Nyquist criterion based design of continuous time

Nyquist criterion based design of continuous time Σ∆ modulators J. De Maeyer, P. Rombouts and L. Weyten Ghent University, Electronics and Information Systems (ELIS), St.-Pietersnieuwstr. 41, 9000 Ghent, Belgium jpdmaeye,rombouts,[email protected]

Abstract— In this paper we propose a way to design continuous time Σ∆ modulators. The method is based on the Nyquist stability criterion. Based on this criterion we propose to use the vector gain margin as a robust stability margin. Using this margin as a design criterion we design a robustly stable modulator. Finally, we also show how this margin can be used to evaluate the relative stability of a designed modulator.

y(n)

y(n) Fig. 2.

H D A C(s)

H (s)

z(n) T

H d(z)

z(n)

Equivalence between DT loop filter and CT loop filter.

I. I NTRODUCTION Fig. 1 shows a linearized model of a continuous time (CT) Σ∆ modulator. It consists of a loop filter H(s), a DAC pulse shaper HDAC (s), a sampler with sampling period T and a quantizer. For the rest of this paper the sampling period will be normalized to 1. The loop filter H(s) is a rational function, for which we can write (non-optimized NTF-zeros): an−1 sn−1 + an−2 sn−2 + · · · + a1 s + a0 (1) sn Where n is the order of the Σ∆ modulator. For the DAC −s pulse shaper we assume a NRZ pulse (HDAC (s) = 1−es ). In this paper we concentrate on Σ∆ modulators with a multibit internal quantizer. This way the loop can be analyzed accurately by replacing the quantizer by an additive source of white quantization noise (Q in Fig. 1). A delay of τ is inserted in the loop, to take into account the required time to perform dynamic element matching, as well as the finite quantizer decision time. H(s) =

the n + 1 poles freely with the n coefficients (an−1 . . . a0 ) of H(s). Moreover, if the nominal loop filter is implemented, then the modulator with delay τ might be unstable. One way to circumvent this problem is to add a second feedback DAC which is directly connected to the input of the quantizer (e.g. [2]). Now, the gain of this extra feedback DAC together with the n coefficients (ai ) provides the freedom to design the n-order modulator according to [3]. As will be shown, it is perfectly feasible to realize stable modulators without the need for this extra DAC. In the next section we will define robust stability and later we will propose to use the vector gain margin (VGM) as a robust stability measure. This margin indicates the ‘level of stability’. Next, we will propose a design example where we show the usefulness of the VGM in the design of CT Σ∆ modulators. In this design example the coefficients ai are chosen while optimizing the VGM and fulfilling a certain performance requirement.

Q x(n)

z(n)

H (s)

II. ROBUST STABILITY y(n)

T H D A C (s) Fig. 1.

-sτ e

Linearized model of a CTSD modulator

For the moment assume the delay τ is zero. If we open the feedback loop after the sampler as shown in Fig. 2, then there exists a discrete time (DT) equivalent loop filter Hd (z) and an equivalent noise transfer function NTF(z) [4]. Today, often the design of CT Σ∆ modulators is based on choosing the loop filter H(s) such that its equivalent NTF(z) equals a NTF designed using [3]. However, a nonzero delay τ augments the order of the modulator from n to n + 1. It is obvious, we cannot set

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To define robustness we need a nominal system, in our case a CT Σ∆ modulator. The model of the nominal system includes all the dynamic behavior we can (easily) model or we think is essential for the behavior of the system. However, this system is liable to parasitics and/or uncertainties. Therefore, the actual system belongs in reality to a family of systems. This family consists of the nominal system perturbed with parasitics. Then by definition the system is said to be robustly stable if all members of the family are stable. Let us illustrate this for the third order CT Σ∆ modulator of Fig. 3. First we define our nominal system, the aspects of the modulator we take into account in our nominal model are: 1) the delay in the loop. We make this delay part of our nominal modulator since we know it has a major influence on the stability. Next, we also assume this

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Q (z)

+

c1 s

c2 s

+

c3 s

+

+

FF3

+

FF2 FF1 H D A C (s)

Fig. 3.

-sτ e

Feedforward topology of third order CT Σ∆ modulator.

delay to be constant in time and known. The latter can be very well approximated by the introduction of a synchronization latch. 2) the GBW of the first integrator. As is known, in most modulators the first integrator is the main power consumer and its power consumption is directly proportional to the GBW of the operational amplifier. Hence, we select this GBW to be a design parameter which we want to make as small as possible. The model for the gain of an operational amplifier with a finite GBW is:
 rad Adc GBW = Adc ωA A(s) = s (2) s ωA + 1 Due to this GBW effect, the transfer function of the first integrator becomes [1] (Adc
) : ITFGBW ≈

c1 · s

GBW GBW+c1 s GBW+c1 +

(3)

1

In conclusion our nominal loop filter has the form of eq. (4) at the bottom of this page. Where, the subscript n stands for nominal value. Next we define the uncertainties to which we want our modulator to be robustly stable. 1) In all implementations the parameters ci,n are realized by the product of a resistor (or the inverse of a transconductance gm ) and a capacitor value. In a typical technology the mismatch of this product can be up to about 30 %. So, the actual value of parameter ci = ci,n (1 ± α), (α ≤ 0.3). In our example we assume that all the integrators are scaled versions of a unit version. Due to the relative matching of resistors, capacitors and transconductances in the different integrators, α does not depend on i. 2) The GBW in the operational amplifier is typically set by the ratio between a transconductance gm and a (compensation) capacitor. Again a possible deviation of 30 % is expected. 3) The third uncertainty is the presence of an extra pole (ωA2 ) and zero (ωz ) in the first operationel amplifier. We assume these to be: ωA2 ≥ 2 · ωA

ωz ≥ 5 · ωA

c1,n · Hnom (s) = s

GBWn GBWn +c1,n s GBWn +c1,n +

(5)

This yields an operational amplifier with a reasonable phase margin of at least 55°. 4) The last uncertainty takes the GBW of the other integrators into account, which we assume to be the same or higher than the one of the first integrator. Note that the parameters FFi are typically implemented as the ratio of resistors or capacitors. This means a matching of about 1% can easily be obtained using unit elements. So, we assume their values are exact. Although, in this paper we assume the delay is exactly known, it should be clear to the reader that it is possible to take a deviation from this nominal value into account as another uncertainty. This is not only true for the delay. It is actually up to the designer to make a reasonable choice for its nominal system as well as the definition of the uncertainties. It is also possible to shift some of the parasitic effects into the nominal model e.g. if more is known about the GBW of the second integrator. With the definition of the nominal modulator and the uncertainties, the family of modulators is defined. Next, we can investigate the robust stability of the modulator. The result of this analysis will be whether the modulator exhibits robust stability or not. However, the model of the nominal system nor the model of the nominal system extended with the uncertainties is an ‘exact’ model for the actual system. This means we are also interested in a margin that indicates the robust stability of the modulator, i.e. the relative robustness of the modulator. This measure of robust stability can then be used for three purposes: 1) First, it gives us an indication on how large other unmodelled uncertainties may become before the actual system become unstable. 2) Second, it provides us a solution for our initial problem: the coefficients ai (or equiv. ci,n ) will be chosen such that our nominal modulator exhibits the largest robust stability margin. 3) Third, we will be able to compare the robustness of an actual implementation of the modulator with the ideal nominally designed modulator. (see later) III. N YQUIST CRITERION In this section we will propose a robust stability margin, based upon the Nyquist criterion. A. Continuous time The stability of continuous time feedback systems is often analyzed using the frequency response of the open loop gain G(s). It is commonly known that the stability of the closed loop system is determined by the location of the roots of the characteristic function 1 + G(s). A stable system requires the

FF1,n s2 + c2,n FF2,n s + c2,n c3,n FF3,n −sτ ·e 1 s2 ·

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(4)

roots of the characteristic function to be located in the left half of the s-plane. The Nyquist stability criterion relates the number of unstable closed loop poles, to the relative location of the polar plot of G(s), the Nyquist curve. This curve is drawn for every s on the contour D in the s-plane, as shown 1 in Fig. 4. As an example the Nyquist curve of G(s) = s(s+1) 2 is shown in Fig. 4 as well. Note the small encirclement of D of the point 0, as s = 0 is a pole of G(s). As is well known the stability of the closed loop system is studied by the encirclements of the Nyquist curve of the critical point -1. d

Hd (z) =

Hd (z) ≈

d

a

c

Fig. 4.

Nyquist theorem for a continuous time feedback system.

B. Discrete time The above concept can readily be extended to the determination of stability of sampled-data feedback systems. To do this, open the feedback loop similar to Fig. 2. Hence, the open loop gain can be described by a discrete transfer function Gd (z). Now, z is chosen on the contour Dd of Fig. 5 and the Nyquist curve corresponds to the polar plot of Gd (ejω ). For the example of the previous subsection G(s) preceded by a NRZ DAC and followed by a sampler, we plotted the Nyquist curve of the corresponding Gd (z), in Fig. 5. This time, note the small encirclement of the point 1 (ω = 0), as z = 1 is a pole of the open loop filter Gd (z). In general the Nyquist stability criterion for a stable open loop Gd (z) demands, the number of encirclements of the point -1 (critical point) by the Nyquist curve to be zero.

N 

(H(s + jkωs )HDAC (s + jkωs )) .

The above result is very useful, since it only requires knowledge of the Bode diagram of H(s), e.g. obtained by a SPICE simulation of the implemented loop filter. Based on this Bode diagram we can easily check the stability of the modulator. Moreover, as will be shown in the next section, plotting the corresponding Nyquist curve provides a simple indication on the robustness degradation of the implemented loop filter compared to the nominal one. This is the third use of the margin as indicated in section II. IV. ROBUST STABILITY MARGIN Based on the Nyquist theorem, three simple robust stability margins are readily available. As the stability of the closed loop is determined by the number of encirclements of the critical point, we could somehow indicate how close the Nyquist curve is about to encircle this critical point. Two such margins are the gain margin (GM) and phase margin (PM), which are indicated in Fig. 6. These two margins measure the distance of the Nyquist curve to the critical point, in two specific directions, i.e. on the real axis resp. the unit circle. They can be interpreted as the robustness of the system against extra gain in the loop, resp. the robustness against extra phase shift.

8

Dd

1 GM

b c

1 a X

PM

d

Rm in

1 VGM

a

Fig. 5.

(7)

d

z-plane

b c

(6)

k=−N

b c

a

(H(s + jkωs )HDAC (s + jkωs ))

where ωs is the sampling pulsation. The infinite sum results from the well known periodic frequency extension (aliasing) as a result of the sampling operation. Noting that for s → ∞ H(s) → 0, we obtain an approximation for Hd (z):

D 0

∞  k=−∞

8

s-plane b

is that it requires an expression of the open loop H(s) as a transfer function, as it is the case for the nominal modulator. However, even if such an expression is not available it is still possible to determine Hd (z) [5]:

Nyquist theorem for a discrete time feedback system. Fig. 6.

C. Practical ways to determine Nyquist curve The Nyquist curve hence provides an easy way to check the stability of the CT Σ∆ modulator based on the open loop Hd (z) without the need for calculating the closed loop poles. This Hd (z) can e.g. be determined using the impulse invariant transformation [4]. The disadvantage of this method however

Robustness margins.

It may be obvious that in a real system the uncertainties are much more complicated. Therefore, we propose to use the vector gain margin (VGM) as a robust stability margin. The VGM is defined as the gain margin in the direction of the worst possible phase. It is hence related to the minimum

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distance from the Nyquist curve to the critical point -1. This distance equals |1 + Hd (z)|. Now, we define: Rmin = min |1 + Hd (z)|,

(8)

ω

then the VGM is defined as: 1 (9) 1 − Rmin Once, we obtained the Nyquist curve this margin is not only simple to be calculated, it also has a nice interpretation. Maximizing the VGM, means, maximizing Rmin (note that Rmin ≤ 1). From the definition of Rmin and the NTF, we conclude that maximizing the VGM is equivalent to minimizing the maximum absolute value of the NTF. Finally, note that we need the VGM of Hd (z) and not of H(s). Only then we take all the aliasing effects (due to the sampler) into account. As also shown in Fig. 4 and Fig. 5, the Nyquist curves of G(s) and Gd (z) might look similar, but the margins are totally different. V GM =

is robustly stable. So, we can state that the designed modulator without any tuning of coefficients and perturbed with this set of rather harsh uncertainties is guaranteed stable. However, the small VGM of 1.04 for the worst case modulator of the set, indicates that unmodelled uncertainty might result in a robust stability problem (i.e. a yield problem). Also Fig. 8 indicates making GBWn even smaller would result in a non-robustly stable modulator. 2 1.5 1 0.5 0 −0.5 −1 −1.5

V. D ESIGN E XAMPLE

−2 −3

In this section we propose the design of a third order modulator with oversampling ratio (OSR) of 16. We aim for an in band RMS value of the NTF of -28.5 dB. This corresponds to a 13-bit modulator if a 5-bit internal quantizer is used. For our nominal modulator, we take τ = 0.25T and we want to investigate whether we can design a modulator, with a GBWn as low as 0.5 · 2πfs . The design is based on finding the parameters ci,n in eq. (4) (FFi = 1) such that the performance requirement is met and such that the VGM is maximized. This optimization yields the following coefficients: c3,n = 1.253, c2,n = 0.266, c1,n = 0.137. The Nyquist curve of the resulting nominal loop filter is shown in Fig. 7, it exhibits a VGM of about 1.67. 1 1.5 1 0.5

−2

−1

0

1

Fig. 8. Detail of Nyquist curve of 250 members of the defined family of modulators.

We performed extensive time and frequency domain analysis on the designed nominal modulator (i.e. with τ = 0.25T , GBWn = 0.5 · 2πfs and the designed parameters ci,n ) with a 5-bit internal quantizer. The modulator indeed achieved the desired 13-bit performance. VI. C ONCLUSIONS In this paper we applied the concept of robust stability to CT Σ∆ modulators. Based on the Nyquist stability criterion, we propose to use the vector gain margin as a measure for robust stability. We show that using the VGM as a simple design criterion allows to design a non-trivial CT Σ∆ modulator with good performance and stability. We also indicate how we can use the Nyquist curve and the VGM to evaluate the robustness of a modulator based on a simple Bode diagram (e.g. obtained with a SPICE simulation) of the loop filter.

0

ACKNOWLEDGMENT J. De Maeyer is supported by a fellowship of the Fund for Scientific Research - Flanders (F.W.O.-V., Belgium.)

-0.5 -1 -1.5 -2.5

-2

-1.5

-1

-0.5

0

R EFERENCES

0.5

Fig. 7. Detail of the Nyquist curve of the designed modulator. The concentric circles correspond to increasing values of Rmin

To check whether the modulator is robustly stable, the Nyquist curves of 250 randomly selected members of the previously defined family of modulators are determined. The result is shown in Fig. 8. As can be seen, none of these modulators is unstable, which means the designed modulator 1 As required for stability, the netto number of encirclements of this Nyquist curve of the critical point -1, is indeed zero if we think of the behavior for ω ≈ 0 (i.e. the three zeros at 1).

[1] M. Ortmans, F. Gerfers, and Y. Manoli, “Compensation of Finite GainBandwidth Induced Errors in Continuous-Time Sigma-Delta Modulators,” IEEE Trans. Circuits Syst.-I, vol. 51, no. 6, pp. 1088–1099, Jun. 2004. [2] S. Paton, A. Di Giandomenico, L. Hern`andez, A. Wiesbauer, T. P¨otscher, and M. Clara, “A 70-mW 300-MHz CMOS Continuous-Time Σ∆ ADC With 15-MHz Bandwidth and 11 Bits of Resolution,” IEEE J. Solid-State Circuits, vol. 39, no. 7, pp. 1056–1063, Jul. 2004. [3] R. Schreier, “An Emperical Study of Higher-Order Single-Bit DeltaSigma Modulators,” IEEE Trans. Circuits Syst.-II, vol. 40, no. 8, pp. 461–466, Aug. 1993. [4] R. Schreier and B. Zhang, “Delta-Sigma Modulators Employing Continuous-Time Circuitry,” IEEE Trans. Circuits Syst.-I, vol. 43, no. 4, pp. 324–332, Apr. 1996. [5] J. T. Tou, Digital and Sampled-Data Control Systems. McGraw-Hill Book Company, Inc., 1959.

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