-order circuit contains dependent sources.
   An integrating amplifier consists of an ideal op amp, a capacitor in the negative feedback branch, and
a resis-tor in series with the signal source. It outputs the inte-gral of the signal source, within
specified limits that avoid saturating the op amp.

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