# Fabricating HF Opamp Filters

Filters have been formed from operational amplifiers (opamps) for audio circuits going back many decades. The use of opamp filters above 1 MHz has been limited due to the roll off of the amplifier's open-loop gain at higher frequencies. However, recent advances in current-feedback amplifiers have enabled the use of opamp designs to create amplifiers with bandwidths exceeding 1 GHz. This report shows the capabilities and limitations of using these amplifiers for lowpass filters to 150 MHz.

Opamp filters have long been a staple at audio frequencies. Where their open-loop gain is large, they can provide nearly ideal frequency responses. Furthermore, opamp filters do not require any inductors. At low frequencies, the inductors required for a passive filter can be very large and often need to be hand wound. Eliminating these inductors is a huge advantage.

Until recently, opamp filters were not practical at MHz frequencies. Opamp open-loop gain rolled off at frequencies too low to preserve the virtually infinite gain assumption that is essential to opamp designs. But with the availability of current-feedback amplifiers (similar to but not exactly like opamps) with useful gain beyond 1 GHz, it is possible to design opamp filters into the MHz frequency range.

Why use opamp filters even if they are possible at tens or hundreds of MHz? Lumped-element inductive-capacitive (LC) ladders are practical at these higher frequencies, and any advantage of eliminating inductors is eliminated since the inductors required for these circuits are small and available in standard values as surface-mount components. Unfortunately, LC ladders are not ideal. The internal losses of the components (mostly the inductors) can cause the frequency response in the passband to droop by 1 dB or more. Opamp filters will not have this problem. Also, opamp filters can often be designed around a single capacitor value that might be available with a 1% tolerance. In combination with the availability of 1% resistors, it is possible to design a very accurate filter, provided that the opamp performance is nearly ideal.

One further advantage of opamp filters at high frequencies is that the design accuracy achievable can provide for an inexpensive matched pair of lowpass filters. This design objective is useful for filters that follow direct-conversion quadrature demodulators where maintaining the amplitude and phase balance between the in-phase and quadrature channels is essential to the system performance. An all-pole lowpass filter will have a transfer function of the form:

T(s) = A/{(s + ω_{0})~~2 + (ω~~_{1}/Q_{1})s + ωsup>2_{1}>2 + (ω^{2}/Q_{2})s + ω^{2}_{2}>}(1)

for a filter with an odd number of poles and:

T(s) = A/{~~2 + (ω~~_{1}/Q_{1})s + ω^{2}_{1}}>2 + (ω_{2}/Q_{2})s + ω^{2}_{2}>}(2)

for a filter with an even number of poles. The locations of the poles are placed to realize a particular filter response such as a Butterworth or Chebyshev response. These transfer functions can be realized with a passive LC ladder, although the path from the desired response to the L and C values is not obvious. For details, see ref. 1.

For an opamp filter, the real pole (s + ω0), when present, is realized with an RC network, followed by a voltage follower (Fig. 1). The complex poles are realized with the Sallen-Key circuit of Fig. 2. The overall filter response is achieved by simply cascading the stages together.

For the real pole circuit of Fig. 1, the opamp is merely serving as a buffer between the RC pole and the following stage. The nonideal frequency response of the opamp simply adds some distant poles to the cascaded response. For the Sallen-Key circuit, the effect of the opamp poles is not so simple. The frequency response of the opamp appears in the feedback loops that create the desired response. If the transfer function of the circuit in Fig. 2 is calculated, the following relationship results:

T(s) = Kω_{0}/~~2 + (3 K)ω~~_{0}s + ω_{0}^{2} + (R_{f}/Z)(s^{2} + 3ω_{0}s + ω_{0}2)>(3)

where quantity Z is the transimpedance of the opamp. If Z >> Rf, the last term in the denominator goes to zero, and the transfer function matches the desired second-order response. At high frequencies, however, the transimpedance rolls off with two real poles, at 1/τ1 and 1/τ2. Including these poles in the transfer function yields:

T(s) = Kω_{0}/~~2 + (3 K)ω~~_{0}s + ω_{0}^{2} + (R_{f}/z_{0})(s^{2} + 3ω_{0}s + ω_{0}^{2})(1 + sτ_{1})(1 + sτ_{1})>(4)

where Z0 represents the low-frequency transimpedance of the amplifier. It may be apparent that the high-frequency effects can be minimized by minimizing the value of Rf. This observation is true to some extent, but the current-feedback amplifiers that work at these frequencies become unstable when the feedback resistance is too low.

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The opamp used (in simulation only) for this study was a model THS3202 from Texas Instruments (www.ti.com). Modeling the opamp response shows that the low-frequency transimpedance is around 300 kΩ, and the time constants for the two real poles are around 0.16 and 160 ns. If these values are plugged in along with the ideal design values for the Sallen-Key circuit, and then the denominator of the nonideal response is factored in, it will be found that there are two additional high-frequency poles. It will also be found that the desired pole frequency, Ω0, shifts to a lower frequency and the resulting Q is higher than desired.

A numerical example may help to illustrate this effect. In this example, a pair of complex-conjugate poles is desired at 150 MHz with Q of 2. Resistor R_{f} is chosen to be 300 Ω to minimize high-frequency effects while maintaining stability. For a Q of 2, K will be 2.5. Inserting all of these values into the transfer function yields:

T(s) = (9.30188 10^{28})/17 + 3.42014 10^{8}s + s^{2})(5.45445 10^{19} + 8.77489 10^{9}s + s^{2}>(5)

The second term in the denominator reveals shows that a new pair of conjugate poles has been gained at Ω0 = (5.45445 x 1019)0.5 rads/s (1.175 GHz). Those poles shouldn't affect the desired response by much. The first term shows that the desired pole frequency has shifted to (6.43554 x 1017)0.5 rads/s or 128 MHz, and the Q has increased to /(3.42014 x 108) = 2.34, a significant difference.

If this calculation is performed for a variety of design values, it is possible to calculate a series of curves to find the necessary compensations needed to achieve the desired filter responses. Figure 3 shows the adjustments necessary to achieve a pole frequency for a number of Q values. For example, to design a stage with a pair of complex poles at 150 MHz and a Q of 1, it would be necessary to follow the green curve and add 25 MHz to the pole frequency, or to design for poles at 175 MHz.

Figure 4 shows the adjustment necessary for the desired Q value at a number of frequencies. If, as in the previous example, it was desired to obtain a pair of complex poles at 150 MHz with a Q of 1, the blue curve would be followed to the point where the actual Q is 1, and it would be necessary to design for a Q that is 0.15 lower in value, or 0.85. Putting the two pieces together, so as to obtain the desired poles at 150 MHz with a Q of 1, it is necessary to design for poles at 175 MHz with a Q of 0.85.

The curves of Fig. 3 and Fig. 4 give a good visual indication of the limits of high-frequency opamp filters. For poles below 50 MHz, the adjustments are small. The Sallen-Key circuit will behave much as expected. As the poles increase beyond 100 MHz, the adjustments are large and the precise sizes of the adjustments are going to depend on the accuracy of the opamp model. Also, the use of higher Qs and higher-frequency poles will require larger corrections. Filter designs require the highest Q's for the highest frequency poles. Therefore, using opamps for lowpass filters is not recommended for designs above 50 to 100 MHz.

Despite these recommendations, it might be educational to examine a five-pole, 150-MHz Chebyshev filter with 0.25-dB ripple to illustrate the design techniques. This filter has a real pole at 65.55 MHz, a pair of complex poles at 109.8 MHz with Q of 1.036, and a pair of complex poles at 157 MHz with Q of 3.876. Figure 5 shows the ideal filter response and the actual response if none of the corrections were applied for the nonideal opamp response. The effects of the downward shift in pole frequency (lower bandwidth) are apparent, as is the upward shift in Q (excess peaking).

Referring to Fig. 3 and Fig. 4, it can be found that to obtain a pair of poles at 109.8 MHz with a Q of 1.036, it is necessary to design for poles at 124 MHz with Q of 0.936. To obtain poles at 157 MHz with Q of 3.876, it is necessary to design for poles at 197 MHz with Q of 3.326. The real pole does not require any adjustment. These adjustments produce the response shown in Fig. 6. It is closer to the desired response, but still calls for some adjustments.

Making some manual adjustments to the design, the response of Fig. 7 can be achieved by moving the poles to 120 MHz with Q of 0.95, and 187 MHz with Q of 2.3. The complete adjusted circuit appears in Fig. 8. As can be seen, a 150-MHz opamp filter design is possible, but requires manual adjustments which can be quite sensitive and, in fact, most like quite sensitive to the characteristics of a particular opamp product. Therefore, designs at these high frequencies are not recommended.

Opamp lowpass filters are practical to about 50 MHz. Operation beyond 100 MHz is possible, but requires careful design to achieve the required frequency response in the presence of the opamp rolloff. Even when practical, the case for high-frequency opamp filters is not compelling. Lumped-element LC ladders are quite practical at these frequencies. The main advantage of high-frequency opamp filters is probably the ability to make a matched pair of filters for use below 50 MHz.

**Reference**

- M. E. Van Valkenburg, Analog Filter Design, Holt, Rinehart, and Winston, 1982.