Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

FREQUENCY
RESPONSE, BODE
PLOT, RESONANCE
Gavax Joshi
1

2

Frequency Response
Def: Transfer function is the ratio of the
output voltage phasor to the input
voltage phasor for any frequency q (in
radians per second)
Note: The definition of transfer function is almost the same as
voltage gain
...








The function H(jω) called the transfer function determines the
amplitude and phase angle of the system
The gain and phase functions are frequency dependent and
together reveal how a circuit responds to input sinusoids of
different frequencies
...


3

4

TYPICAL FRQUENCY RESPONSE
PLOT
The frequency associated with
the transition from a passband
to an adjacent stopband is
called the cutoff frequency,
denoted as ωC or fC
...

The range of frequencies
with significantly reduced
gain is called a
stopband
...

The first row in Table shows that the
impedance of a resistor does not
change with frequency
...

In the last row we see that an inductor
acts like a short circuit at dc and an
open circuit at infinite frequency
...

For this reason frequency-response
plots almost always use a logarithmic
scale for the frequency variable
...

Bode diagrams are plots of the gain
|H(jω)| dB and phase θ(ω)versus
log-frequency
...


|H(jω)|

|H(jω)| dB

EXAMPLE

10

EXAMPLE

11

12

BODE PLOT/ BODE DIAGRAMS








The use of log-frequency scales involves some special
terminology
...

For example, the frequency range from 10 Hz to 20 Hz is
one octave, as is the range from 20 MHz to 40 MHz
...
3 GHz to 3GHz
...


Bode plot

13

Bode plot

14

Bode plot

15

Bode plot

16

Bode Plot (Magnitude
and phase response) for
transfer function of the
form
TF =

1
1+𝑗ωτ

the value of tau (τ) is
simply a time constant
determined
by
the
components
of
the
circuit
...


 We

will see that “frequency is being
adjusted until resonance occurs” ; it is
also possible to change the physical
parameters of the system and achieve
resonance but this may not be so easily
achievable
...

Thus, a network is in resonance (or resonant)
when the voltage and current at the network
input terminals are in phase
...


RESONANCE
Parallel RLC

19

I 1 
1 
Y = = + jωC −

V R 
ωL 
At resonance the imaginary term is zero, therefore

RESONANCE
Parallel RLC
 The

20

resonance frequency is

RESONANCE
Parallel RLC

21

22

RESONANCE and the voltage
response: ||EL RLC

23

@RESONANCE: ||EL RLC
 Current

is also present in L and C at
resonance

 For

the inductor

 For

the capacitor

24

@RESONANCE: ||EL RLC

25

The Quality Factor Q: ||EL RLC
The height of the response curve depends
only upon the value of R, the width of the
curve (BW) or the steepness of the sides
depends upon the other two element values
also
...


maximum energy stored
Q = quality factor ≡ 2 π
total energy lost per period

For the parallel RLC circuit, the
quality factor at resonance is

Q0 = 2π f0RC = ω0RC

26

The Quality Factor Q: ||EL RLC
PR total average power absorbed by all the resistor

27

The Quality Factor Q: ||EL RLC
Thus , Quality factor is equal to

Decreasing ‘R’ decreases ‘Q’
Increasing ‘C’ increases ‘Q’
Increasing ‘L‘ decreases ‘Q’

28

BANDWIDTH: ||EL RLC
(half-power) bandwidth of a resonant
circuit is defined as the difference of
these two half-power frequencies
...
Thus

30

BANDWIDTH: ||EL RLC
in terms of Q

We can also show that ω0 is exactly equal to the geometric
mean of the half-power frequencies

31

RESONANCE: Series RLC
V = VM ∠0
V

R

+

L
I

_

The input impedance is given by:

Z=
R + j ( wL −

1
)
wC

The magnitude of the circuit current is;
=
I |=
I |

Vm
R 2 + ( wL −

1 2
)
wC

C

32

RESONANCE: Series RLC
 By

again applying Ohm’s law, we find the
voltage across each of the elements in
the circuit as follows:

33

RESONANCE: Series RLC
 We

determine the average power
dissipated by the resistor and the reactive
powers of the inductor and capacitor as
follows:

34

RESONANCE: Series RLC
Resonance occurs when,

1
wL =
wC
At resonance we designate w as wo and write;

1
wo =
LC
This is an important equation to remember
...


35

RESONANCE: Series RLC

Due to the changing impedance of the circuit, we conclude
that if a constant amplitude voltage is applied to the series
resonant circuit, the current and power of the circuit will not
be constant at all frequencies
...


|I|

Vm
R

Vm
2R
Half power point

w1 wo w2

Bandwidth:

BW = wBW = w2 – w1

w

37

RESONANCE: Series RLC
2

The peak power delivered to the circuit is; P = Vm
R

...


wo
BW =
Q

R
L

40

RESONANCE: Series RLC




For any resonant circuit, we define the quality factor, Q,
as the ratio of reactive power to average power,
namely,

Because the reactive power of the inductor is equal to
the reactive power of the capacitor at resonance, we
may express Q in terms of either reactive power
...


BW
w2 =
wo +
2

42

RESONANCE: Series RLC
An Observation
By using Q = woL/R in the equations for w1and w2 we have;
2
 −1



1
w1 =wo 
+ 
+ 1

 2Q

 2Q 



and
2
 1



1
w2 =wo 
+ 
+ 1

2Q 
 2Q





43

RESONANCE: Series RLC
SELECTIVITY If the bandwidth of a circuit is kept very

narrow, the circuit is said to have a high
selectivity, since it is highly selective to
signals occurring within a very narrow range
of frequencies
...


44

RESONANCE: Series RLC
SELECTIVITY
The elements of a series resonant circuit
determine not only the frequency at
which the circuit is resonant, but also the
shape (and hence the bandwidth) of the
power response curve
...
We
find that by increasing the ratio of L/C, the
sides of the power response curve
become steeper
...
Inversely,
decreasing the ratio of LC causes the
sides of the curve to become more
gradual, resulting in an increased
bandwidth
...

Figure 21–11 shows how
the shape of the selectivity
curve is dependent upon
the value of resistance
...


SUMMARY

46

Consider the circuits shown below:
V
I

R

C

L

R
V

L

I

C

1
1 
I = V  + jwC +

R
jwL




1 
V = I  R + jwL +

jwC



USING DUALITY
1
1 
I = V  + jwC +

R
jwL



47


1 
V = I  R + jwL +

jwC



We notice the above equations are the same provided:

I

V

R

1
R

L

C

USING DUALITY

48

What this means is that for all the equations we have derived
for the parallel resonant circuit, we can use for the series
resonant circuit provided we make the substitutions:

R

replaced be

L replaced by
C replaced by

1
R

C
L

Summary

49

50

APPROXIMATE EXPRESSIONS
Q >= 10

EXAMPLE 1

51

Determine the resonant frequency for the circuit below
...


52

EXAMPLE 1

(− w LRC + jwL )
(1 − w LC ) + jwRC
2

2

For zero phase;

wL
wRC
=
(− w LCR ) (1 − w LC
2

2

This gives;

w LC − w R C =1
2

2

2

2

or

1
w=
( LC − R C )
o

2

2

EXAMPLE 2

53

A series
parallel
RLC
RLC
resonant
resonant
circuit
circuit
has
has
a resonant
a resonant
frequency
frequency
admittance
admittance
of of
2x10-2 S(mohs)
...
Calculate the values of R, L, and C
...

First, R = 1/G = 1/(0
...

Second, from Q =
find L = 0
...

Third, we can use

w L , we solve for L, knowing Q, R, and w to
o
R
O

Q
50
C=
=
=100 µ F
w R 10,000 x50
O

54

EXAMPLE 2

Fourth: We can use

wo 1x10 4
BW = =
= 200 rad / sec
Q
50

and
Fifth: Use the approximations;
w1 = wo - 0
...
5wBW = 10,000 + 100 = 10,100 rad/sec

55

SERIES TO PARALLEL RL AND RC
CONVERSION
equivalence is only valid at a single frequency, ω
...


59

PARALLEL TO SERIES RL
CONVERSION
 If

we are given a parallel RL network, it is
possible to show that the conversion to an
equivalent series network is accomplished
by applying the following equations:

60

CONVERSION FOR RC CIRCUIT
 Although

we have performed conversions
between series and parallel RL circuits, it is
easily shown that if the reactive element is
a capacitor, the conversions apply
equally well
...
The Q of the network is
determined by the ratios



---