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Mid-II - Announcements
Course
: Chapters 12 & 13
Paper
: 4 questions (2 from each chapter)
Difficulty Level
: Tough
Formula Sheet(s
) will be provided attached to question papers.
Books/Notes NOT ALLOWED
.
Marking
: No marks for matching formulas to questions. Concise and
clear-cut solution is required.
Obviously
partial attempts
will get partial marks but that partial attempt
should point towards a correct solution.
Myth Busted:
Writing something will get some marks!
Mistakes
: Stupid mistakes (like I = Z/V) are heavily punished. Calculation
mistakes are taken lightly as they can be done by the most genius!
An unfinished attempt can get
maximum marks
. A finished attempt may
get lesser marks due to some mistakes.
Transfer function and the Steady-State Sinusoidal response
Transfer function is used to relate the steady-stateresponse to the excitation source
x(t) = ACos(
t +
x(t) = ACos
tCos
tSin
X(s) =(ACos
)s/(s
2
2
) - (ASin
/(s
2
2
=A(sCos
Sin
)/(s
2
2
Y(s) = H(s)A(sCos
Sin
)/(s
2
2
Y(s) = K
1
/(s-j
) + K
1
/(s+j
) + terms generated by
poles of H(s)
Transfer function and the Steady-State Sinusoidal response
y(t) = A|H(j
)|Cos[
t +
]
The terms generated by poles of H(s) have been ignored
Amplitude of the response equals the amplitude of thesource ‘A’ times the magnitude of the transfer functionH(j
)
Phase angle of the response is equal to the sum of phaseangles of the source and transfer function
Transfer function is evaluated at source frequency
)
( j
j
j
1
e )
j (
H
A^2
Ae
e ) j ( H K
Example
v
g
Steady Stateexpression for v
o
Frequency is 5000rad/s
6
2
x
s
s
s (
s (
H
6
6
x
j (
x
j (
j (
H
t
Cos
0
Impulse function in circuitanalysis
Impulse functions occur in circuits
switching operation
circuit is excited by an impulsive source
Laplace Transform used to predict
The impulsive currents and voltages due toswitching
Response of a circuit to an impulsive source
Switching Operation
C
1
is charged to v
o
C
2
is not charged
At t = 0 i(t)?
C
e
is series combination
of C
1
& C
2
sC /
1 (
sC /
1 (
R
s /
V
I
2
O^1
RC
s
R
V
I
e
O
Switching Operation
Area under i vs. t curve represents the total chargetransferred to C
2
(Q=IT)
Total charge transferred to C
2
is independent of R
As R
0
i
V
O
C
e
(t)
When R = 0 a finite amount of charge is transferred to C
2
instantaneously
e
O
0
RC / t
O
C V ) t ( d e R
V
q
e
Switching Operation
Contradiction
when R is absent and switch is closed, voltageacross C
2
is not V
O
but V
2
= C
1
V
O
/(C
1
+C
2
Using Laplace transform
i = Ce
V
O
(t)
O
e
2
1
O
2
1
2
1
O
V
C
C
C
V
C
C
sC /
1 (
sC /
1 (
s /
V
I
Switching Operation
vo
contains an impulse term as second term
is an improper rational function
s
) 5. 7 s (
s ( s
) 5. 7 s (
V
O
s
s 60
s
s
s 60
V
O
) t ( u ) e
t ( δ
v
t 5
O
Switching Operation
At t = 0 when the switchis opened
) t ( u ) e 2 4 ( i
t 5
s
4 s
s
s
4 s
s
6 ) 5 s ( s
s 5
s /
I
Impulsive Sources
Impulse function occur in
Circuit Sources
Circuit Responses
Impulsive source driving a circuit
Imparts energy to the circuit instantaneously
The natural response of the circuit determines theoutput of the circuit
Analogy to striking a bell
Impulsive Source
An impulsive source V
O
(t) applied to a series
connection of a resistor and inductor
There is no voltage drop across R and the impulsivevoltage appears across L
Current through circuit is
i(
) = V
o
/L A
Energy stored in inductor is ½LI
2
= ½(V
o
2
/L) J
Current decays to zero according to the naturalresponse of the circuit
t 0
O
dx )
x ( δ
V
1 L
i
) t ( u e L V i
) R / L /( t
O
Multiple Impulses
An external signal impulsesource is appliedsimultaneously with aninternally generated impulsesignal
A 50
(t) source is
connected in series with the100 V source produces animpulse signal at t = 0
Switch is opened at t = 0 togenerate an internallygenerated impulse signal
δ
Multiple Impulses
At t < 0
i
1
= 10 A, i
2
= 0 A
s
s
I
s
4 s
s
s
4 s
s
s ( s
) t ( u ) e
i
t 5
s 60
s
s ( s
) 5. 7 s (
s
) 5. 7 s (
I
s 2
V
O
t (
u )
e
t ( δ
v
t 5
o
δ
