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Develop analog high-pass filters without capacitors in the signal path (Part 1 of 2)

Many analog signal-chain circuits require ac-coupling to eliminate unwanted dc voltages or offsets. The simplest means to ac-couple is to use a capacitor in series with the signal path, thus forming a single-pole high-pass filter (HPF). In this article, we'll review a common method of implementing this high-pass function without placing a capacitor in the signal path. Then we will extend this method to create second- and higher-order high-pass filters.

In many applications, a series capacitor is all that one needs for ac-coupling. In other situations, this simple method can cause problems In audio circuits, for example, the HPF pole often needs to be in the sub-10 Hz range. With low impedances desired for noise reasons, a large-value capacitor is required. Such capacitors usually have attributes that cause problems with audio signals.

Still other applications, such as the heart-sensor circuit in automatic external defibrillators (AED), have input dc and circuit-induced offsets that must be removed before an analog/digital converter (ADC).

Many precision applications use an instrumentation amplifier (INA) in the signal chain. Often, an input capacitor is not practical in these circuits. INAs have exceptionally high common mode rejection due to good balance between the two inputs. Figure 1 shows a typical INA circuit using the INA217 configured for a gain of 40 dB. Because of its characteristics, this circuit has very low noise and distortion. However, the offset voltage characteristics are not quite as good.


Figure 1: An instrumentation amplifier
(Click on image to enlarge)

To remove dc offsets at the input, add a capacitor in series with each input terminal. However, the input offset of the INA itself is also amplified by the gain. In this circuit, the output offset could be as high as 30 mV. If this offset must be removed as well, then another capacitor in series with the output will be required.

Using “servo feedback” to remove dc offsets
Servo feedback is a common technique to provide ac-coupling in INA circuits. By adding a low-offset op amp (for example, the OPA277) configured as an integrator (Figure 2 ), we get a first-order high-pass function with a cutoff frequency of 16 Hz (Reference 1 ).


Figure 2: Traditional “servo feedback”
(Click on image to enlarge)

The circuit's offset with the added dc-correction stage typically is 15 µV, or 30 µV maximum. This three-orders-of-magnitude improvement in offset occurs because the modified circuit corrects for the offset at the output of the entire circuit. The output offset in this implementation is determined by the input offset voltage of the op amp, plus the offset voltage caused by the bias current being drawn through R1. It also removes offsets applied at the input of the INA, as long as the offset on the INA input stage, multiplied by the gain, is smaller than the maximum output voltage of this stage, and smaller than the output range of the servo amp.

The pole frequency of the high-pass filter formed by this circuit and the overall response of the output stage of the INA is easily calculated.

Take the gain of the output stage alone as K. The gain of this stage with the servo feedback is (Equation 1 ):


which is the original gain of the amplifier with a high-pass filter with a 3-dB point at (Equation 2a ):


For our example (Equation 2b ):


Another advantage of this method is that the circuitry used to create the high-pass function is out of the signal path. With proper-quality passive components, it has little effect on performance at frequencies above the high-pass cutoff. Similarly, this technique often can be used to provide the desired ac-coupling without adversely affecting the common-mode rejection of the INA.

Typically, instrumentation amps are used where good common-mode rejection (CMR) is required. Since the INA input structure is well balanced, it provides exceptional CMR. If, instead, we ac-couple using capacitors in series with the inputs, we have substantially degraded the CMR at frequencies within a decade or so of the cutoff frequency. This is because capacitors simply are not as well matched as the resistors within the INA. This impedance mismatch caused by these capacitors is typically several orders of magnitude worse than the INA itself.

Since ideally the servo-amplifier feedback stage has no effect on the signals in the forward path for frequencies above the pole frequency, we can choose the op amp in the feedback path primarily for its dc specifications. However, be aware of how the dc-specified op amp behaves with the higher frequencies in the passband of the forward path. Frequencies in the signal that are beyond the passband of this op amp may be rectified by the input transistors in the op amp and show up as dc offsets.

Using the servo-amp technique we get a single-pole, high-pass filter response compared to the second-order, simple-pole high-pass response that would have been created by using ac-coupling capacitors at both the input and output.

Servo feedback for more complex circuits
In long signal chains, often there are multiple offset contributors that call for multiple ac-coupling capacitors. In many applications, the composite high-pass filter formed by all these ac-coupling stages can result in a higher-order filter than desired.

Fortunately, the servo technique can be readily extended to remove dc errors generated in just about any amplifier. Judiciously using the servo-feedback technique can minimize the number of ac-coupling stages by extending a single servo-feedback stage across multiple signal-path stages.

The servo-feedback technique can be used in complex circuits that do more than just provide gain. For example, consider an amplifier chain with a high-order low-pass filter. As long as the cutoff frequency of the low-pass filter and the cutoff of the high-pass function created by the integrator feedback are separated by more than a decade or so in frequency, there will be virtually no interaction between the two.

The amplifier/low-pass filter circuit can be designed ignoring the fact that the high-pass function will be implemented. Then add the high-pass function without being concerned with the characteristics of the low-pass filter, other than its pass-band gain (Figure 3 ).


Figure 3: A servo amplifier ac-coupling mMultiple stages
(Click on image to enlarge)

It is common to ac-couple the input and output of such a circuit. However, with the servo technique, we only need to implement a single-order HPF function.

The offset at the circuit output, as mentioned above, is determined by the input offset of the feedback integrator. It will even remove the offset of the input signal as long as it is not of a magnitude that would require a correction voltage beyond the capabilities of the feedback op amp.

As with the INA example, the pole of the HPF function created here is determined by the R-C time constant of the feedback integrator, and the forward stage gain from the left side of R3 to the output. This gain is (Equation 3 ):


where Glpf is the absolute value of low frequency gain of the low-pass filter. This gain is negative, resulting in negative feedback around the loop. The composite circuit results in a HPF filter with a pole (3-dB point) of (Equation 4 ):


While we have reduced the number of high-pass filters using this technique, all high-pass filters in each example are implemented as single-pole filters. Combining several of these into one path results in a bunch of simple poles. Almost any practical multi-pole filter involves complex pole pairs to optimize one or more of these important characteristics:

  • Frequency selectivity
  • Settling time
  • Phase/group delay response
  • In-band ripple
  • And more

It is very rare that an all-simple-pole filter is the best choice.

We can add traditional high-pass filter stages with complex pole-pairs in series with the other circuits in the chain in order to devise more complex filter functions. However, simply extending the servo-amplifier technique can do this without adding any series filter stages, while retaining the other advantages of the servo-feedback technique.

Extending the servo technique to synthesize complex pole pairs
Using a simple, inverting op amp gain stage to demonstrate, a logical first step is to add another integrator in the feedback path in series with the one in the earlier circuits. However, this circuit will have a very high-Q and, thus, a sizable peak in its response.

Adding a resistor in series with one of the integrating capacitors, R2 in Figure 4 , allows us to damp out the resonance and set the filter characteristics to what we really want.


Figure 4: Adding a second integrator with resistor for second-order filter
(Click on image to enlarge)

As with the first-order servo example, our second-order high-pass circuit has a dc characteristic determined by the feedback op amp, and ac characteristics determined by the forward dc-coupled amplifier. Except at frequencies below the high-pass cutoff, and within a decade or so above, the feedback circuit virtually has no effect on the transfer function.

The transfer function of this circuit is given in Equation 5 .


Equation 6 and Equation 7 give the frequency of the high-pass-filter poles, and the 'Q' which determines the amount of peaking in the frequency response of the filter.




When working with filters implementing complex poles or zeros, pay close attention to the sensitivity of the filter characteristics to component variations to prevent implementing an unreliable circuit that cannot be reproduced in production.

The sensitivities (References 2, 3, 4 and 5 ) of F0 and Q to the components are (Equation 8 through Equation 13 ):


Note that all of these sensitivities are constants, and less than or equal to a magnitude of one. It is very unusual to achieve sensitivities any lower than this. Also note that the sensitivity of F0 to R2 is zero. This means that R2 can be used to modify Q without having any affect on F0 . Equation 6 confirms this, as R2 is not a factor in the equation for F0 , while it is a factor in Equation 7 for Q.

If this circuit is redrawn with all amplifiers in a row, it looks very similar to an old, commonly used filter topology, Figure 5 :


Figure 5: The same filter, drawn in-line
(Click on image to enlarge)

Figure 5 is our same filter drawn in-line. We have been using input 3 and output 3. R7 and R8 were added to show a generalized version of this filter with three different inputs and three different outputs. Table 1 shows what kind of filter we can get with each combination of input and output.


Table 1: Filter types available for Figure 5
(Click on image to enlarge)

Table 1 indicates that input 3 also provides a band-pass function at output 1, simultaneous with our high-pass output.

The old cousin of this filter is called the Tow-Thomas filter, or simply 'TT' (Reference 2 and Reference 3 ). See Figure 6 for a generalized version of this circuit.


Figure 6: Tow-Thomas filter
(Click on image to enlarge)

The only difference between these two is that the first integrator in the TT filter has a resistor in parallel with the capacitor, and the second integrator in our new filter has a resistor in series with the capacitor.

Table 2 shows the filter types available using all three inputs and outputs of the TT filter.


Table 2: TT filter types available
(Click on image to enlarge)

This filter topology can provide a BPF and LPF for a single input, but cannot provide an HPF function.

In fact, many circuit design text books have used the following variation of the TT (Figure 7 ) to implement a HPF.


Figure 7: Tow-Thomas HPF
(Click on image to enlarge)

Our new circuit, Figure 5, clearly is less complex than this HPF.

As with the first-order servo technique, we can use this technique to add a second-order high-pass function to just about any gain block. The added two-integrator circuitry is 'wrapped around' this gain block and adds a high-pass function without adding anything, in particular capacitors, in series with the signal path.

As with the first-order servo technique, an amplifier stage can be designed without considering the high-pass function as long as there are no poles or zeros within a decade or so of the intended high-pass poles. Then add the feedback circuitry to create the high-pass function.

For example, this circuit can be easily adapted to work with a non-inverting amplifier configuration (Figure 8 ).


Figure 8: Non-inverting gain stage
(Click on image to enlarge)

This variation has very high input impedance, which is an important feature for some applications.

We have successfully extended the servo-feedback technique to create a second-order high-pass filter topology with the same advantages, and have shown how to use it with both inverting and non-inverting gain stages. This topology can be used to advantage in many different applications.

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. Click on www.planetanalog.com/features/showArticle.jhtml;?articleID=207403049 to see Part 2.

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. Tow, J., “A step-by-step active-filter design,” IEEE Spectrum , Vol. 6, pp. 64-68, December 1969.
3. Thomas, L. C., “The Biquad: Part I ” Some Practical Design Considerations,” IEEE Transactions on Circuit Theory , vol. CT-18, pp. 350-357, May 1971.
4. Fortunato, M., “Circuit Sensitivity; With Emphasis On Analog Filters,” Texas Instruments Developer Conference 2007 , March 2007: www.ti.com/tidc2007
5. Huelsman, L.P. and Allen, P.E., Introduction to the Theory and Design of Active Filters , McGraw-Hill, New York, 1980.
6. Budak, Aram, Passive and Active Network Analysis and Synthesis , Houghton Mifflin company, Boston, 1974.
7. Ghausi, M.S. and Laker, K.R., Modern filter Design: Active RC and Switched Capacitor , Prentice-Hall, Englewood Cliffs, N.J., 1981.

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. .

1 comment on “Develop analog high-pass filters without capacitors in the signal path (Part 1 of 2)

  1. ojsdofj
    September 1, 2015

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