# Develop analog high-pass filters without capacitors in the signal path (Part 2 of 2)

1 Comment

In **Part 1** (click here), we learned how an old technique known as servo feedback can remove dc offsets in a dc-coupled gain block, without putting any components, especially capacitors, in the forward-signal path. We developed a second-order equivalent of this technique, and showed how it can be adapted to work with both inverting and non-inverting gain stages.

In **Part 2**, we will review a practical example, explore refinements to the basic architecture, and generalize its use to generate more complex high-pass filter functions.

**Servo feedback and microphone circuits**

**Figure 1** is a circuit schematic using an electret (condenser) microphone. The microphone is modeled as an ideal current source for demonstration purposes. An electret microphone must be pulled up to a dc voltage with a resistor. Thus, there is an inherent offset determined by the dc voltage, pull-up resistor, and current range of the microphone. Usually, such a microphone is followed by an ac-coupling capacitor.

If we are to apply the signal from the microphone to an analog-to-digital converter (ADC), then the microphone's offset voltage must be removed. Most common ADCs today are unipolar and require a specific dc offset be added to any bipolar signal.

This circuit provides the desired high-pass function, removing the microphone's dc offset and adds in the dc offset required by the ADC. The high input impedance of the non-inverting operational amplifier (op amp) prevents the circuit from loading down the microphone.

*Figure 1: Replacing the inherent offset from an electret microphone*

*(Click on image to enlarge)*

Running a transient simulation on this circuit, in **Figure 2** (left) Vin has an offset of 4 V, while Vout is centered at 1.25 V. An ac analysis, Figure 2 (right), shows the second-order high-pass filter function.

*Figure 2: Transient and frequency response of electret microphone circuit*

*(Click on image to enlarge)*

The circuit as drawn provides 6 dB gain (a gain of two). Let's assume the application requires a gain of 10. We can simply change the gain stage as we like, then recalculate the component values for the feedback circuit to keep the pole locations unchanged. **Figure 3** shows a circuit modified for this gain. **Figure 4** shows the transient and AC analyses.

In order to adjust for a change in gain (change in the ratio R_{5}/R_{4}), keep the pole locations unchanged by dividing R_{6} and R_{3}, each by the square-root of the change in the R_{5}/R_{4} ratio.

*Figure 3: Electret circuit with increased gain*

*(Click on image to enlarge)*

*Figure 4: Transient and frequency response with increased gain*

*(Click on image to enlarge)*

**Selecting and accommodating the 'right' capacitor**

Many capacitor types have capacitances with undesirable voltage coefficients. These capacitors can cause significant distortion in signals at or near the HPF roll-off frequency. Ceramic NPO capacitors, as well as mica and many of the metal film types of capacitors, can generally resolve this issue. However, these capacitors tend not to be economical in larger capacitance values. If the capacitors required by the design are of larger value than desired, one can increase the values of the resistors to allow a reduction in value of the capacitors. Another option is to throw away some loop gain in the feedback path. This is not much of a compromise as two op amps in series provide a lot of gain.

The circuit in **Figure 5** has the same response as our non-inverting example with 20 dB of gain. R_{a} and R_{b} are added to provide an attenuation factor of ten in the feedback path allowing us to decrease the values of both capacitors by √10. Of course, we must increase R_{2} by this same factor to keep the compensating zero at the same frequency. This has the effect of maintaining the Q of the circuit at the desired value.

*Figure 5: Modification for lower capacitor values*

*(Click on image to enlarge)*

The added resistors could have easily been placed in the first feedback stage, between the output and R_{6}. Doing this, however, adversely affects the offset at the output. No current (except bias) flow from R_{6} into OpAmp_{2} or C_{2}. Therefore, neglecting the small bias currents, the dc voltage on each side of R_{6} is the same. If the attenuator is placed ahead of R_{6}, then the feedback loop ensures that the output has an offset voltage equal to the op amp input voltage divided by the attenuation. In our example, this results in ten times the output voltage offset (attenuation = 1/10).

If larger resistor values are adequate for the application, then we can increase the values of R_{6}, R_{3}, R_{a} and R_{b} by some factor, then decrease both Cs by the same factor. Once again, R_{2} needs to be increased by the same factor that C_{1} is decreased, to keep the zero at the correct frequency. **Figure 6** shows the circuit modified this way using a factor of √10.

*Figure 6: Lowering capacitor values further, by increasing resistor values*

*(Click on image to enlarge)*

In these last two steps, we have decreased capacitor values by a factor of ten without changing the frequency response.

Inverting amplifiers simplified

Another useful trick can be used when the gain block is intended to be inverting. Inverting op amp gain blocks are commonly used in audio and other circuits to minimize distortion. In a non-inverting circuit, the common-mode signal at the inputs of the op amp includes the ac input signal being amplified, whereas in an inverting circuit, the op amp inputs only include the dc common-mode signal and the small error signal. This modulation of the op amp common-mode signal in the non-inverting amplifier can cause increased distortion.

We start with the circuit from Part 1, Figure 4, repeated below as **Figure 7**.

*Figure 7: The basic non-inverting circuit (Figure 4 in Part 1)*

*(Click on image to enlarge)*

In general, the original gain block comprises OpAmp1 and R_{5}. R_{1} is unchanged by adding the feedback circuit, other than creating the high-pass function. However, the presence of R_{4} reduces the loop gain of the OpAmp_{1} gain block. While the nominal gain of -R_{5}/R_{1} remains unchanged, the gain rolls off at a lower frequency.

If R_{4} = R_{1}, the result is 67 percent of the bandwidth compared to the same circuit without R_{4}. The end result is that, with R_{4}, the op amp's effective gain-bandwidth-product (GBWP) is decreased.

We can eliminate this GBWP degradation by applying the feedback signal to the non-inverting terminal of OpAmp1 and eliminating R_{4} as shown in **Figure 8**.

*Figure 8: Alternate feedback for inverting amplifier--too many zeros*

*(Click on image to enlarge)*

Note that feeding back to the non-inverting terminal of OpAmp_{1} causes the feedback to become positive, and thus unstable. Therefore, we also changed OpAmp_{2} to be non-inverting to maintain negative feedback.

Note, however, that the gain of the OpAmp_{2} block changed from **Equation 1**:

to **Equation 1a**:

We have added another zero to the feedback path. Therefore, we no longer need the zero we created with the addition of R_{2}. We can get rid of R_{2} resulting in the final topology of **Figure 9**.

*Figure 9: Alternate feedback for inverting amplifier, final version*

*(Click on image to enlarge)*

The transfer function for this variation is shown in **Equation 2, 3, and 4**:

The equations for this topology are a little more difficult to work with. We no longer have R_{2}, which can be used to adjust the Q independent of F_{0}.

We can set F_{0} using the above equations fairly easily. It is the same as with our original topology, except the R_{4}/R_{5} term is replaced by R1/(R1+R_{5}). This term is not adjustable without changing the gain stage (OpAmp1), whereas in the original implementation, R_{4} had no direct affect on the gain block. The equation for Q (4) also has this "untouchable" factor, R1/(R1+R_{5}), and does not have a unique component that only affects Q.

The remaining parameters that can be modified to adjust both F_{0} and Q are R_{3}, R_{6}, C_{1}, and C_{2}. The product of these terms determines F_{0}, while the ratios of the resistors and capacitors determines Q.

**Figure 10** illustrates an implementation with the same frequency response as our previous circuits using this modified topology.

*Figure 10: An implementation with feedback to non-inverting terminal*

*(Click on image to enlarge)*

Note that the non-inverting input of OpAmp_{2} is subject to the full output swing of the circuit. In some applications this could cause complications due to common mode input-voltage limitations of the op amp used. In this case, we can configure OpAmp_{2} back to the way it was, and change the configuration of OpAmp_{3} to be non-inverting as in **Figure 11**.

*Figure 11: Eliminating CMV concerns in OpAmp*

_{2}*(Click on image to enlarge)*

**Creating higher-order high-pass filters**

Now that since we have both first-order and second-order circuits which provide high-pass functions without adding anything in the forward-path gain block, we can design and cascade a number of such circuits to get higher-order high-pass filters.

We can combine two of the above examples to demonstrate implementing a third order high-pass function distributed across a larger circuit. **Figure 12** is a diagram of a third-order high-pass function distributed across two gain blocks. **Figure 13** illustrates the transfer function of each stage and the combined circuit.

*Figure 12: Distributed third-order, high-pass filter function*

*(Click on image to enlarge)*

*Figure 13: Frequency response of distributed HPF*

*(Click on image to enlarge)*

More second-order stages can be cascaded to implement higher-order filter functions.

This new, high-pass variation of the venerable Tow-Thomas Filter provides us with another low-sensitivity topology for second-order high-pass filters, as well as an easy means of extending the established first-order servo-feedback technique for ac-coupling to second- and higher-order functions.

Simply use this topology to implement a three op amp, second-order high-pass filter. Or, because the input and output of the overall filter are also the input and output of a simple gain block that is one of the three stages of the filter, you can use this topology to add a high-pass (dc-blocking) function to nearly any gain block, without affecting the gain or higher-frequency behavior, and without adding any circuitry in the signal path.

This new filter topology can be a useful tool in the analog designer's "bag of tricks."

**References: **

1. Stitt, R. M., "AC Coupling Instrumentation And Difference Amplifiers," *TI Document SBOA003*, 1990, focus.ti.com/general/docs/techdocsabstract.tsp?abstractName=sboa003

2. Fortunato, M., "Circuit Sensitivity with Emphasis on Analog Filters," *Texas Instruments Developer Conference 2007*, March 2007, focus.ti.com/lit/ml/sprp524/sprp524.pdf.

3. Huelsman, L.P. and Allen, P.E., *Introduction to the Theory and Design of Active Filters*, McGraw-Hill, New York, 1980.

4. Budak, Aram, *Passive and Active Network Analysis and Synthesis*, Houghton Mifflin company, Boston, 1974.

5. Ghausi, M.S. and Laker, K.R., *Modern filter Design: Active RC and Switched Capacitor*, Prentice-Hall, Englewood Cliffs, N.J., 1981.

6. Tow, J., "A step-by-step active-filter design," *IEEE Spectrum*, Vol. 6, pp. 64-68, December 1969.

7. Thomas, L. C., "The Biquad: Part I " Some Practical Design Considerations," *IEEE Transactions on Circuit Theory*, vol. CT-18, pp. 350-357, May 1971.

**About the Author**

For the last five years, ** Mark Fortunato** has been the Southwest Analog Field Applications Manager for Texas Instruments. When not working with customers, Mark enjoys reading, coaching youth sports, listening to his son perform live jazz and Latin music.