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Step natural

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Natural and step response Ahsan Khawaja
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Page 1: Step natural

Natural and step response

Ahsan Khawaja

Page 2: Step natural

Cont. inductor

• Where V= voltage between inductor in (volt)L= inductor in (henery) = rate of change of current flow in amper = rate of change of time in second

Vt

Ldi

d

dtdi

Page 3: Step natural

Cont. capacitor

• Mathematical relation

• Wherei= current that in capacitor in (ampere)C= capacitance in (farad) = rated of change of voltage. = rate of change of time in second

it

Cdv

d

.

dv

dvdt

Page 4: Step natural

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.

–+

vs L

R

–+

vs C

R

i i

Page 5: Step natural

6 Different First-Order CircuitsThere are six different STC circuits. These

are listed below.• An inductor and a resistance (called RL

Natural Response).• A capacitor and a resistance (called RC

Natural Response).• An inductor and a Thévenin equivalent

(called RL Step Response). • An inductor and a Norton equivalent

(also called RL Step Response).• A capacitor and a Thévenin equivalent

(called RC Step Response). • A capacitor and a Norton equivalent

(also called RC Step Response).

These are the simple, first-order cases. Many circuits can be reduced to one of these six cases. They all have solutions which are in similar forms.

LX RX RXCX

+

-

vS

RXLX

+

-

vS

RXCX

LXRXiS

RXiS CX

Page 6: Step natural

6 Different First-Order CircuitsThere are six different STC circuits. These

are listed below.• An inductor and a resistance (called RL

Natural Response).• A capacitor and a resistance (called RC

Natural Response).• An inductor and a Thévenin equivalent

(called RL Step Response). • An inductor and a Norton equivalent

(also called RL Step Response).• A capacitor and a Thévenin equivalent

(called RC Step Response). • A capacitor and a Norton equivalent

(also called RC Step Response). These are the simple, first-order cases. They all have solutions which are in similar forms.

LX RX RXCX

+

-

vS

RXLX

+

-

vS

RXCX

LXRXiS

RXiS CX

These are the simplest cases, so we handle them first.

Page 7: Step natural

The Natural Response of a Circuit

• The currents and voltages that arise when energy stored in an inductor or capacitor is suddenly released into a resistive circuit.

• These “signals” are determined by the circuit itself, not by external sources!

Page 8: Step natural

Step Response

• The sudden application of a DC voltage or current source is referred to as a “step”.

• The step response consists of the voltages and currents that arise when energy is being absorbed by an inductor or capacitor.

Page 9: Step natural

Circuits for Natural Response

• Energy is “stored” in an inductor (a) as an initial current.

• Energy is “stored” in a capacitor (b) as an initial voltage.

Page 10: Step natural

General Configurations for RL

• If the independent sources are equal to zero, the circuits simplify to

Page 11: Step natural

Natural Response of an RL Circuit

• Consider the circuit shown.• Assume that the switch has been closed “for a

long time”, and is “opened” at t=0.

Page 12: Step natural

What does “for a long time” Mean?

• All of the currents and voltages have reached a constant (dc) value.

• What is the voltage across the inductor just before the switch is opened?

Page 13: Step natural

Just before t = 0

• The voltage across the inductor is equal to zero.

• There is no current in either resistor.• The current in the inductor is equal to IS.

Page 14: Step natural

Just after t = 0• The current source and its parallel resistor R0

are disconnected from the rest of the circuit, and the inductor begins to release energy.

Page 15: Step natural

The Source-Free RL Circuit

• A first-order RL circuit consists of a inductor L (or its equivalent) and a resistor (or its equivalent)

0 RL vvBy KVL

0 iRdt

diL

Inductors law Ohms law

dtL

R

i

di LtReIti /

0 )(

Page 16: Step natural

The Source-FreeRC Circuit

• A first-order circuit is characterized by a first-order differential equation.

• Apply Kirchhoff’s laws to purely resistive circuit results in algebraic equations. • Apply the laws to RC and RL circuits produces differential equations.

Ohms law Capacitor law

0 dt

dvC

R

v0 CR iiBy KCL

Page 17: Step natural

Natural Response of an RL Circuit• Consider the following circuit, for which the switch is closed

for t < 0, and then opened at t = 0:

Notation:0– is used to denote the time just prior to switching0+ is used to denote the time immediately after switching

• The current flowing in the inductor at t = 0– is Io

LRo RIo

t = 0 i +

v

Page 18: Step natural

Solving for the Current (t 0)• For t > 0, the circuit reduces to

• Applying KVL to the LR circuit:

• Solution:

LRo RIo

i +

v

tLReiti )/()0()( = I0e-(R/L)t

Page 19: Step natural

Solving for the Voltage (t > 0)

• Note that the voltage changes abruptly:

tLRoeIti )/()(

LRo RIo

+

v

I0Rv

ReIiRtvt

vtLR

o

==>

=

+

-

-

)0(

)( 0,for

0)0()/(

Page 20: Step natural

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.

R

L

tLRo

tLRo ReItveIti )/()/( )( )( and

(sec)

Page 21: Step natural

The Source-Free RL Circuit

/0)( teIti

R

L

A RL source-free circuit

where /0)( teVtv RC

A RC source-free circuit

where

Comparison between a RL and RC circuit

Page 22: Step natural

The Complete Solution

0( ) , 0

Rt

Li t I e t

Page 23: Step natural

The voltage drop across the resistor

0

0

, 0 .

(0 ) 0

(0 )

Rt

L

v iR

v I Re t

v

v I R

Page 24: Step natural

The Power Dissipated in the Resistor

2

2

22

0, 0

Rt

L

vp vi i R

R

p I Re t

Page 25: Step natural

The Energy Delivered to the Resistor

22

00 0

22

0

2

0

1(1 ), 0.

2

1,

2

Rt t xL

Rt

L

w pdx I Re dx

w I R e tRL

t w LI

Page 26: Step natural

The Source-Free RL 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% of its initial value.

• i(t) decays faster for small t and slower for large t.• The general form is very similar to a RC source-free circuit.

/0)( teIti

R

L

A general form representing a RL

where

Page 27: Step natural

The Source-Free RC Circuit

• The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.

• The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.

• v decays faster for small t and slower for large t.

CRTime constantDecays more slowly

Decays faster

Page 28: Step natural

Natural Response SummaryRL Circuit

• Inductor current cannot change instantaneously

• time constant

RC Circuit

• Capacitor voltage cannot change instantaneously

• time constantR

L

/)0()(

)0()0(teiti

ii

R

i

L

+

v

RC

/)0()(

)0()0(tevtv

vv

RC

Page 29: Step natural

General Solution for Natural and Step Responses of RL and RC Circuits

0( )

0( ) [ ( ) ]t t

f fx t x x t x e

Final Value

Initial Value

Time Constant

Determine the initial and final values of the variable of interest and the time constant of the circuit.

Substitute into the given expression.

Page 30: Step natural

Example

V190 V

R1

400kOhm

C0.5uF

R260 Ohm

R3

20 Ohm

V240 V

J1

Key = Space

b a

• What is the initial value of vC?

• What is the final value of vC?• What is the time constant when the switch is in

position b?• What is the expression for vC(t) when t>=0?

+

vC(t)

-

Page 31: Step natural

Initial Value of vC

V190 V

R1

400kOhm

C0.5uF

R260 Ohm

R3

20 Ohm

V240 V

J1

Key = Space

b a

• The capacitor looks like an open circuit, so the voltage @ C is the same as the voltage @ 60Ω.

60(0) 40 30

20 60Cv V V

+

vC(0)

-

+

V60

-

Page 32: Step natural

Final Value of vC

• After the switch is in position b for a long time, the capacitor will look like an open circuit again, and the voltage @ C is +90 Volts.

V190 V

R1

400kOhm

C0.5uF

R260 Ohm

R3

20 Ohm

V240 V

J1

Key = Space

b a

+

vC(∞)

-

Page 33: Step natural

The time constant of the circuit when the switch is in position b

• The time constant τ = RC = (400kΩ)(0.5μF)• τ = 0.2 s

V190 V

R1

400kOhm

C0.5uF

R260 Ohm

R3

20 Ohm

V240 V

J1

Key = Space

b a

Page 34: Step natural

The expression for vC(t) for t>=0

0.2

5

( ) ( ) [ (0) ( )]

( ) 90 [ 30 90]

( ) 90 120

t

C C C C

t

C

tC

v t v v v e

v t e

v t e V

Page 35: Step natural

The expression for i(t) for t>=0

• Initial value of i is (90 - - 30)V/400kΩ = 300μA• Final value of i is 0 – the capacitor charges to +90 V

and acts as an open circuit• The time constant is still τ = 0.2 s

V190 V

R1

400kOhm

C0.5uF

R260 Ohm

R3

20 Ohm

V240 V

J1

Key = Space

b a

i(t)-

30V

+

Page 36: Step natural

The expression for i(t) (continued)

6 0.2

5

( ) ( ) [ (0 ) ( )]

( ) 0 [300 10 0]

( ) 300

t

t

t

i t i i i e

i t e

i t e A

Page 37: Step natural

How long after the switch is in position b does the capacitor voltage equal 0?

5

5

5

( ) 90 120 0

120 90

90

12090

5 ln 0.28768120

0.05754 57.54

tC

t

t

v t e

e

e

t

t s ms

Page 38: Step natural

Plot vC(t)

Page 39: Step natural

Plot i(t)


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