4  Biquads

5 Architectural Options for High-Order Filters

Cascades of (active) first and second-order sections

Ladder filters (passive or emulated using active components)

  • Specialized architectures, typically emphasizing low complexity
    • Watch out for sensitivity issues (more later)

[1]

6 Building Block Options

Kuhn, IEEE TCAS II, 10/2003 [1]

7 Example

  • An interesting filter that combines three different approaches
    • Passive LC
    • Active RC
    • Switched capacitor

Cascade the filter with an equalizer block that compensates the delay variation of the filter.

Schreier, JSSC 12/2002 [1]

8 The Challenge

  • Way too many options available.
  • Deciding on which implementation is best may only be possible once several options have been thoroughly compared.
    • In terms of both first-order properties and second-order non-idealities , which aren’t always easy to understand.
  • The following discussion starts from the most basic ideas, and derives some of the most popular solutions used in practice.
  • For now, we will focus on the realization of second order sections.
    • The treatment of second order sections will help us understand, why we may ultimately want to go for a ladder implementation.

[1]

9 Passive LC Lowpass Filter

[1]

10 On-Chip Capacitors

  • Typically 1-2 fF/μm2 (10-20 fF/μm2 for advanced structures)
    • For 1 fF/μm2, a 10 pF capacitor occupies ~100μm x 100μm
  • Both MIM and VPP capacitors have good electrical properties
    • Mostly worry about parasitic caps
    • Series and parallel resistances are often not a concern

Metal-Insulator-Metal (MIM)

Vertical Parallel Plate (VPP)

Aparicio, JSSC 3/2002, doi: 10.1109/4.987091

Ng et al., IEEE Trans. Electron Dev., 7/2005, doi: 10.1109/TED.2005.850642

[1]

11 On-chip Inductors

Many nonidealities, hard to model, low “Q”

Area inefficient, typically achieve L < 10nH

Sometimes bondwires are used as an alternative, L ~ 1nH/mm

Mohan, JSSC 10/1999

Bevilacqua, ISSCC 2004

[1]

12 Inductor Quality Factor

In general:

On-chip inductors typically achieve QL ≈ 5-10 at 1 GHz

Generally unusable below 100 MHz

They are very useful for RF, though!

13 LC Lowpass Example

[1]

14 Summary

On-chip capacitors are great, even though they’re usually not as large as we would like them to be.

On-chip inductors tend to be avoided whenever possible, and are typically not useful in a filter with poles at frequencies below 200 MHz to 500 MHz.

The solution to this problem is to simulate the inductors using active components.

15 Gyrators

Gyrators are active inductors .

[2]

The above circuit is not all that useful for our lowpass filter; we need a floating inductor.

Don’t want the inductance to be ground referenced .

[1]

16 Floating Gyrator

Floating gyrators are pretty complex (and sensitive to parasitics).

There must be a better way to solve this problem …

[1]

17 Integrators

A circuit that we can build without much sweat is an active integrator, e.g. using an opamp

Assuming the availability of an ideal op-amp, we have

[1]

18 State-Space Realization

State variables (integrator outputs)

[1]

19 Signal Flow Graph (SFG)

Looks promising, but the problem with this realization is that the first integrator takes a voltage at the input and produces a current at the output.

We need the opposite if we want to realize the circuit with an RC integrator.

[1]

20 Modified (Equivalent) SFG

[1]

21 Implementation

  • One remaining issue is that the transfer function is inverted
    • We could fix that (if needed) using a fourth op-amp
    • Or by pushing A2 toward the input, and utilizing both its inverting and non-inverting input
      • The latter trick is used in the so-called KHN biquad

[1]

22 KHN Biquad

KHN biquad, Deliyannis et al., Continuous-Time Active Filter Design, 1998

[1]

23 Highpass and Bandpass Output

An interesting feature of some biquads (including the KHN) is that they provide additional highpass and bandpass outputs for “free”

[1]

24 The General KHN Biquad

Implements arbitrary poles and zeros

[1]

25 Tow-Thomas Biquad

General biquad using only three op-amps

P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” Proc. IEEE, vol. 61, pp. 662-3, May 1973.

[1]

26 Tow-Thomas Transfer Functions

Vo2/Vin implements a general biquad section with arbitrary poles and zeros

Vo1/Vin and Vo3/Vin realize the same poles but are limited to at most one finite zero

[1]

27 Tow-Thomas Design Equations

[1]

28 Sallen-Key LPF

Sallen and Key, IRE Trans. Circuit Theory, Vol. CT-2, pp. 74-85, 1955

Deliyannis et al., Continuous-Time Active Filter Design , ch. 4.5.2

Single gain element, typically 1 ≤ K ≤ 10

Poles only, no zeros

Sensitive to parasitic capacitances

Versions exist for HP, BP, etc.cnf. https://en.wikipedia.org/wiki/Sallen–Key_topology

Popular choice of parameters:

[1]

29 Tow-Thomas or Sallen-Key?

Suppose we now wanted to realize a biquad that has poles only

Should we use a Tow-Thomas or Sallen-Key realization?

Clearly, from the perspective of complexity, we would probably want to go for Sallen-Key

Unfortunately, the Sallen-Key realization comes with disadvantages in terms of sensitivity to component variations

Let’s take a closer look …

[1]

30 Bode Sensitivity

The sensitivity of any variable y to any parameter x is defined as Schaumann et al., Analog Filter Design , 2nd Ed., 2011, ch. 12.1

In order to relate fractional changes in y to fractional changes in x we can then write

Common sense: sensitivity is a first order approximation, accurate only for “small” errors (Taylor series expansion at operation point).

[1]

31 Parameter Variations (1)

Discrete resistors and capacitors

  • Come in many different shapes and sizes and accuracies
    • e.g. metal film resistors, ~0.1% accurate, 5ppm/°C
    • e.g. C0G dielectric capacitors, 2% accurate, very small temperature dependence

[1]

Integrated resistors and capacitors

  • Important to distinguish between
    • Global process variations  On the order of +/- 20% !
    • Device-to-device mismatch  On the order of 1% or less

[1]

32 Global Process Variations

[1]

33 Device-to-Device Mismatch

Upon closer inspection, device parameters not only vary from lot-to-lot or wafer-to-wafer, but there are also differences between closely spaced, nominally identical devices on the same chip.

These differences are called mismatch .

[1]

34 Statistical Model

Experiments over the past decades have shown that device-to-device mismatch for properly laid out devices is typically “random” and well-described by a Gaussian distribution.

With zero mean and a standard deviation that depends on the process and the size of the device.

Empirically, the standard deviation of the mismatch between two closely spaced devices is modeled using the following expression:where represents the area of the device, and X is the device parameter under consideration.

Sometimes referred to as “Pelgrom’s rule” with as “Pelgrom’s coefficient”Ref. Pelgrom et. al., “Matching Properties of MOS Transistors,” JSSC, Oct. 1989

[1]

35 Global vs. Device-to-Device Variations

  • Global Variations
    • Resistors, capacitors: ~ +/- 20 %
    • Time constants, RC: ~ +/- 30 %
    • Determines absolute pole frequencies
    • Use trimming or tuning for better accuracy
  • Device-to-Device Variations:
    • Resistors, small capacitors: < 1 %
    • Large capacitors & “careful layour”: < 0.1 %
    • Relative variations are much smaller than global variations:
      • All poles in a filter (on a chip) move together (with < 1 % error).
  • Hmm … could we build filters that depend only on component ratios?
    • Switched capacitor filters: Poles set by ratio of capacitors and a clock

[1]

36 Sensitivity to Global Variations

  • QP is independent of global variations in both realizations
    • Assuming all R and C use the same device structure, respectively
  • ωP varies with the RC product of the components

[1]

37 Sensitivity to Mismatch (Sallen-Key)

Sensitivity depends on QP and “component spread” i.e. the ratios of the resistors and capacitors, respectively.

[1]

38 Example (1)

  • Want to design a Sallen-Key filter with QP=10
  • Choice 1 : All R and C are the same: G = 3 -(1/QP) = 2.9
    • Very nice from the perspective of component spread, but bad for sensitivity, e.g.
  • Choice 2 : Reduce sensitivity by accepting large component spread

[1]

39 Example (2)

For G=1, we have

and it follows that

Unfortunately, in this case

Bottom line: The Sallen-Key realization suffers from a strong tradeoff between sensitivity and component spread

[1]

40 Sensitivity to Mismatch (Tow-Thomas)

  • Constant sensitivities, independent of Q and component spread
    • Much nicer!

[1]

41 Conclusions

  • Biquads can be realized in numerous different ways.
  • Implementation and component sizing have a big impact on sensitivity to variations.
  • No theory for finding a low-sensitivity architecture.
    • Use proven circuits and check!
  • Tow-Thomas biquad
    • Arbitrary poles and zeros, three amplifiers
    • Well-behaved sensitivities.
  • Sallen-Key biquad

[1]