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Resistors values select cut-offs for 2nd and 4th order analog filters*

An active filter can be expected to provide amplification along with band limiting to eliminate components in the signal path. Though the use of oversampling audio converters has significantly simplified filtering, signal sub-harmonics from neighboring systems can intrude on a converter's allowed bandwidth, and in the end, totally redefine the filtering requirement. To solve this problem without adding complex circuitry, the simple lowpass filter originally placed in front of the converter must perform band rejection with repeatable depth from unit to unit in high volume production.

An analog filter like LT1568 meets provides a relatively simple solution. It requires only a few external passive components to satisfy complex filtering requirements, including band rejection. Most important, though, is that the performance of the LT1568 is precisely repeatable from device to device, making it possible to manufacture systems in volume without the need for costly production trimming.

Figure 1 shows a block diagram of the LT1568. The part is configured as dual 2nd order matched building blocks, which can be used in a variety of ways for different applications. The two filter blocks can be interconnected to obtain 4th order or higher filter transfer functions with an optional notch.

Each individual filter block uses two low noise (1.5nV/Hz input voltage noise) operational amplifiers, where one op amp is configured as an integrator (with capacitor C1 connected in its feedback loop) and the other one is connected as an inverter. The inverter senses the output signal and drives the bottom plate of capacitor C2. The inverter op amp output is pinned out, and it can be used in applications requiring differential outputs or an output with phase reversal. This simple building block, when used with a few external passive components, can provide various 2nd order or 3rd order filter functions.

A simple case is shown in the block diagram of Figure 1, where a few resistors are the only external components required to make a dual lowpass filter.

One of the features of this device is repeatable AC performance from part to part. This is achieved by trimming the internal C1 and C2 capacitors to a better than 1 percent tolerance, and by trimming the gain bandwidth product (GBW) of the internal op amps.


Figure 1: LT1568 block diagram. The configuration of external components shown is for a dual 2nd order lowpass filter.



Figure 2: Basic 2nd order lowpass filter.Resistors R1, R2, R3 and capacitors C1, C2 dictate response.


Dual 2nd Order Lowpass Filter

Key lowpass filter responses are include Q, the center frequency (f0), and the passband gain (H0). These parameters are defined by the value of three external resistors, R1, R2 and R3, where



and


The values of the external resistors (R1, R2, and R3) for a few classical filter responses are given in the LT1568 data sheet. Alternately, the resistor values for a wide selection of filter responses can be determined by using the LT1568 Design Guide spreadsheet, available at www.linear.com.

For a 2nd order Butterworth lowpass filter with a unity passband gain and a Q of 0.707, C1 = 105.7pF, C2 = 141.3pF

R1 = R2 = R3 = 1.28k-ohm (1MHz/fC)

This expression can be used for cutoff frequencies up to 10MHz.

For instance, for a 2.5MHz cutoff frequency, all three resistors should be 512 ohms (or 511 ohms, the standard resistor value). Unequal values of these three resistors allow gains other than unity, and points to other filter response types, unity gain or not.

The second order lowpass algorithm, illustrated above, can be used to calculate resistor values for all applications where f0 Q H0 2MHz.

For applications ABOVE 2MHz, the resistor values can still be calculated with this simple algorithm ” but it may produce responses that will deviate from an ideal textbook filter response. This is a result of the finite GBW of the internal op amps. Nevertheless, as the low noise op amps and the internal capacitors of the LT1568 are trimmed, these higher frequency responses still remain repeatable from part to part. (The online LT1568 Design Guide automatically corrects for the finite GBW of the LT1568 internal op amps.)

Table 1 shows a sample of recommended resistor values for a 2nd order Butterworth response and for a 2nd order 0.25dB Chebyshev responses. Arbitrary values of gains are chosen. The range of cutoff frequencies and gain the LT1568 can provide is dictated by the spread of the values of the external resistors.


Table 1: Recommended resistor values for 2nd order filters..


The lowest recommended center frequency is dictated by the value(s) of the required external resistors. High external resistor values add to the overall output DC offset of the filter. For instance a 200kHz, 2-pole Butterworth response dictates R11 = R12 = R13 = 6.4k. The output DC offset created by these resistor values is 9.6mV, assuming IBIAS = 0.5uA for the high speed internal op amp.

For lowpass responses, the recommended set of center frequency, Q, and Gain, should obey the following: f0 Q H0 10MHz, with Q 5 and f0 0.1MHz

4th Order, All Pole, Lowpass Filter


Figure 3: Dual 2nd order lowpass filter. Cascading the filter stages (with the red wire shown) converts the device to a 4th order lowpass filter..


The two identical sections of the LT1568 can be cascaded to create a 4th order lowpass filter response. This can be accomplished by closing the loop between R12 (VIN2) and pin 4 of the LT1568 (VOUTA) (the “red line” dotted line in Figure 3). Each 2nd order section of the filter should still comply with the f0, Q and Gain limitations described above. (To find the value of the center frequency, f0's, and Q's for the desired lowpass function you can use FilterCAD, available for download from www.linear.com.)

Example: design a lowpass filter that processes signals from DC to 5MHz, and provides an attenuation of 25dB or more to signals above 10MHz. The ripple of the 5MHz passband should not exceed 0.5dB (or 1dB peak-to-peak) and, the passband gain should be 3dB, or 1.414V/V. Note, for all-pole 4th order filters, the attenuation at twice cutoff cannot get much better than 24dB to 30dB.

The theoretical response of a Chebyshev lowpass filter has a 0.6dB peak-to-peak ripple (Figure 4). A lower ripple, 0.6dBP”P ripple vs 1dBP”P, is deliberately chosen as a guard-band for external resistor tolerances and PC board parasitics.


Figure 4: Frequency response for 4th order Chebyshev filter.


Note, for a Butterworth response instead of the Chebyshev response, in order to maintain the stated passband and stopband specifications, a 6th order filter would be required with a “3dB filter cutoff frequency of 5.86MHz. Hence, a Chebyshev design can save components and cost.

Table 2 shows the calculated resistor values to be used in conjunction with the LT1568 filter shown in Figure 3. For this particular exercise the LT1568 Design Guide was used with the Custom Filter Design feature of the spreadsheet.
Adding a Stop-Band Notch


Table 2: Resistor values for 4th order Chebyshev..


Sometimes, the gain roll-off of an all-pole lowpass filter is insufficient to reject a specific frequency outside the system passband. This can happen when an interfering signal, although outside the system cutoff frequency, is still close enough to affect performance. A seemingly simple solution to this problem is the addition of a notch in the filter's transition band, which, before the arrival of the LT1568, was easier said than done.

The unique architecture of the LT1568 allows the addition of a notch by simply adding an external capacitor between the summing node of the first stage and the summing node of the second stage, as shown in Figure 5. Also, for this notch topology, the lowpass sections should be cascaded via the inverting output (OUTA) of the first section. This proprietary technique is described in detail in the LTC1562 data sheet (also available at www.linear.com.)


Figure 5: A capacitor loop adds a notch to the 4th order lowpass filter.


Capacitor CN and resistor R12 are the only external passive components affecting the accuracy and repeatability of the notch frequency”the notch frequency is inversely proportional to the square root of the product of CN and R12.

As very low tolerance resistors are readily available, so the burden for frequency accuracy lies on the tolerance of CN. For instance, if CN has a 5% tolerance, the notch variability is approximately 2.5%, so a capacitor with 2% tolerance is recommended to reduce the variability to ~1%.

The LT1568 Design Guide can be used to calculate the external component values required to reliably produce this type of filter response. Use the Fixed Filter Response section. This design offers a choice of two types of lowpass filters with a stopband notch:

? Type 1: A 4th order lowpass with 0.25dB passband ripple and a notch such that the attenuation at 1.35 filter cutoff is >20dB.

? Type 2: Same passband as type 1 but the notch is moved to a higher frequency such that the minimum attenuation at 2 filter cutoff is at least 36dB.

Figure 6 illustrates the two types of filters. Both filters are normalized to a 1MHz cutoff.


Figure 6: Frequency response for the two types of lowpass filters with a stopband notch.


It is recommended that the maximum frequency of the notch remain below 10MHz.

For example, for a lowpass filter with 2.5MHz cutoff frequency and an attenuation of better than 20dB at 3.5MHz, a Type 1 filter (from Figure 6) is chosen and the component values of Figure 5 are derived using the LT1568 Design Guide spreadsheet, and are shown in Table 3.


Table 3: Resistor values for 2.5MHz notch filter.


Conclusion

The LT1568 offers the versatility and compactness demanded by the latest applications. It can be used as a simple dual matched 3rd order lowpass filter in I/Q applications, or it can be used as a sharp and accurate bandpass filter. The ability to add a notch in the filter transition band adds selectivity without adding complexity. Careful trimming of its active and passive components gives the LT1568 unprecedented predictability and repeatability, which can save significant costs in the volume production of high frequency devices.

*A version of this article appeared in Linear Technology's internal magazine.

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