Transient Excitation of First-Order Circuits

1
Transient Excitation of First-Order Circuits

1. What is transient excitation and why is it
   important?
2. What is a first-order circuit?
3. What are natural response and step response?
4. Transients in RL circuits (briefly)
5. Transients in RC circuits ? application to
   computer circuits

2
Types of Circuit Excitation
Steady-State Excitation
OR
(DC Steady-State)
Sinusoidal (Single- Frequency) Excitation ?AC
Steady-State
Transient Excitation
3
First-Order Circuits

• A circuit that contains only sources, resistors
  and an inductor is called an RL circuit.
• A circuit that contains only sources, resistors
  and a capacitor is called an RC circuit.
• RL and RC circuits are called first-order
  circuits because their voltages and currents are
  described by first-order differential equations.

R
R
i
i

vs

vs
L
C
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Review (Conceptual)

• Any first-order circuit can be reduced to a
  Thévenin (or Norton) equivalent connected to
  either a single equivalent inductor or capacitor.
• In steady state, an inductor behaves like a short
  circuit
• In steady state, a capacitor behaves like an open
  circuit

RTh

VTh
C
L
RN
IN
5

• The natural response of an RL or RC circuit is
  its behavior (i.e., current and voltage) when
  stored energy in the inductor or capacitor is
  released to the resistive part of the network
  (containing no independent sources).
• The step response of an RL or RC circuit is its
  behavior when a voltage or current source step is
  applied to the circuit, or immediately after a
  switch state is changed.

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Natural Response of an RL Circuit

• Consider the following circuit, for which the
  switch is closed for t lt 0, and then opened at t
  0
• Notation
• 0 is used to denote the time just prior to
  switching
• 0 is used to denote the time immediately after
  switching
• The current flowing in the inductor at t 0 is
  Io

t 0
i
v
L
Ro
R
Io
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Solving for the Current (t ? 0)

• For t gt 0, the circuit reduces to
• Applying KVL to the LR circuit yields first-order
  D.E.
• Solution

i
v
L
Ro
R
Io
I0e-(R/L)t
8
Solving for the Voltage (t gt 0)
v
L
Ro
R
Io

• Note that the voltage changes abruptly (step
  response)

-

v
0
)
0
(
-


gt
)
/
(
t
L
R
Re
I
iR
t
v
t
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(

0,
for
o


Þ
I0R
v
)
0
(


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Time Constant t

• In the example, we found that
• Define the time constant
• At t t, the current has reduced to 1/e (0.37)
  of its initial value.
• At t 5t, the current has reduced to less than
  1 of its initial value.

(sec)
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Transient response of RC circuits and application
to computer circuits driven by binary voltage
pulses
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Capacitors and Stored Charge

• So far, we have assumed that electrons keep on
  moving around and around a circuit.
• Current doesnt really flow through a
  capacitor. No electrons can go through the
  insulator.
• But, we say that current flows through a
  capacitor. What we mean is that positive charge
  collects on one plate and leaves the other.
• A capacitor stores charge. Theoretically, if we
  did a KCL surface around one plate, KCL could
  fail. But we dont do that.
• When a capacitor stores charge, it has nonzero
  voltage. In this case, we say the capacitor is
  charged. A capacitor with zero voltage has no
  charge differential, and we say it is
  discharged.

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Capacitors in circuits

• If you have a circuit with capacitors, you can
  use KVL and KCL, nodal analysis, etc.
• The voltage across the capacitor is related to
  the current through it by a differential equation
  instead of Ohms law.

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CAPACITORS
(
C
i(t)
capacitance is defined by
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Charging a Capacitor with a constant current
V(t)

?
(
C
i
voltage
time
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Discharging a Capacitor through a resistor

?
V(t)
i
R
C
i
This is an elementary differential equation,
whose solution is the exponential
Since
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Voltage vs time for an RC discharge
Voltage
Time
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Natural Response of an RC Circuit

• Consider the following circuit, for which the
  switch is closed for t lt 0, and then opened at t
  0
• Notation
• 0 is used to denote the time just prior to
  switching
• 0 is used to denote the time immediately after
  switching
• The voltage on the capacitor at t 0 is Vo

t 0
Ro
v
?
R
Vo
C
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Solving for the Voltage (t ? 0)

• For t gt 0, the circuit reduces to
• Applying KCL to the RC circuit
• Solution

i
v
Ro
?
C
R
Vo
19
Solving for the Current (t gt 0)
i
v
Ro
?
C
R
Vo

• Note that the current changes abruptly

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Time Constant t

• In the example, we found that
• Define the time constant
• At t t, the voltage has reduced to 1/e (0.37)
  of its initial value.
• At t 5t, the voltage has reduced to less than
  1 of its initial value.

(sec)
21
RC Circuit Model for a Digital Logic Circuit

• The capacitor is used to model the response of a
  digital circuit to a new voltage input
• The digital circuit is modeled by
• a resistor in series with a capacitor.
• The capacitor cannot
• change its voltage instantly,
• as charges cant jump instantly
• to the other plate, they must go through the
  circuit!

R
Vout


Vin
Vout
C
_
_
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RC Circuits Abound in Computers
We compute with pulses We send beautiful pulses
in
But we receive lousy-looking pulses at the output
Capacitor charging effects are responsible!

• Every node in a circuit has natural capacitance,
  and it is the charging of these capacitances that
  limits real circuit performance (speed)

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RC Circuit Model

• Every digital circuit has natural resistance and
  capacitance. In real life, the resistance and
  capacitance can be estimated using
  characteristics of the materials used and the
  layout of the physical device.
• The value of R and C
• for a digital circuit
• determine how long it will
• take the capacitor to change its
• voltagethe gate delay.

R
Vout


Vin
Vout
C
_
_
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RC Circuit Model
R

• With the digital context in mind, Vin will
  usually be a time-varying voltage that switches
  instantaneously between logic 1 voltage and logic
  0 voltage.
• We often represent this switching voltage with a
  switch in the circuit diagram.

Vout


Vin
Vout
C
_
_
t 0
i
Vout
?
Vs 5 V
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Analysis of RC Circuit
R
Vout

• By KVL,
• Using the capacitor I-V relationship,
• We have a first-order linear differential
  equation for the output voltage

Vin
Vout
I
C
_
_
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Analysis of RC Circuit
R
Vout

• What does that mean?
• One could solve the
• differential equation to get

Vin
Vout
I
C
_
_
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Insight

• Vout(t) starts at Vout(0) and goes to Vin
  asymptotically.
• The difference between the two values decays
  exponentially.
• The rate of convergence depends on RC. The
  bigger RC is, the slower the convergence.

Vout
Vout
Vout(0)
Vin
bigger RC
Vout(0)
Vin
0
0
time
time
0
0
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Time Constant

• The value RC is called the time constant.
• After 1 time constant has passed (t RC), the
  above works out to
• So after 1 time constant, Vout(t) has completed
  63 of its transition, with 37 left to go.
• After 2 time constants, only 0.372 left to go.

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Transient vs.Steady-State
R
Vout


Vin
Vout
I
C
_
_

• When Vin does not match up with Vout , due to an
  abrupt change in Vin for example, Vout will begin
  its transient period where it exponentially
  decays to the value of Vin.
• After a while, Vout will be close to Vin and be
  nearly constant. We call this steady-state.
• In steady state, the current through the
  capacitor is (approx) zero. The capacitor
  behaves like an open circuit in steady-state.
• Why? I C dVout/dt, and Vout is constant in
  steady-state.

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General RC Solution

• Every current or voltage (except the source
  voltage) in an RC circuit has the following form
• x represents any current or voltage
• t0 is the time when the source voltage switches
• xf is the final (asymptotic) value of the
  current or voltage
• All we need to do is find these values and plug
  in to solve for any current or voltage in an RC
  circuit.

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Solving the RC Circuit

• We need the following three ingredients to fill
  in our equation for any current or voltage
• x(t0) This is the current or voltage of
  interest just after the voltage source switches.
  It is the starting point of our transition, the
  initial value.
• xf This is the value that the current or
  voltage approaches as t goes to infinity. It is
  called the final value.
• RC This is the time constant. It determines
  how fast the current or voltage transitions
  between initial and final value.

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Finding the Initial Condition

• To find x(t0), the current or voltage just after
  the switch, we use the following essential fact
• Capacitor voltage is continuous it cannot jump
  when a switch occurs.
• So we can find the capacitor voltage VC(t0) by
  finding VC(t0-), the voltage before switching.
• We can assume the capacitor was in steady-state
  before switching. The capacitor acts like an
  open circuit in this case, and its not too hard
  to find the voltage over this open circuit.
• We can then find x(t0) using VC(t0) using KVL
  or the capacitor I-V relationship. These laws
  hold for every instant in time.

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Finding the Final Value

• To find xf , the asymptotic final value, we
  assume that the circuit will be in steady-state
  as t goes to infinity.
• So we assume that the capacitor is acting like an
  open circuit. We then find the value of current
  or voltage we are looking for using this
  open-circuit assumption.
• Here, we use the circuit after switching along
  with the open-circuit assumption.
• When we found the initial value, we applied the
  open-circuit assumption to the circuit before
  switching, and found the capacitor voltage which
  would be preserved through the switch.

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Finding the Time Constant

• It seems easy to find the time constant it
  equals RC.
• But what if there is more than one resistor or
  capacitor?
• R is the Thevenin equivalent resistance with
  respect to the capacitor terminals.
• Remove the capacitor and find RTH. It might help
  to turn off the voltage source. Use the circuit
  after switching.

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Natural Response Summary

• RL Circuit
• Inductor current cannot change instantaneously
• time constant

• RC Circuit
• Capacitor voltage cannot change instantaneously
• time constant

i
v
R
L
R
C
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RC Circuit Transient Analysis Example

• The switch is closed for t lt 0, and then opened
  at t 0.
• Find the voltage vc(t) for t 0.

t 0
3 kW
i
vc
?
2 kW
5 V
10 mF

• Determine the initial voltage vc(0)

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3 kW
i
vc
?
2 kW
5 V
10 mF
2. Determine the final voltage vc(8) 3.
Calculate the time constant t
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