ANALOG FILTERS
FREQUENCY TRANSFORMATIONS
5.51
SECTION 5-5: FREQUENCY TRANSFORMATIONS
Until now, only filters using the lowpass configuration have been examined. In this
section, transforming the lowpass prototype into the other configurations: highpass,
bandpass, bandreject (notch) and allpass will be discussed .
Lowpass to Highpass
The lowpass prototype is converted to highpass filter by scaling by 1/s in the transfer
function. In practice, this amounts to capacitors becoming inductors with a value 1/C, and
inductors becoming capacitors with a value of 1/L for passive designs. For active designs,
resistors become capacitors with a value of 1/R, and capacitors become resistors with a
value of 1/C. This applies only to frequency setting resistor, not those only used to set
gain.
Another way to look at the transformation is to investigate the transformation in the s
plane. The complex pole pairs of the lowpass prototype are made up of a real part, α, and
an imaginary part, β. The normalized highpass poles are the given by:
and:
A simple pole, α0, is transformed to:
Lowpass zeros, ωz,lp, are transformed by:
In addition, a number of zeros equal to the number of poles are added at the origin.
After the normalized lowpass prototype poles and zeros are converted to highpass, they
are then denormalized in the same way as the lowpass, that is, by frequency and
impedance.
As an example a 3 pole 1dB Chebyshev lowpass filter will be converted to a highpass
filter.
α
α2 + β2 αHP =
βHP =
β
α2 + β2
αω,HP =
1
α0
ωZ,HP = 1
ωZ,LP
Eq. 5-43
Eq. 5-44
Eq. 5-45
Eq. 5-46
OP AMP APPLICATIONS
5.52
From the design tables of the last section:
This will transform to:
Which then becomes:
A worked out example of this transformation will appear in a later section.
A highpass filter can be considered to be a lowpass filter turned on its side. Instead of a
flat response at DC, there is a rising response of n × (20dB/decade), due to the zeros at
the origin, where n is the number of poles. At the corner frequency a response of
n × (–20dB/decade) due to the poles is added to the above rising response. This results in
a flat response beyond the corner frequency.
Lowpass to Bandpass
Transformation to the bandpass response is a little more complicated. Bandpass filters
can be classified as either wideband or narrowband, depending on the separation of the
poles. If the corner frequencies of the bandpass are widely separated (by more than 2
octaves), the filter is wideband and is made up of separate lowpass and highpass sections,
which will be cascaded. The assumption made is that with the widely separated poles,
interaction between them is minimal. This condition does not hold in the case of a
narrowband bandpass filter, where the separation is less than 2 octaves. We will be
covering the narrowband case in this discussion.
As in the highpass transformation, start with the complex pole pairs of the lowpass
prototype, α and β. The pole pairs are known to be complex conjugates. This implies
symmetry around DC (0Hz.). The process of transformation to the bandpass case is one of
mirroring the response around DC of the lowpass prototype to the same response around
the new center frequency F0.
=
βLP1 =
αLP2 =
.2257
.8822
.4513
αLP1 αHP1=
βHP1=
αHP2=
.2722
1.0639
2.2158
F01=
α=
Q=
F02=
1.0982
.4958
2.0173
2.2158
ANALOG FILTERS
FREQUENCY TRANSFORMATIONS
5.53
This clearly implies that the number of poles and zeros is doubled when the bandpass
transformation is done. As in the lowpass case, the poles and zeros below the real axis are
ignored. So an nth order lowpass prototype transforms into an nth order bandpass, even
though the filter order will be 2n. An nth order bandpass filter will consist of n sections,
versus n/2 sections for the lowpass prototype. It may be convenient to think of the
response as n poles up and n poles down.
The value of QBP is determined by:
where BW is the bandwidth at some level, typically –3dB.
A transformation algorithm was defined by Geffe ( Reference 16) for converting lowpass
poles into equivalent bandpass poles.
Given the pole locations of the lowpass prototype:
and the values of F0 and QBP, the following calculations will result in two sets of values
for Q and frequencies, FH and FL, which define a pair of bandpass filter sections.
Observe that the Q of each section will be the same.
The pole frequencies are determined by:
Each pole pair transformation will also result in 2 zeros that will be located at the origin.
QBP =
F0
BW
-α ± jβ
C = α2 + β2
D =
E =
G = E2 - 4 D2
Q =
2α
QBP
C
QBP
2 + 4
√
E + G
2 D2 √
M =
W = M + M2 - 1
FBP2 = W F0
α Q
QBP
√
FBP1 = F0W
Eq. 5-47
Eq. 5-48
Eq. 5-49
Eq. 5-50
Eq. 5-51
Eq. 5-52
Eq. 5-53
Eq. 5-54
Eq. 5-55
Eq. 5-56
Eq. 5-57
OP AMP APPLICATIONS
5.54
A normalized lowpass real pole with a magnitude of α0 is transformed into a bandpass
section where:
and the frequency is F0.
Each single pole transformation will also result in a zero at the origin.
Elliptical function lowpass prototypes contain zeros as well as poles. In transforming the
filter the zeros must be transformed as well. Given the lowpass zeros at ± jωZ , the
bandpass zeros are obtained as follows:
Since the gain of a bandpass filter peaks at FBP instead of F0, an adjustment in the
amplitude function is required to normalize the response of the aggregate filter. The gain
of the individual filter section is given by:
where:
A0 = gain a filter center frequency
AR = filter section gain at resonance
F0 = filter center frequency
FBP = filter section resonant frequency.
Again using a 3 pole 1dB Chebychev as an example:
A 3 dB bandwidth of 0.5Hz. with a center frequency of 1Hz. is arbitrarily assigned. Then:
QBP = 2
Q =
QBP
α0
M =
W = M + M2 - 1
FBP1 =
FBP2 = W F0
α Q
QBP
√
F0W
=
βLP1 =
αLP2 =
.2257
.8822
.4513
αLP1 AR = A0 1 + Q2 F0
FBP
FBP
F0
√ ( -
)
2
Eq. 5-58
Eq. 5-59
Eq. 5-60
Eq. 5-61
Eq. 5-62
Eq. 5-63
ANALOG FILTERS
FREQUENCY TRANSFORMATIONS
5.55
Going through the calculations for the pole pair the intermediate results are:
C = 0.829217 D = 0.2257
E = 4.207034 G = 4.098611
M = 1.01894 W = 1.214489
and:
FBP1 = 0.823391 FBP2 = 1.214489
QBP1 = QBP2 = 9.029157
And for the single pole:
FBP3 = 1 QBP3 = 4.431642
Again a full example will be worked out in a later section.
Lowpass to Bandreject (Notch)
As in the bandpass case, a bandreject filter can be either wideband or narrowband,
depending on whether or not the poles are separated by 2 octaves or more. To avoid
confusion, the following convention will be adopted. If the filter is wideband, it will be
referred to as a bandreject filter. A narrowband filter will be referred to as a notch filter.
One way to build a notch filter is to construct it as a bandpass filter whose output is
subtracted from the input (1 – BP). Another way is with cascaded lowpass and highpass
sections, especially for the bandreject (wideband) case. In this case, the sections are in
parallel, and the output is the difference.
Just as the bandpass case is a direct transformation of the lowpass prototype, where DC is
transformed to F0, the notch filter can be first transformed to the highpass case, and then
DC, which is now a zero, is transformed to F0.
A more general approach would be to convert the poles directly. A notch transformation
results in two pairs of complex poles and a pair of second order imaginary zeros from
each lowpass pole pair.
First, the value of QBR is determined by:
where BW is the bandwidth at – 3dB.
Given the pole locations of the lowpass prototype
-α ± jβ
QBR =
F0
BW
Eq. 5-64
Eq. 5-65
OP AMP APPLICATIONS
5.56
and the values of F0 and QBR, the following calculations will result in two sets of values
for Q and frequencies, FH and FL, which define a pair of notch filter sections.
The pole frequencies are given by:
where F0 is the notch frequency and the geometric mean of FBR1 and FBR2.
A simple real pole, α0, transforms to a single section having a Q given by:
with a frequency FBR = F0. There will also be transmission zero at F0.
In some instances, such as the elimination of the power line frequency (hum) from low
level sensor measurements, a notch filter for a specific frequency may be designed.
C = α2 + β2
D =
E =
F = E2 - 4 D2
G = + + D2 E2
H =
K = (D + H)2 + (E + G)2
Q =
α
QBRC
β
QBRC
+ 4
F √ F2
√ 2 4
D E
G
12
√
K
D + H
Q = QBR α0
FBR1 =
FBR2 = K F0
FZ = F0
F0
K
F0 = √ FBR1*FBR2
Eq. 5-66
Eq. 5-67
Eq. 5-68
Eq. 5-69
Eq. 5-70
Eq. 5-71
Eq. 5-72
Eq. 5-73
Eq. 5-74
Eq. 5-75
Eq. 5-76
Eq. 5-77
Eq. 5-78
ANALOG FILTERS
FREQUENCY TRANSFORMATIONS
5.57
Assuming that an attenuation of A dB is required over a bandwidth of B, then the
required Q is determined by:
A 3 pole 1 dB Chebychev is again used as an example:
A 3dB bandwidth of 0.1 Hz. with a center frequency of 1Hz. is arbitrarily assigned.
Then:
QBR = 10
Going through the calculations for the pole pair yields the intermediate results:
C = 0.829217 D = 0.027218
E = 0.106389 F = 4.079171
G = 2.019696 H = 0.001434
K = 1.063139
and
FBR1 = 0.94061 FBR2 = 1.063139
QBR1 = QBR2 = 37.10499
and for the single pole
FBP3 = 1 QBP3 = 4.431642
Once again a full example will be worked out in a later section.
Lowpass to Allpass
The transformation from lowpass to allpass involves adding a zero in the right hand side
of the s plane corresponding to each pole in the left hand side.
In general, however, the allpass filter is usually not designed in this manner. The main
purpose of the allpass filter is to equalize the delay of another filter. Many modulation
schemes in communications use some form or another of quadrature modulation, which
processes both the amplitude and phase of the signal.
Allpass filters add delay to flatten the delay curve without changing the amplitude. In
most cases a closed form of the equalizer is not available. Instead the amplitude filter is
designed and the delay calculated or measured. Then graphical means or computer
programs are used to figure out the required sections of equalization.
ω0
B √ 10.1 A - 1
Q =
αLP1 =
βLP1 =
αLP2 =
.2257
.8822
.4513
Eq. 5-79
OP AMP APPLICATIONS
5.58
Each section of the equalizer gives twice the delay of the lowpass prototype due to the
interaction of the zeros. A rough estimate of the required number of sections is given by:
n = 2 ΔBW ΔT + 1
Where ΔBW is the bandwidth of interest in hertz and ΔT is the delay distortion over ΔBW
in seconds.
ANALOG FILTERS
FILTER REALIZATIONS
5.59
SECTION 5-6: FILTER REALIZATIONS
Now that it has been decided what to build, it now must be decided how to build it. That
means that it is necessary to decide which of the filter topologies to use. Filter design is a
two step process where it is determined what is to be built (the filter transfer function)
and then how to build it (the topology used for the circuit).
In general, filters are built out of one-pole sections for real poles, and two-pole sections
for pole pairs. While you can build a filter out of three-pole, or higher order sections, the
interaction between the sections increases, and therefore, component sensitivities go up.
It is better to use buffers to isolate the various sections. In addition, it is assumed that all
filter sections are driven from a low impedance source. Any source impedance can be
modeled as being in series with the filter input.
In all of the design equation figures the following convention will be used:
H = circuit gain in the passband or at resonance
F0 = cutoff or resonant frequency in Hertz
ω0 = cutoff or resonant frequency in radians/sec.
Q = circuit “quality factor”. Indicates circuit peaking.
α = 1/Q = damping ratio
It is unfortunate that the symbol α is used for damping ratio. It is not the same as the α
that is used to denote pole locations (α ± jβ). The same issue occurs for Q. It is used for
the circuit quality factor and also the component quality factor, which are not the same
thing.
The circuit Q is the amount of peaking in the circuit. This is a function of the angle of the
pole to the origin in the s plane. The component Q is the amount of losses in what should
be lossless reactances. These losses are the parasitics of the components; dissipation
factor, leakage resistance, ESR (equivalent series resistance), etc. in capacitors and series
resistance and parasitic capacitances in inductors.
OP AMP APPLICATIONS
5.60
Single Pole RC
The simplest filter building block is the passive RC section. The single pole can be either
lowpass or highpass. Odd order filters will have a single pole section.
The basic form of the lowpass RC section is shown in Figure 5-37(A). It is assumed that
the load impedance is high (> ×10), so that there is no loading of the circuit. The load will
be in parallel with the shunt arm of the filter. If this is not the case, the section will have
to be buffered with an op amp. A lowpass filter can be transformed to a highpass filter by
exchanging the resistor and the capacitor. The basic form of the highpass filter is shown
in Figure 5-37(B). Again it is assumed that load impedance is high.
Figure 5-37: Single pole sections
The pole can also be incorporated into an amplifier circuit. Figure 5-38(A) shows an
amplifier circuit with a capacitor in the feedback loop. This forms a lowpass filter since
as frequency is increased, the effective feedback impedance decreases, which causes the
gain to decrease.
Figure 5-38: Single pole active filter blocks
Figure 5-38(B) shows a capacitor in series with the input resistor. This causes the signal
to be blocked at DC. As the frequency is increased from DC, the impedance of the
capacitor decreases and the gain of the circuit increases. This is a highpass filter.
The design equations for single pole filters appear in Figure 5-66.
(A)
LOWPASS
(B)
HIGHPASS
+
-
+
-
(A)
LOWPASS
(B)
HIGHPASS
ANALOG FILTERS
FILTER REALIZATIONS
5.61
Passive LC Section
While not strictly a function that uses op amps, passive filters form the basis of several
active filters topologies and are included here for completeness.
As in active filters, passive filters are built up of individual subsections. Figure 5-39
shows lowpass filter sections. The full section is the basic two pole section. Odd order
filters use one half section which is a single pole section. The m derived sections, shown
in Figure 5-40, are used in designs requiring transmission zeros as well as poles.
Figure 5-39: Passive filter blocks (lowpass)
Figure 5-40: Passive filter blocks (lowpass m-derived)
A lowpass filter can be transformed into a highpass (see Figures 5-41 and 5-42) by simply
replacing capacitors with inductors with reciprocal values and vise versa so:
and
CHP = 1
LLP
LHP = 1
CLP
Eq. 5-80
Eq. 5-81
(A)
HALF SECTION
(B)
FULL SECTION
(A)
HALF SECTION
(B)
FULL SECTION
OP AMP APPLICATIONS
5.62
Transmission zeros are also reciprocated in the transformation so:
Figure 5-41: Passive filter blocks (highpass)
Figure 5-42: Passive filter blocks (highpass m-derived)
The lowpass prototype is transformed to bandpass and bandreject filters as well by using
the table in Figure 5-43.
For a passive filter to operate, the source and load impedances must be specified. One
issue with designing passive filters is that in multipole filters each section is the load for
the preceding sections and also the source impedance for subsequent sections, so
component interaction is a major concern. Because of this, designers typically make use
of tables, such as in Williams's book (Reference 2).
ω Z ,HP =
1
ω Z ,LP
Eq. 5-82
(A)
HALF SECTION
(B)
FULL SECTION
(A)
HALF SECTION
(B)
FULL SECTION
ANALOG FILTERS
FILTER REALIZATIONS
5.63
Figure 5-43: Lowpass → bandpass and highpass → bandreject transformation
Integrator
Any time that you put a frequency-dependent impedance in a feedback network the
inverse frequency response is obtained. For example, if a capacitor, which has a
frequency dependent impedance that decreases with increasing frequency, is put in the
feedback network of an op amp, an integrator is formed, as in Figure 5-44.
Figure 5-44: Integrator
The integrator has high gain (i.e. the open loop gain of the op amp) at DC. An integrator
can also be thought of as a low pass filter with a cutoff frequency of 0Hz.
1
ω0
C1 = 2 L1
1
ω0
Ca = 2 La
1
ω0
L2 = 2 C2
1
ω0
Lb = 2 Cb
1
ω0
L = 2 C
1
ω0
C = 2 L
LOWPASS
BRANCH
BANDPASS
CONFIGURATION
CIRCUIT
VALUES
HIGHPASS
BRANCH
BANDREJECT
CONFIGURATION
CIRCUIT
VALUES
C
L
La
Cb
C
L
L1 C2
L C
La Ca
Lb
Cb
L1 C1 C2
L2
+
-
OP AMP APPLICATIONS
5.64
General Impedance Converter
Figure 5-45 is the block diagram of a general impedance converter. The impedance of this
circuit is:
By substituting one or two capacitors into appropriate locations (the other locations being
resistors), several impedances can be synthesized (see Reference 25).
发表于 2011-12-31 11:47:19
