Switched capacitor filters are growing
increasingly popular because they have many advantages over active
filters. Switched capacitor filters don't require external
precision capacitors like active filters do. Their cutoff
frequencies have a typical accuracy of ±0.3% and they are less
sensitive to temperature changes. These characteristics allow
consistent, repeatable filter designs.
Another distinct advantage of switched capacitor filters is that
their cutoff frequency can be adjusted by changing the clock
frequency. Switched capacitor filters offer higher integration at a
lower system cost. Center frequencies of up to 150-kHz with Q
values up to 20 are achievable.
Effects of Sampling
Switched capacitor filters, as opposed to digital filters, are
sampled-data systems. The signal remains in the
analog domain because the charge is sampled on a capacitor, not
converted to a number. Whether using an analog or digital
sampled-data system, however, the effects of sampling the signal
must be considered.
Figure 1: The time-domain input and output signal of an
analog sampled-data system
In the ideal case, the sampled-data system samples the input
signal instantaneously, with an impulse function. The amplitude of
each sample is equal to the instantaneous amplitude of the input
signal. The output is a series of narrow pulses, each separated by
time T, the sampling period.
Because an impulse function in the time domain corresponds to a
flat spectrum in the frequency domain, the input spectrum is
exactly reproduced in the frequency domain. However, in reality the
sampling signal is periodic and has a finite pulse width. When
convoluting a finite pulse width with an input spectrum:
with unity amplitude, the result is found to be:
From this equation, the gain is a continuous function of
frequency defined by:
is the sample pulse width in seconds, T is the sample period in
is the frequency in radians per second.
Figure 2: The time and frequency domain plots for the
finite pulse width sampled signal
Figure 2 is a plot of the previous equations where the frequency
spectrum is formed around multiples of the sampling frequency. As
long as the adjacent spectra do not overlap (aliasing distortion),
the continuous signal can be reconstructed from the discrete
To evaluate the amplitude distortion caused by having a finite
pulse width, you can simply solve Equation 1. For the case of the
Micro Linear ML2111,
/T is unity because it has a zero-order hold. Assuming a 7.5-MHz
sampling frequency and a bandwidth of 150-kHz, the amplitude
distortion or attenuation is 5.7 x 10-3 dB.
The equation shows that when the sampling frequency is 40-50
times greater than the bandwidth, the aperture effects are
Another potential source for distortion in a sampled-data system
is aliasing. Aliasing distortion occurs when the input signal to a
sampled-data system contains frequency components above one half
the sampling frequency. These higher frequency components beat with
the sampling frequency and are reflected back into the baseband
causing aliasing distortion.
The additional spectral components caused by sampling the input
signal are the sum and differences of the input frequencies with
multiples of the sampling frequency.
Figure 3: Aliasing distortion
For example, assume the input to a sampled-data system is a sine
wave with a frequency of 100-kHz (fi) sampled at 250-kHz
(fS) as Figure 3a demonstrates. The first few spectral
components will be at:
fi = 100-kHz, original signal
fS - fi = 150-kHz
fS + fi = 350-kHz
2fS - fi = 450-kHz
2fS + fi = 600-kHz
Now assume fi has a second harmonic, which would be
at 200-kHz. Figure 3b shows the resulting spectral components. If
our bandwidth of interest were from DC to fS/2, then the
fS - 2fi component interferes with the
original signal. If we were to reconstruct the original signal by
lowpass filtering it, we could not separate the aliased component,
fS - 2fi = 50-kHz, from the original
If our bandwidth of interest is a passband, the aliased
component may not interfere. For example, if the switched-capacitor
filter were to be used as a four-pole bandpass filter with a center
frequency at 100-kHz and a Q of 10 (Figure 3c), then the aliasing
components in the above example would be filtered out (Figure 3d).
But if the switched-capacitor filter were to be used as a low-pass
filter, then the fS - 2fi aliased component
would not be filtered out by the switched-capacitor filter, and an
anti-aliasing filter would be needed.
If the input signal is not band-limited, and the aliasing
components fall within the bandwidth of interest, then a lowpass
filter—or anti-aliasing filter—must be placed in front
of the switched-capacitor filter. This filter must be a continuous
filter rather than a sampled-data filter. However, the complexity
of this filter is typically much less than the switched-capacitor
filter filters and its frequency response is less critical,
allowing for relaxed component tolerances.
Because no frequency component can be totally eliminated, you
must determine the acceptable amplitude of the aliasing components
that will not impact the signal to noise ratio of your system.
The higher the ratio of sampling frequency to input bandwidth,
the lower the requirements on the anti-aliasing filter.
Figure 4: The effects of sampling rate on the separation
of sampled signal spectra
Note that the amount of overlap increases as the sampling
frequency is decreased for a fixed input signal bandwidth. In
general, the higher the sampling frequency, the less aliasing
distortion. Because the ML2111's sampling frequency is typically
either 50 or 100 times greater than the input bandwidth, the
aliasing distortion may be negligible.
Signal to Noise Ratio and Aliasing Distortion
To determine whether aliasing distortion could be a problem, you
must first determine the SNR of the overall system. Aliasing
distortion less than the SNR is of no concern.
The data sheet specifies noise based on Q and bandwidth. From
these specs you can deduce the SNR of one bi-quad in the ML2111.
Using a simplified example, a bandpass filter with a Q = 10 and a
system-clock to center-frequency ratio of 50:1 has noise that is
262-µVRMS over a 750-kHz bandwidth.
To determine the maximum input signal amplitude, you must
consider the slew rate spec. The typical value is 2-V/µs;
however a comfortable safety margin is 1.495-V/µs for the
commercial temperature range and 1.256-V/µs for the military
temperature range. The slew rate = 2
fA, where f is the maximum input frequency, and A is the peak
amplitude in volts. Therefore:
and the SNR = 78 dB.
Based on a 100-kHz bandpass filter with a Q of 10, the ratio of
fCLK to fO equal to 50:1, and a SNR of 78 dB,
what sort of anti-aliasing filter would suffice? You must first
look at the spectrum of the input signal, particularly in the
4.895- to 4.905-MHz frequency range because this is the range that
will be reflected back into the bandwidth of interest, 95- to
If the frequency components in the 4.895- to 4.905-MHz are below
78 dB, they will have a minimal impact on the SNR. Let's assume
that these frequency components are down only 20 dB. Then the
anti-aliasing filter will have to attenuate the frequencies in the
4.895- to 4.905-MHz range by 78 dB - 20 dB = 58 dB, and pass the
frequencies in the 95- to 105-kHz frequency range with no
attenuation. A simple two-pole Butterworth filter with a cutoff
frequency of 170-kHz will suffice, however there will be an
attenuation of about 0.5 dB at 100-kHz because of this filter.
Figure 5: Figure 5a shows a Sallen-Key active filter
capable of implementing two poles, and Figure 5b shows a Rauch
filter also implementing two poles. These two active filters are
good examples to use for anti-aliasing and reconstruction
Using the Rauch filter for the above example, C5=400pF, C8=90
pF, and R = R4 = R6 = R7 = 5-k.
Fortunately the cutoff frequency for the antialiasing and
reconstruction filters are not critical because capacitors can vary
5% and resistors can vary 1%. Taking into account component
tolerance for our example, the cutoff frequency can vary in the
worst case from 152- up to 178-kHz.
The DC gain is:
The transfer function is:
Choose Butterworth response for example:
The important aspects to note are that you must first determine
the SNR in the bandwidth of interest. Based on this bandwidth, are
there any frequencies that will be reflected back into the
bandwidth of interest, and if so how much will they need to be
Remember that frequency components reflected back outside of the
bandwidth of interest will be filtered by the switched-capacitor
filter. Because the ratio of the sampling frequency to the center
frequency is large for the ML2111, most designs will not need an
anti-aliasing filter; and if they do, a simple two-pole Butterworth
The output signal of a switched capacitor filter contains higher
frequency components because it is a sampled signal. Many systems
can tolerate these higher frequency components; however, if they
interfere with the system's performance, then a signal
reconstruction filter can be employed.
Figure 6: A time domain and frequency domain plot of the
output from the switched-capacitor filter
The output signal changes amplitude every clock period. These
sharp transitions elicit high frequency components in the output
Once again, the fact that the ratio of the sampling frequency to
the input bandwidth is high reduces these distortion effects. As a
result of the sin(x)/x envelope, the higher frequency components
are attenuated. For example, assuming the input bandwidth is
100-kHz and the sampling rate is 5-MHz, the frequencies around
4.9-MHz are down 34 dB, and they degrade towards zero as the
frequency reaches 5-MHz.
A single-pole reconstruction filter with a cutoff frequency at
200-kHz would add an additional attenuation of 27 dB at 4.9-MHz but
would attenuate the output by only 1 dB at 100-kHz. A two-pole
Butterworth as in Figure 5a or 5b would yield 58 dB of attenuation
at 4.9-MHz and only 0.5 dB at 100-kHz.
The layout of any board with analog and digital circuitry
combined mandates careful consideration. The most important steps
in designing a low noise system are:
- All power-source leads should have a bypass capacitor to ground
on each printed circuit board (PCB). At least one electrolytic
bypass capacitor (50 pF or more) per board is recommended at the
point where all power traces from the switched-capacitor filter
join prior to interfacing with the edge connector pins assigned to
the power leads.
- Lay out the traces such that analog signal and capacitor leads
are far from the digital clock as possible.
- Both grounds and power-supply leads must have low resistance
and inductance. This should be accomplished by using a ground plane
wherever possible. Either multiple or extra-large plated-through
holes should be used when passing the ground connections through
- Use a separate trace for the clock ground and connect it to the
edge connector's board ground.
- Use ground planes on both sides of PC board.
- All power pins on ICs should have 0.1- and a 0.01-µF
capacitors in parallel tied to ground and as close to the power
pins as possible.
It is important to properly terminate the switched-capacitor
filter's clock input to prevent overshoot. Each pin of the ML2111
has protection diodes against electro-static discharge (ESD) and
any overshoot of more than 0.3 to 0.5V will be injected directly
into the device's ground or supplies. Matching the characteristic
impedance of the line will prevent any ringing thus reduce clock
One particularly nice feature of sampled-data filters is the
fact that the filter's center frequency is directly related to the
clock frequency. For a lowpass filter, increasing the clock
frequency increases the cutoff frequency. Even though the center
frequency increases proportionally with the clock, Q stays
constant. Therefore in a bandpass filter, increasing the clock
frequency increases the center frequency as well as the
A good rule of thumb for the maximum rate that a filter can be
swept is that the sweep rate should be less than the square of the
bandwidth of the filter. This will reduce attenuation of the
passband as a result of sweeping the filter.
The theoretical derivation of this approximation is: assume you
have a bandpass filter with an in-band signal that starts at t = 0.
The output of the filter will exponentially increase until it
reaches the steady state gain of the passband. After four time
), the output sine wave will be at 98% of its final amplitude.
Sweeping a filter is analogous to keeping the filter constant
and sweeping the input frequency. To prevent the filter from
attenuating the sweeping input signal by more than 2% or 0.16
Sweep Rate <>
but the time constant can be approximated by:
Q = f0/BW or BW = f0/Q (4)
and BW into equation (2) results in:
Sweep Rate <>
A Flexible Building Block
Figure 7: The block diagram of a second order section
that includes both a complex pole pair and a complex zero pair
The poles are provided by the ML2111 and the zeros realized by
one and sometimes two external op amps. This building block uses
the device's mode 1c, which allows the poles to have a center
frequency based on external resistors as well as the clock. Plus it
can be used in higher frequency filters because the op amp is
outside of the resonant loop.
The same feedforward circuit can be used on other modes as well,
but for high frequency filters where each complex pole pair has a
different center frequency, mode 1c is the best choice. Only when
Butterworth filters are desired, use mode 1 to achieve higher
frequencies and a higher dynamic range. Equation 6 is the transfer
function for the flexible building block.
At least one and sometimes two external op amps are required to
realize the zeros. The first op amp serves as an inverter, while
the second one sums the input signal with the lowpass and bandpass
outputs. A fast op amp should usually be used with greater than
10-MHz bandwidth to minimize signal phase shifts. Depending on the
application, sometimes a slower amplifier will suffice. In some
cases no external op amp is necessary and the second op amp in the
ML2111, if not being used, will suffice.
With the flexible building block a lowpass, highpass, notch, and
allpass section can be realized by properly positioning the zero
locations. Zero locations are chosen by selecting the appropriate
resistors. The difference between the lowpass output provided by
the ML2111 in mode 1c and the lowpass function realized by the
flexible building block is that the response is monotonically
decreasing, while the Flexible Building Block has a complex zero
pair that inserts a ripple in the stop band and flattens out at
Because the flexible building block uses mode 1c, the pole
equations remain the same whether there is feedforward or not. What
changes is the zero location and the DC gain. The following
equations are used to determine the pole locations and Q for the
flexible building block.
A handy set of equations to convert pole and zero locations
given in rectangular coordinates to f0 and Q values
Complex pole =
By cascading several of these building blocks, complex high
frequency elliptical filters can be realized.
For a lowpass design with a notch, the zeros should be placed on
axis at frequencies greater than the poles' center frequency. In
the numerator of the transfer function for Equation 6, the
coefficient for s
should be set to zero, leading to:
determines the center frequency of the zero. In this form it is
always greater than one, therefore the center frequency of the zero
is always greater than the center frequency for the poles; hence a
lowpass filter. The pole/zero location and the frequency response
And the equations for the low-pass configuration are:
The ratio of the zero to the pole frequency determines the DC to
high frequency attenuation.
When the zeros are at the same frequency as the poles, the
bi-quad becomes a notch and there is no difference between the high
frequency and low frequency gain. The larger the difference between
the pole and zero frequencies, the greater the rejection.
Figure 8: An illustration of the relationship between
pole/zero location and gain. Varying fZ and keeping
fO and Q constant.
For a highpass filter, the zeros must be less than the center
frequency for the poles. The pole/zero plot and frequency plot for
a highpass filter are:
To place the zeros at a lower frequency than the poles the
must be less than one. This can be done by removing the inverter
in Figure 7, which makes the sign of R19 negative. To place the
zeros on the
axis, once again:
The Equations for the highpass configuration are:
Even though the ML2111's mode 1c provides a notch-filter output,
the notch realized by the flexible building block achieves 0 dB of
gain at DC and at high frequencies regardless of the Q value. The
problem with the notch filter of mode 1c is that
That is, as Q increases HON1,2 must decrease
otherwise the IC's bandpass output node (BP pin 2 or 19) will
saturate. The restriction for the ML2111 is that H0BP =
1 = -R3/R1.
To realize a notch filter using the Flexible Building Block, the
zeros must be placed on the j
axis at the same resonant frequency as the poles. Therefore from
Setting R19 equal to infinity means removing it from the
circuit; which saves an op amp and a few resistors. HOBP
still must equal 1. However with the Flexible Building Block
version of the notch filter the gain at DC and fCLK/2 is
independent of Q. And
Tuning R18 adjusts the depth of the notch.
An allpass filter linearizes the filter's phase response. A
linear phase response results in a constant group delay. An allpass
filter keeps the gain constant and just shifts the phase. To keep
the gain constant and only shift the phase, the poles and zeros
must be equal but on opposite sides of the s-
Figure 9: s-plane representation of a second order
The Flexible Building Block can function as an allpass when:
The Transfer function for the allpass is:
Design Method for Complex Filters
The previous sections described how to use the Flexible Building
Block to implement lowpass, highpass, notch, and allpass
second-order sections. Higher order filters are achieved by
cascading these second-order sections.
Figure 10: An elliptical notch is accomplished by
cascading lowpass and highpass sections
An elliptical bandpass is also a combination of highpass and
lowpass sections, except for a bandpass filter, the cutoff
frequency for the highpass bi-quads are lower than the cutoff
frequency for the lowpass.