Stability Analysis of Continuous Time Sigma Delta Modulators

Abstract

Several methods have been developed for predicting the stability of continuous time sigma delta modulators (CT ΣΔMs). In this paper, an analytical root locus method is used to determine the stability criteria for CT ΣΔMs that include exponential functions in their signal transfer functions and noise transfer functions. This method can be used to determine the minimum quantizer gains that keep a CT ΣΔM stable. Using the minimum quantizer gains, a CT ΣΔM’s maximum input amplitude that guarantees stability can be determined.

1 Introduction

Many electronic systems, including instrumentation, signal processing systems and portable communication systems, digitize analog signals using analog to digital converters (ADCs). Both continuous time (CT) and discrete time (DT) sigma delta modulator (ΣΔM) ADCs are often used for such applications because they use relatively simple, low power, analog circuitry and a low order quantizer in a feedback loop to achieve high speed, high resolution and low power signal conversion. Although CT ΣΔMs and DT ΣΔMs have many similarities such as the ability to increase their resolution by increasing their quantizer’s sampling rate, increasing the number of bits in their ΣΔM’s quantizers and increasing the orders of their loop filters, CT ΣΔMs have some advantages which include having inherent antialiasing filtering in the CT ΣΔM’s signal transfer function (STF) and operating at higher frequencies than DT ΣΔMs because CT ΣΔMs don’t have settling time requirements in their loop filters [1]; however, because DT ΣΔMs are entirely implemented using discrete time, or clocked, components while CT ΣΔMs are implemented using both analog and discrete components, DT ΣΔMs are simpler to analyze and simulate than CT ΣΔMs. Stability analysis in particular is more difficult to perform for CT ΣΔMs than it is for DT ΣΔMs.

Because a ΣΔM’s output is typically the output of the ΣΔM’s quantizer, ΣΔMs cannot be unstable in the bounded input bounded output (BIBO) sense. Instead, a ΣΔM is considered to have become unstable when the amplitude of a ΣΔM’s input is increased over a value which causes the ΣΔM’s output signal to quantization noise ratio (SQNR) to decrease dramatically and the ΣΔM’s output SQNR cannot be restored to its previous values even when the ΣΔM’s input is decreased to its previous amplitudes. Other phenomenon, such as input overload, can also cause a ΣΔM’s output SQNR to decreased dramatically when the ΣΔM’s input is increased over a certain value; however, in these cases, the ΣΔM’s output SQNR can be restored to its previous values when the ΣΔM’s input is decreased to its previous amplitudes.

In a general, a CT ΣΔM can be modeled by the canonical feedback loop shown in Fig. 1 where X(s) and Y(s) are the Laplace transforms of the input signal and the output signal, respectively, and F(s), G(s) and H(s) are the system functions of the pre-filter stage, the feedforward path and the feedback path, respectively. The quantizer block represents a clocked quantizer, and the DAC block represents a digital to analog converter (DAC). The quantizer delay and DAC delay are often represented by a single delay block as they are in Fig. 1, and the combination of these two delays is often referred to as the excess loop delay.

Fig. 1

The canonical form of the feedback system

Fig. 2 shows a linear model of the CT ΣΔM shown in Fig. 1 where the quantizer has been modeled by a variable gain, K, for the ΣΔM’s STF, Y(s)/X(s), and by an additive quantization noise, E(s) for the ΣΔM’s noise transfer function (NTF), Y(s)/E(s). The ΣΔM’s STF can be written as

$$ STF(s)=\frac{K\cdot F(s)\cdot G(s)}{1+ K\cdot {e}^{- sD}\cdot G(s)\cdot H(s)\cdot DAC(s)} $$

(1)

and the ΣΔM’s NTF can be written as

$$ NTF(s)=\frac{1}{1+{e}^{- sD}\cdot G(s)\cdot H(s)\cdot DAC(s)} $$

(2)

where the exponential function, e ^-sD is the Laplace transform of the excess loop delay, D. Although the DAC is not explicitly modeled, a typical zero order hold (ZOH) DAC would have the system function

$$ DAC(s)=\frac{1-{e}^{- sT}}{sT} $$

(3)

where T is the ΣΔM’s sampling period. Most other DACs are also typically modeled using exponential functions.

Fig. 2

A linear model of the CT ΣΔM

Root locus methods have been successfully used to determine the stability of DT ΣΔMs; however, because the denominator terms of both the STF and NTF contain exponential functions, traditional root locus methods cannot be used for determining the stability of CT ΣΔMs. Instead, several other methods have been developed for predicting the stability of CT ΣΔMs. One such method models the nonlinear quantizer using two linear gains, one for the signal gain and one for quantization noise gain [2]. This approach has not received much attention because of its complexity and because it cannot predict stability for several classes of ΣΔMs. Other approaches predict CT ΣΔM stability by assuming that the ΣΔM’s has a DC input and then by performing a simple stability analysis. These methods are effective for predicting stability for lower order ΣΔMs but not for higher order ΣΔMs [3–9]. Another method attempts to determine ΣΔM’s stability by using a one-norm of the ΣΔM’s NTF to determine stability in a BIBO sense. It has been shown that the one-norm condition is available only for second order lowpass modulators [10]. Therefore, a mixture of one-norm, two-norm and infinity-norm constraints have been proposed to predict the stability of higher order modulators [11]. Lee’s rule is another method used to determine the stability of ΣΔMs [12]. Lee’s rule states that a ΣΔM will be stable if the gain of the ΣΔM’s NTF is less than two for all frequencies. It has been shown that Lee’s rule is neither a necessary nor a sufficient condition to ensure stability in ΣΔMs [13].

In this paper, an analytical root locus method is used to determine the stability criteria for CT ΣΔMs that include exponential functions in their characteristic equations. This root locus method determines the range of quantizer gains for which a CT ΣΔM is stable. These values can then be used to determine input signal and internal signal ranges that prevent ΣΔM from becoming unstable. A circuit designer can then take measures to prevent the ΣΔM from becoming unstable. Examples of 3^rd order CT ΣΔMs illustrate this method.

2 ΣΔM Stability Analysis using an Analytical Root Locus Method

Root locus analysis is a method for examining how the poles of a system change as function of a certain system parameter. This method is commonly used to determine the stable region of feedback systems as a function of open loop gain by plotting the poles of the system’s closed loop transfer function as a function of the system’s open loop gain. As shown in (1), CT ΣΔMs typically have characteristic equations of the form

$$ 1+ K\cdot {e}^{- sD}\cdot G(s)\cdot H(s)\cdot DAC(s)=0 $$

(4)

where D is the ΣΔM’s excess loop delay and DAC(s) contains at least one exponential function.

When D = 0 and DAC(s) = 1, root locus analysis of the characteristic equation in (4) can be performed using standard graphical analysis methods [14] or using an analytical method [15,16]. When D ≠ 0 and DAC(s) contains at least one exponential function, root locus analysis of the characteristic equation in (4) can be performed using an extended graphical analysis method [17–19] or using the analytical method in [20]. In this paper, the analytical method in [20] is used to determine the quantizer gains that allow CT ΣΔΜ to remain stable.

To illustrate this method, the term e ^-sD∙G(s)∙H(s)∙DAC(s) in (4) is written as

$$ {e}^{- sD}\cdot G(s)\cdot H(s)\cdot DAC(s)=\frac{N(s)}{D(s)} $$

(5)

which implies that (4) can be written as

$$ D(s)+ K\cdot N(s)=0\ . $$

(6)

Solving (6) for K,

$$ K=-\frac{D(s)}{N(s)}=-\frac{\mathrm{Re}\left\{ D(s)\right\}+ j\mathrm{Im}\left\{ D(s)\right\}}{\mathrm{Re}\left\{ N(s)\right\}+ j\mathrm{Im}\left\{ N(s)\right\}}\ . $$

(7)

In standard form, (7) can be written as

$$ \begin{array}{c} K=\frac{-\mathrm{Re}\left\{ D(s)\right\}\cdot \mathrm{Re}\left\{ N(s)\right\}-\mathrm{Im}\left\{ D(s)\right\}\cdot \mathrm{Im}\left\{ N(s)\right\}}{{\left(\mathrm{Re}\left\{ N(s)\right\}\right)}^2+{\left(\mathrm{Im}\left\{ N(s)\right\}\right)}^2}\\ {}\kern2.25em + j\frac{\mathrm{Re}\left\{ D(s)\right\}\cdot \mathrm{Im}\left\{ N(s)\right\}-\mathrm{Im}\left\{ D(s)\right\}\cdot \mathrm{Re}\left\{ N(s)\right\}}{{\left(\mathrm{Re}\left\{ N(s)\right\}\right)}^2+{\left(\mathrm{Im}\left\{ N(s)\right\}\right)}^2}\end{array} $$

(8)

Because the quantizer’s variable gain, K, is real, (8) implies that

$$ K=\frac{-\mathrm{Re}\left\{ D(s)\right\}\cdot \mathrm{Re}\left\{ N(s)\right\}-\mathrm{Im}\left\{ D(s)\right\}\cdot \mathrm{Im}\left\{ N(s)\right\}}{{\left(\mathrm{Re}\left\{ N(s)\right\}\right)}^2+{\left(\mathrm{Im}\left\{ N(s)\right\}\right)}^2} $$

(9)

and that

$$ \mathrm{Re}\left\{ D(s)\right\}\cdot \mathrm{Im}\left\{ N(s)\right\}-\mathrm{Im}\left\{ D(s)\right\}\cdot \mathrm{Re}\left\{ N(s)\right\}=0\ . $$

(10)

Plotting (10) in the s-plane renders the root locus of (4) for -∞ < K < ∞.

3 Examples

ΣΔMs achieve high resolution by using a feedback loop to attenuate quantization noise in the frequency band of interest while passing the input signal to the output. Because of the importance of attenuating the quantization noise over the frequency band of interest, the ΣΔM’s NTF is designed before the STF. After determining an NTF and STF, the NTF and STF coefficients need to be implemented in a hardware structure, such as a cascade of resonators feedback (CRFB), cascade of resonators feedforward (CRFF), cascade of integrator feedback (CIFB), and cascade of integrator feedforward (CIFF) implementations. In the following example, 3^rd order Chebyshev Type 2 NTFs are implemented using a CIFB implementation.

Fig. 3 shows the linear model of a 3^rd order lowpass CT ΣΔM using a CIFB implementation. In Fig. 3, the signals, X(s), E(s) and Y(s), are the Laplace transforms of the input signal, the quantization noise signal and the output signal, respectively. The quantizer has been modeled by a variable gain, K, an additive quantization noise, E(s), and a delay. The DAC block represents a digital to analog converter (DAC), and the excess loop delay is represented by the Delay block as it is in Fig. 2. The blocks with the symbols, a ₀, a ₁, a ₂, a ₃, b ₀, b ₁, b ₂, b ₃, c ₀, c ₁, c ₂ and g ₀ represent scalar multiplication with gains associated by the blocks’ respective symbols.

Fig. 3

3^rd order low pass CT ΣΔ block diagram

The STF and NTF of the ΣΔM shown in Fig. 3 can be calculated from the block diagram by calculating the states and the output as

$$ {Q}_1(s)=\left\{{b}_0 X(s)-{a}_0\cdot M(s)\cdot Y(s)\right\}\frac{1}{s} $$

(11)

$$ {Q}_2(s)=\left\{{b}_1 X(s)-{a}_1\cdot M(s)\cdot Y(s)+{c}_0{Q}_1(s)+{g}_0{Q}_3(s)\right\}\frac{1}{s} $$

(12)

$$ {Q}_3(s)=\left\{{b}_2 X(s)-{a}_2\cdot M(s)\cdot Y(s)+{c}_1{Q}_2(s)\right\}\frac{1}{s} $$

(13)

$$ \varPsi (s)={b}_3 X(s)-{a}_3\cdot M(s)\cdot Y(s)+{c}_2{Q}_3(s) $$

(14)

$$ Y(s)= K\cdot \varPsi (s)+ E(s) $$

(15)

where M(s)=DAC(s)∙e ^-sD. Substituting (11) into (12), (12) into (13), (13) into (14 14) and (14) into (15), the STF and the NTF can be written as

$$ STF(s)=\frac{\frac{K\cdot \left\{{b}_3{s}^3+{b}_2{c}_2{s}^2+\left({b}_1{c}_1{c}_2-{g}_0{b}_3{c}_1\right) s+{b}_0{c}_0{c}_1{c}_2\right\}}{s\left({s}^2-{g}_0{c}_1\right)}}{1 + \frac{K\cdot M(s)\cdot \left\{{a}_3{s}^3+{a}_2{c}_2{s}^2+\left({a}_1{c}_1{c}_2-{g}_0{a}_3{c}_1\right) s+{a}_0{c}_0{c}_1{c}_2\right\}}{s\left({s}^2-{g}_0{c}_1\right)}} $$

(16)

and

$$ NTF(s)=\frac{1}{1 + \frac{M(s)\cdot \left\{{a}_3{s}^3+{a}_2{c}_2{s}^2+\left({a}_1{c}_1{c}_2-{g}_0{a}_3{c}_1\right) s+{a}_0{c}_0{c}_1{c}_2\right\}}{s\left({s}^2-{g}_0{c}_1\right)}}\kern0.5em . $$

(17)

Comparing (1) with (16) and (2) with (17), it can be seen that

$$ F(s)={b}_3{s}^3+{b}_2{c}_2 s+\left({b}_1{c}_1{c}_2-{g}_0{b}_3{c}_1\right) s+{b}_0{c}_0{c}_1{c}_2 $$

(18)

$$ G(s)=\frac{1}{s\left({s}^2-{g}_0{c}_1\right)} $$

(19)

$$ H(s)={a}_3{s}^3+{a}_2{c}_2{s}^2+\left({a}_1{c}_1{c}_2-{g}_0{a}_3{c}_1\right) s+{a}_0{c}_0{c}_1{c}_2\ . $$

(20)

Assuming that the DAC is implemented using a ZOH, DAC(s) can be written as shown in (3). The gains, a ₀, a ₁, a ₂, a ₃, b ₀, b ₁, b ₂, b ₃, c ₀, c ₁, c ₂ and g ₀ can be determined by equating the STF coefficients in (16) and the NTF coefficients in (17) with the desired STF and NTF coefficients, respectively. For example, if the NTF is a high pass Chebyshev Type 2 filter and the STF has a lowpass characteristic, then the rational function describing the Chebyshev Type 2 filter is set equal to the rational NTF in (17) and the rational function describing the lowpass STF is set equal to the rational STF in (16).

Using (5), (19), (20) and (3), N(s) and D(s) can be determined to be

$$ N(s)=\left\{{a}_3{s}^3+{a}_2{c}_2{s}^2+\left({a}_1{c}_1{c}_2-{g}_0{a}_3{c}_1\right) s+{a}_0{c}_0{c}_1{c}_2\right\}\cdot \left(1-{e}^{- sT}\right)\cdot {e}^{- sD} $$

(21)

and

$$ D(s)=\left({s}^2-{g}_0{c}_1\right)\cdot {s}^2\cdot T\ . $$

(22)

Substituting σ + jω for s where σ = Re{s} and ω = Im{s},

$$ \mathrm{Re}\left\{ D(s)\right\}= T\cdot \upsigma \left({\upsigma}^3+ a\upsigma -3\upsigma {\upomega}^2\right)- T\cdot \upomega \left(-{\upomega}^3+ a\upomega +3{\upsigma}^2\upomega \right) $$

(23)

$$ \mathrm{Im}\left\{ D(s)\right\}= T\cdot \upomega \left({\upsigma}^3+ a\upsigma -3\upsigma {\upomega}^2\right)+ T\cdot \upsigma \left(-{\upomega}^3+ a\upomega +3{\upsigma}^2\upomega \right) $$

(24)

$$ \begin{array}{c}\mathrm{Re}\left\{ N(s)\right\}=\left( b{\upsigma}^2- b{\upomega}^2+ c\upsigma + d\right)\\ {}\kern5em \left\{{e}^{- D\upsigma} \cos \left( D\upomega \right)-{e}^{-\left( T+ D\right)\upsigma} \cos \left(\left( T+ D\right)\upomega \right)\right\}\\ {} - \left(2 b\upsigma \upomega + c\upomega \right)\left\{-{e}^{- D\upsigma} \sin \left( D\upomega \right)+{e}^{-\left( T+ D\right)\upsigma} \sin \left(\left( T+ D\right)\upomega \right)\right\}\end{array} $$

(25)

and

$$ \begin{array}{c}\mathrm{Im}\left\{ N(s)\right\}=\left( b{\upsigma}^2- b{\upomega}^2+ c\upsigma + d\right)\\ {}\kern5em \left\{-{e}^{- D\upsigma} \sin \left( D\upomega \right)+{e}^{-\left( T+ D\right)\upsigma} \sin \left(\left( T+ D\right)\upomega \right)\right\}\\ {} + \left(2 b\upsigma \upomega + c\upomega \right)\left\{{e}^{- D\upsigma} \cos \left( D\upomega \right)-{e}^{-\left( T+ D\right)\upsigma} \cos \left(\left( T+ D\right)\upomega \right)\right\}\ .\ \end{array} $$

(26)

Substituting (23), (24), (25) and (26) into (10), the root locus of the 3^rd order CT ΣΔΜ shown in Fig. 3 can be plotted for -∞ < K < ∞.

Fig. 4 (a), (b) and (c) show the plot of (10), or the root locus, for 3^rd order CT ΣΔΜs with a sampling frequency, f _s where f _s =1/T, of 1GHz and Chebyshev Type 2 NTF with 47dB, 37dB, and 30dB attenuation in the stopband for D = 0, D = T/2, and D = T, respectively. The plots in Fig. 4 include both the positive gain (K > 0) root locus and the negative gain (K < 0) root locus. According to the plots, the CT ΣΔΜs with D = 0, D = T/2, and D = T are stable for 0.326 < K < 3.734, 0.359 < K < 2.001, and 0.376 < K < 1.648, respectively. Although the root locus plots show that the CT ΣΔΜs with D = 0, D = T/2, and D = T are unstable for K > 3.734, K > 2.001, and K > 1.648, respectively, none of the modulators show a degradation in SQNR when K enters those ranges because when the modulator enters unstable regions for large values of quantizer gain, K, the feedback signal increases which reduces the quantizer gain, K, and moves the poles back into a stable region. However, when K < 0.326, K < 0.359, and K < 0.376 for the CT ΣΔΜs with D = 0, D = T/2, and D = T, respectively, the modulator shows a degradation in SQNR because when the modulator enters those unstable regions the feedback signal increases which further reduces the quantizer gain, K, and consequently moves the poles further from the stable region. Therefore, a CT ΣΔΜ will remain stable if its quantizer gain, K, remains above its minimum value, K _min, as determined from its CT ΣΔΜs root locus plot.

Fig. 4

The root locus of 3^rd order CT ΣΔΜs that uses Chebyshev Type 2 NTFs and a sampling frequency of 1GHz for (a) D = 0 (b) D = T/2 (c) D = T.

Using this stability criterion, the maximum quantizer input, ψ_max, that prevents the modulator from becoming unstable is 1/K _min. Therefore, the CT ΣΔΜs with D = 0, D = T/2, and D = T are unstable when ψ_max > 3.07, ψ_max > 2.78, and ψ_max > 2.66, respectively.

Assuming that the ΣΔΜ’s input, output and quantization noise signals have means of zero, the maximum power, σ ² _x , of the ΣΔΜ’s input signal can be calculated from

$$ {\sigma}_y^2={\sigma}_x^2{\displaystyle {\int}_{- fs/2}^{f s/2}\left| STF(f)\right|{}^2} df+\frac{\sigma_e^2}{f_s}{\displaystyle {\int}_{- fs/2}^{f s/2}\left| NTF(f)\right|{}^2} df $$

(27)

where σ ² _e is the quantization noise power when the maximum input signal power is applied to the input and σ ² _y is the output signal power which equals one. If the input signal is sinusoidal, then σ ² _x = x _max ²/2 where x _max is the ΣΔΜ’s maximum input amplitude. Assuming a sinusoidal input, the maximum input values are 0.432 (-7.2 dB), 0.447 (-7.0 dB), and 0.376 (-8.5 dB) for the CT ΣΔΜs with D = 0, D = T/2, and D = T, respectively.

All three CT ΣΔΜs were simulated using the method described in [21]. Fig. 5(a) shows SQNR and the minimum gain, K _min, as a function of input signal amplitude for the ΣΔΜs with D = 0. As shown in the Fig. 5(a), the ΣΔΜ’s SQNR increases linearly until the input signal’s amplitude is less than -8 dB, or K _min = 0.391. As the input signal’s amplitude is increased above -8 dB, the ΣΔΜ’s SQNR no longer increases linearly. As the input signal’s amplitude is increased above -6 dB, the ΣΔΜ’s SQNR degrades dramatically and the ΣΔΜ’s SQNR cannot be restored to its previous values even when the ΣΔM’s input is decreased to its previous amplitudes. Fig. 5 (b) and (c) show the SQNR and the minimum gain, K _min, as a function of input signal amplitude for the ΣΔΜs with D = T/2 and D = T, respectively. From Fig. 5 (b), when the input signal’s amplitude is increased above -7 dB, or K _min is less than 0.359, the ΣΔΜ’s SQNR begins to degrade. Similarly, the ΣΔΜ’s SQNR with D = T is degraded when the ΣΔM’s input is greater than -9 dB, or K _min is less than 0.376 as shown in Fig. 5 (c).

Fig. 5

Simulated SQNR and the minimum gain (K _min) for the 3^rd order CT ΣΔΜ for (a) D = 0 (b) D = T/2 (c) D = T

4 Conclusion

In this paper, an analytical root locus method was used to determine the minimum quantizer gains that keep a CT ΣΔM stable. It was then shown how to use the minimum quantizer gain to determine the maximum quantizer input amplitude and ΣΔM maximum input amplitude value that prevent ΣΔM from becoming unstable. Using these values, a circuit designer can then take measures to prevent the ΣΔM from becoming unstable. Examples of 3^rd order CT ΣΔM illustrate this method.