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.

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:

(1)

From this equation, the gain is a continuous function of frequency defined by:

where is the sample pulse width in seconds, T is the sample period in seconds, and 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 samples.

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 negligible.

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 (f_i) sampled at 250-kHz (f_S) as Figure 3a demonstrates. The first few spectral components will be at:

f_i = 100-kHz, original signal
f_S - f_i = 150-kHz
f_S + f_i = 350-kHz
2f_S - f_i = 450-kHz
2f_S + f_i = 600-kHz

Now assume f_i 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 f_S/2, then the f_S - 2f_i 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, f_S - 2f_i = 50-kHz, from the original signal.

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 f_S - 2f_i 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.

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-µV_RMS 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 f_CLK to f_O 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 105-kHz.

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 filters.

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 attenuated?

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 should suffice.

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 signal.

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.

1. 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.
2. Lay out the traces such that analog signal and capacitor leads are far from the digital clock as possible.
3. 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 the PCB.
4. Use a separate trace for the clock ground and connect it to the edge connector's board ground.
5. Use ground planes on both sides of PC board.
6. 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 noise.

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 constants ( ), 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 dB:

Sweep Rate < BW/4 (2)

but the time constant can be approximated by:

Q/2 f₀ (3)

and,

Q = f₀/BW or BW = f₀/Q (4)

substituting and BW into equation (2) results in:

Sweep Rate < BW²/2 (5)

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.

(6)

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 high frequency.

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 is:

Complex pole =

By cascading several of these building blocks, complex high frequency elliptical filters can be realized.

Because

the coefficient

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 are:

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 f_Z and keeping f_O and Q constant.

To place the zeros at a lower frequency than the poles the coefficient

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:

That is, as Q increases H_ON1,2 must decrease otherwise the IC's bandpass output node (BP pin 2 or 19) will saturate. The restriction for the ML2111 is that H_0BP = 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 Equation 6:

Setting R19 equal to infinity means removing it from the circuit; which saves an op amp and a few resistors. H_OBP still must equal 1. However with the Flexible Building Block version of the notch filter the gain at DC and f_CLK/2 is independent of Q. And

Tuning R18 adjusts the depth of the notch.

Figure 9: s-plane representation of a second order allpass filter

The Flexible Building Block can function as an allpass when:

The Transfer function for the allpass is:

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.