slyt299.pdf

Amplifiers: Op Amps

Texas Instruments Incorporated

A new filter topology for analog

high-pass filters

By Mark Fortunato

Analog Field Applications Manager

Figure 1. Gain block with high-order LPF

The analog circuit designer today has liter-

ally dozens of circuit topologies available to

implement filters, from the venerable

Sallen-Key (SK) filter, in use for well over

fifty years, 1 to more esoteric and hard-to-

pronounce filters such as the Mikhael-

Bhattacharyya (MB) filter or the Padukone-

Mulawka-Ghausi (PMG) filter. 2 Each of

these filters has advantages and disadvan-

tages relative to its cousins. Nearly all of the

filter topologies used today were developed

in the 1950s, ’60s, and ’70s. 2–6 . Can we

come up with a filter topology that has an advantage over

all the many topologies that have been in use for decades?

For some specific needs, the new topology presented here

has some unique advantages.

Almost all the common high-pass filters (HPFs) tend to

have one thing in common—capacitors in series with the

forward signal path. For most applications, having capaci-

tors in the signal path is not a problem. However, there

are applications where such capacitors can be problematic.

For example, in broadband low-noise circuits such as many

audio circuits, there is a need to keep resistance values,

and thus noise, low. These applications also often call for

high-pass functions that roll off at low frequencies, below

10 Hz in some audio applications. These cases can thus

call for very large capacitor values. Large-value capacitors

tend to be very expensive or have voltage coefficients and

other non-idealities that can ruin the fidelity of the signals

being passed through them.

Another limitation of the HPFs commonly used is that

the entire filter circuit is implemented separately and

R 1

OpAmp1

R 2

–

Output

High-Order, Inverting,

Unity-Gain LPF

Input

placed either before or after another circuit functional

block. Sometimes a filter in front and one after a block is

needed. The technique we will discuss here allows an

engineer to design a circuit without consideration of the

high-pass function required. After the circuit is designed,

another circuit can be “wrapped around” the original one

that will cause the overall circuit to have a high-pass func-

tion without affecting the operation of the original circuit

at frequencies above the high-pass rolloff.

Adding a high-pass function to block DC offsets

Figure 1 is a simplified schematic of a circuit with a gain

block driving a high-order low-pass filter (LPF) in a signal-

processing application. In this example there is an offset in

the input signal and an offset caused by the filter, both of

which must be removed. Typically a designer would place

a capacitor in series with the input and the output as

shown in Figure 2. For many applications this approach is

just fine; but for some applications, this simple AC-coupling

scheme can cause problems. Besides the reasons already

Figure 2. Adding simple AC-coupling stages

R 1

OpAmp1

R 2

–

Output

High-Order, Inverting,

Unity-Gain LPF

Input

C 2

C 1

Stage

R 3

Analog Applications Journal

High-Performance Analog Products

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3Q 2008

Texas Instruments Incorporated

Amplifiers: Op Amps

discussed, the characteristics of these high-

pass stages may adversely affect the signal-

processing function that is the primary

purpose of the overall circuit. Each of these

AC-coupling stages creates a single real

(i.e., simple) pole at the frequency deter-

mined by the applicable RC time constant.

Especially as the number of AC-coupling

stages in the signal chain increases, the

composite high-pass function is unlikely to

be the optimal one for the full circuit and

system. More commonly, a filter with com-

plex pole pairs is necessary to optimize the

HPF response.

Using servo feedback rather than

blocking capacitors

In our example, let’s assume that the input

signal presents the largest of the offsets

and that the filter’s output offset, though

small, is objectionable for the latter stages.

Let’s further assume that the desirable HPF

function is that of a single pole. If we elimi-

nate the input AC-coupling filter, the output

filter will certainly remove all DC offsets for

the subsequent stages; but then that input

offset will cause the signal applied to and

processed by the filter to be significantly

shifted “off center,” which can cause signifi-

cant distortion.

An old technique referred to as “servo

feedback” is often used in cases like this.

This technique provides AC coupling,

removing all offsets at the output of the

circuit without putting any circuitry in

series with the amplifier/filter chain of

Figure 1. Servo feedback is fully covered

in Reference 7.

The feedback path added in Figure 3 is

an inverting integrator. The integrator out-

put is fed to the inverting terminal of the

input amplifier so that the overall loop has

negative feedback. Assuming that the rolloff of the LPF is

at least a decade above the desired high-pass rolloff, we

can treat the LPF as a flat gain block for the purposes of

calculating the high-pass response. A simple analysis

shows that we have added a high-pass function with the

transfer function

Figure 3. Single-pole HPF using servo feedback

1µ 1

10 k

OpAmp3

10 k

OpAmp1

–

High-Order, Inverting,

Unity-Gain LPF

Output

Input

V/V

OUT

–5

–10

–15

–20

100

10 k

Frequency (Hz)

LPF

(2)

2π

Since the LPF in this circuit has a gain of –1, we get the

frequency response shown in Figure 3 with a pole frequency

of 15.92 Hz.

This servo technique solves the problem of putting

capacitors in series with the signal path, eliminates the

need for multiple high-pass stages, and allows a designer

to add a high-pass function to a gain/low-pass block with-

out modifying the block itself. However, this technique is

capable of implementing only simple poles and thus does

S RRC

231

OUT

LPF

=−

(1)

LPF

231

LPF

where G LPF is the absolute value of the LPF gain. G LPF is a

high-pass function with a 3-dB (pole) frequency of

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Figure 4. New second-order high-pass topology

1µF

15 k

10 k

OpAmp2

10 k

OpAmp3

10 k

OpAmp1

Input

–

10 k

Output

(a) Second-order servo feedback

10 k

Input

1µF

15 k

1µF

10 k

OpAmp2

OpAmp3

OpAmp1

–

Output

(b) Same circuit redrawn to be in-line

not allow us to create complex pole pairs. Therefore we

need a similar technique that will provide a complex second-

order function.

A new circuit topology implements a

complex pole pair

Figure 4 shows just such a circuit, drawn two different

ways. Figure 4a shows a gain block with the frequency-

dependent part of the circuit wrapped around it like the

first-order servo filter discussed earlier. Figure 4a is very

similar to the previous schematics except that there are

two integrators in the feedback path, one of which has an

added resistor included to set the Q of the second-order

function. Figure 4b shows the identical circuit just shifted

around to be in-line. Anyone familiar with three-op-amp

biquad circuits such as the Kerwin-Huelsman-Newcomb

(KHN) and Tow-Thomas (TT) filters will see a distinct

similarity. In fact, this topology is the same as the TT filter

except that, rather than having a resistor in parallel with

C 1 , it has R 2 in series with C 2 .

The end result of this subtle change from the TT filter is

that, whereas the TT filter implements an LPF and a band-

pass filter (BPF) but no HPF, this circuit can implement

an HPF and a BPF but no LPF. In our new circuit the HPF

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Amplifiers: Op Amps

output is the node labeled “Output” in both Figures 4 and 5.

Figure 5 identifies the BPF output as “BPF OUT .”

The transfer function, pole frequency, and Q for our

new HPF are respectively given by

RR C

36 1

and Q =

(5)

Using sensitivity analysis as covered in References 2, 5, 6,

and 8, we find that, as with the KHN and TT filters, all

sensitivities of f 0 or Q to the passive components are 1 or

lower. It is quite difficult to get lower sensitivities.

Of course, this filter could be used as a separate filter

block like any other HPF topology. In the first example

given earlier, we could add a three-op-amp circuit like this

to the front of the gain block/filter section and another fol-

lowing the filter. In this case, our new HPF topology would

have no real advantage over the KHN filter, and the only

SCCRR R

1236

OUT

=−

(3)

SCCRR R

SC R

22 1

1236

(4)

CC R R

1236

Figure 5. Transfer functions for the new filter

1µF

15 k

BPF OUT

10 k

OpAmp2

10 k

OpAmp3

10 k

OpAmp1

–

Input

10 1

Output

–10

BPF OUT

–20

Output

–30

–40

–50

100

1 k

Frequency (Hz)

Analog Applications Journal

3Q 2008

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High-Performance Analog Products

Amplifiers: Op Amps

Texas Instruments Incorporated

advantage it would have over any other common topology

is that it can implement an HPF with no capacitors in the

forward signal path.

The unique feature that this filter provides is that it is

easily applied around a gain block to add a high-pass func-

tion without putting any additional circuitry in the forward

signal path. Later we will see that there is a variation of

this circuit that works with an inverting amplifier while

feeding back to the non-inverting terminal, and that there

are other variations that work with non-inverting gain

blocks. All of these variations feed back to only one input.

Applying this technique to a non-inverting

amplifier

If the gain block to which we want to add a high-pass

function is non-inverting, we can use the variation of the

topology shown in Figure 6.

The transfer function for this circuit, Equation 6, is

identical to that of the circuit in Figure 5, except that the

gain term is (R 4 + R 5 )/R 4 rather than –R 5 /R 1 :

Figure 6. Applying the technique to a non-

inverting amplifier

C 1

C 2

R 2

OpAmp2

R 6

OpAmp3

R 3

R 4

R 5

OpAmp1

–

Output

Input

Figure 7. Inverting amplifier with feedback

to non-inverting terminal

SCCRR R

1236

OUT

(6)

SCCRR R

SC R

22 1

1µF

1236

1µF

Since the rest of Equation 6 is the same as Equation 3, the

pole frequency and Q for this HPF variation are also given

by Equations 4 and 5.

Maintaining gain-bandwidth product with an

inverting amplifier

In Figure 5 we demonstrated this filter technique for an

inverting amplifier. Note that both the input signal and the

feedback signal are applied to the inverting terminal via R 1

and R 4 , respectively. While the addition of R 4 to add the

HPF feedback has no effect on the nominal forward gain

for the gain block, it does increase the division ratio of the

feedback path of the gain block from

14 k

OpAmp3

14 k

OpAmp2

10 k

OpAmp1

Input

–

Output

RR R

14 5

(7)

necessary to keep the feedback negative. The resulting

transfer function, along with the pole frequency and Q, are

respectively given by

where R 1 || R 4 represents that R 1 is parallel to R 4 . This

change in local feedback around op amp 1 has the effect

of decreasing the effective gain-bandwidth product

(GBWP) of the op amp by the same amount. In our case,

R 1 = R 4 , which means the GBWP will be decreased by 33%.

For an inverting amplifier, rather than feeding back to the

inverting terminal as in Figure 5, we can feed back to the

non-inverting terminal, thus avoiding decreasing the effec-

tive GBWP of the op amp. Figure 7 shows this variation.

Notice that R 2 is missing in this configuration. Recall that

R 2 was needed to add a zero to the feedback path, which

allowed the Q to be set to a reasonable value; without R2,

the Q of the circuit would always be very high. In this

circuit we get a zero by reconfiguring the op amp 3 stage

from inverting to non-inverting. This reconfiguration was

SCCRR

1236

OUT

(8)

=−

RR SC R

SCCRR

23 1

1236

(9)

CC R R

1236

and Q =

(10)

Analog Applications Journal

High-Performance Analog Products

www.ti.com/aaj

3Q 2008

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