Frequency Response
(I&N Chap 12)
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Introduction & TFs
Decibel Scale & Bode Plots
Resonance
Scaling
Filter Networks
Applications/Design
Based on slides by J. Yan
Slide 3.1
Network Scaling
Considering a Bode plot, there are two ways to scale.
1. For magnitude scaling, the magnitude plot is shifted
up or down while the phase plot is unchanged.
2. For frequency scaling, both magnitude and phase
plots are shifted left or right.
Let’s see how to change component values for resonant
circuits to achieve these types of scaling.
Based on slides by J. Yan
Slide 3.2
Magnitude/Impedance Scaling
To scale the equivalent impedance by a factor K M ,
simply scale each component impedance by K M .
Resistors: Z R = R → R ' = K M R
1
C
Capacitors: Z C =
→C'=
sC
KM
Inductors: Z L = sL → L ' = K M L
For a series RLC circuit, what happens to Q and ω0 with magnitude scaling?
Based on slides by J. Yan
Slide 3.3
Frequency Scaling
To shift the plots by a frequency factor K F , we wish
the impedance at the scaled frequency ω′ = K F ω to
be the same as at the original frequency ω.
Resistors: Z R = R → R ' = R
Capacitors: Z C =
1
1
C
=
→C'=
jωC
jω′C′
KF
Inductors: Z L = jω L = jω′L′ → L ' =
L
KF
For a series RLC circuit, what happens to Q and ω0 with frequency scaling?
Based on slides by J. Yan
Slide 3.4
Magnitude and Frequency Scaling
To simultaneously scale impedance in both magnitude and frequency:
Km L
C
R ' = K m R, L ' =
, C' =
, ω' = K f ω
Kf
Km K f
E.g., a 3rd order Butterworth filter normalized to ωc=1rad/s is shown. Scale the circuit
to a cutoff frequency of 10kHz and use 15 nF capacitors.
Based on slides by J. Yan
Slide 3.5
Filter Networks
Filter
E.g., Low-Pass
Filter
• Filters are used to allow a particular range of
frequencies to pass through while rejecting
others.
• The type of filter can readily be determined by
the magnitude plot of the TF relating the filter
input to the filter output.
• Filters may be classified as passive or active,
depending on whether the filter has any
internal sources of energy.
Based on slides by J. Yan
Slide 3.6
Common Filter Networks
We’ll examine 4 common filters with self-explanatory names.
Low-pass filter
Band-pass filter
High-pass filter
Band-rejection filter
We first focus on passive filters. We need op-amps for active ones.
Based on slides by J. Yan
Slide 3.7
Ideal Filter Characteristics
Q: Consider the ideal low-pass
filter characteristic. Why do
you suppose this isn’t
realizable in practice? (Hint:
What TF would yield this
characteristic?)
Based on slides by J. Yan
Slide 3.8
Passive Low-Pass Filter (LPF)
A simple low-pass filter is the RC
series circuit where capacitor
voltage is taken as the output.
Based on slides by J. Yan
Slide 3.9
Passive High-Pass Filter (HPF)
A simple high-pass filter is the RC series
circuit where resistor voltage is taken as the
output.
Based on slides by J. Yan
Slide 3.10
Passive Band-Pass Filter (BPF)
This RLC circuit gives a band-pass
filter if vR is taken as the output.
Q: Give an intuitive reason for why
both low and high frequencies are
rejected.
Transfer Function :
V
R
sCR
Gv = 0 =
=
V1 R + sL + sC1 s 2 LC + sCR + 1
Gv
s = jω
=
R
R + j (ωL − ω1C )
Center Freq: ω0 =
Based on slides by J. Yan
Cut-off Frequencies:
ωlo , hi =
1
LC
m ( R / L) +
( R / L)
2
+ 4ω 20
2
Bandwidth: BW = ωhi − ωlo =
R
L
Slide 3.11
Band-Rejection Filter
This RLC circuit gives a band-rejection (aka “bandstop” or
“notch”) filter if vL+vC is taken as the output.
Transfer Function:
Cut-off Frequencies:
V0
s 2 LC + 1
Gv =
= 2
V1
s LC + sRC + 1
Center Freq: ω0 =
Based on slides by J. Yan
ωlo, hi =
m ( R / L) +
( R / L ) + 4ω 20
2
2
1
LC
Slide 3.12
Example (efts)
By now, you should realize that the RLC circuit can be used for any of our
four filter types, depending on where the output is taken.
Based on slides by J. Yan
Slide 3.13
Examples
Design an RL lowpass filter that uses a 40 mH coil and has a cutoff frequency of 5
kHz.
Design an RC highpass filter that uses a 20 µF capacitor and has a cutoff frequency
of 3 kHz.
Based on slides by J. Yan
Slide 3.14
Example: Connecting Passive Filters
What is the overall result of attaching the lowpass filter output to the highpass filter
input of the previous slide?
Based on slides by J. Yan
Slide 3.15
Example
Determine the center frequency and BW of these bandpass filters.
Based on slides by J. Yan
Slide 3.16
Passive Filter Limitations
Passive filters have some drawbacks:
• The output voltage driving a load cannot be larger than the
input (i.e., gains are no greater than unity).
• Loading effects mean that they must be reanalyzed when
interconnected.
• Designs sometimes require inductors which are expensive
and inherently lossy.
• Component values may need to be large to achieve desired
specs.
Active filters, using op-amps or transistors,
can overcome these limitations.
Based on slides by J. Yan
Slide 3.17
Active Filter Networks
Using op-amps in filters, we can:
• achieve gains greater than unity
• “buffer” the input signal to avoid loading
effects (especially helpful in “cascading”)
• design all filters with only resistors and
capacitors (no need for inductors)
Based on slides by J. Yan
Slide 3.18
Inverting Op-Amp Configuration
Many filters use the inverting op-amp configuration.
e.g., 1st Order Lowpass filter
Based on slides by J. Yan
e.g., 1st Order Highpass filter
Slide 3.19
Active Band-Pass Filter
One way to design a BPF is to cascade HPF and LPF as shown below.
NB: cf slide 3.15,
cascading passive
filters required system
reanalysis of the entire
circuit. Using active
filters, the design can
be more modular so
the poles of each stage
carry into the final T.F.
Based on slides by J. Yan
Slide 3.20
Active Band-Rejection Filter
One way to design a notch filter is to sum HPF and LPF outputs as shown.
Based on slides by J. Yan
Slide 3.21
Example
Find the voltage gain TF and identify the filter type for the circuit shown.
C2
R2
R1
+
C1
vi(t)
_
Based on slides by J. Yan
+
vo(t)
_
Slide 3.22
Example (Cascaded Filters)
10000
2.5kΩ
Consider the single-stage filter
above for radio tuning. You want
to listen to the 100 MHz station
with little interference by the 98
MHz station. By cascading
filters (connect end-to-end so the
output of one is the input of
another), greater steepness in
response can be achieved
resulting in increased selectivity.
Based on slides by J. Yan
Slide 3.23
Cascading Low-Q Filters
Bode Diagram
1
0
0.9
-10
0.8
-20
0.7
-30
Magnitude (dB)
0.6
0.5
0.4
-40
-50
0.3
-60
0.2
-70
0.1
0
10
8.76
10
8.78
10
8.8
10
8.82
10
8.84
-80
10
8.76
10
8.78
10
8.8
10
8.82
10
8.84
Frequency (rad/sec)
# of Stages
BW (MHz)
Q
Gain @98MHz
Single
2.30
43.4
53.8%
Double
1.38
72.3
28.9%
Triple
1.15
86.9
15.5%
Quadruple
0.69
144.8
8.35%
Based on slides by J. Yan
Slide 3.24
“Take Home Message”
• Passive filters can be simple to design and easy to implement.
We could design four basic filter types using resistors, inductors
and capacitors.
• Active filters utilize op-amps (or OTA) to permit gains greater
than unity, isolation for better cascading effects and designs
which avoid the necessity of inductors.
• Depending on the application, different filter types have been
designed to optimise for passband flatness (Butterworth filters),
for immediate passband-to-stopband transition (Tschebyscheff
filters), and for phase response linearity (Bessel filters).
• Active filters are typically used for frequencies below 100 kHz.
Based on slides by J. Yan
Slide 3.25