Chapter 5 Models of Sigma-Delta Modulator Nonlinear Distortion
5.1 Settling Distortion
Models of Sigma-Delta Modulator Nonlinear Distortion
5.1 Settling Distortion
We analyze incomplete transfer of charge in a SC integrator to obtain analytical models to represent harmonic distortion as function of the operational amplifier finite gain-bandwidth (GBW), slew-rate (SR). The model developed here assumes the effect of the SR in a SC integrator may be interpreted as a nonlinear gain. Consider the integrator operates in the integration phase. As discussed in Chapter 4, there are three settling conditions depending on the absolute value of VS [Mal 03].
We can represent integrator output voltage during the nth integration interval as
) considered here because it is not significant. Note that（5.1）and（5.2）at end of each integration interval can be rewritten as
L
which is the integrator gain. Harmonic distortions are produced at the modulator output when op-amps operate in the partial slewing region, because in the partial slewing region the integrator gain is a function of input VS. In order not to produce harmonic distortion, op-amps should always operate in the linear region. From （5.4）, we can see that if VS ≤VL is satisfied all the time, the modulator always operates in linear region and harmonic distortion would not be produced. VS ≤VL can be further derived as:
We then plot（5.5）as shown in Fig. 5.1 which shows that OSR is inverse proportional to SR and is almost independent to GBW.
Fig. 5.1 3D plot of（5.5）
Fig. 5.1 indicates that if we design SR and GBW above the curve with desired OSR, the modulator would have no harmonic distortion. It shows that the op-amp slew rate needs to be at least 200V/us, then the modulator can have no harmonic distortions with OSR larger than 15. Although op-amps operate in linear region can have no harmonic distortion, it may consume more power dissipation (because large slew rate). Therefore, there has a trade off between power consumption and harmonic distortion. In general, one can choose smaller slew rate to let power consumption lower and have negligible harmonic distortions. In the following, we analyses the influences of slew rate on harmonic distortion when op-amps operate in partial slewing region.
Assume that gi(v) can be approximated by ) (
)
(v a1 1 3v2 5v4
pi = ⋅ α +α +α （5.6）
In this point, the problem of estimating harmonic distortion consists of searching for the curve with the form shown in （5.6） which best fits （5.4） for a specific interval. We will use the
least square method to determine the coefficient α1,α3 and α5 to fit （5.4）. The pi(v) should be fitted through all the points in that specific interval so that the sum of the squares of the distances of those points from the pi(v) is minimum. The sum of the squares is
With this method, the calculation of the coefficients in （5.6） becomes the solution of the following system of linear equations:
[ ]
where Vh is the maximum distribution range of the first integrator input
The amplitudes of the third and fifth harmonics of the modulator output are:
; 16
clear that using Ainis not correct, and our simulation shows that（5.8）is correct and precise.
Next we need to obtain an expression for AVS. X(z) plus high-pass filtered (noise shaped) quantization noise E(z). Therefore,
)
Ignoring the quantization noise and taking the inverse z-transform, one obtains VS(t)=x(t)−x(t−2T)u(t−2T) = Ainsin(ωt)−Ainsin(ω(t−2T))⋅u(t−2T) （5.12） Then, the amplitude of VS can be obtained as
T A T A
T x T V
AVS = S(2 )= (2 )= insin(ω⋅2 )≅2 in⋅ω⋅ （5.13） Note that AVS is not related to quantizer bit number B which can only affect the level of noise floor E(ω). The result（5.13）has been verified by behavior simulation under different B values, as shown in Fig. 5.2. From（5.9）（5.13）, we can see that input signal amplitude Ain, input signal frequency ω and sampling time T are the critical parameters to impact the harmonic distortion.
Fig. 5.2 Spectrum of VS with different quantizer bit number
In order to verify the result in （5.9）, we use SIMULINK to build a second-order Σ∆
modulator with a multi-bit quantizer. The behavioral settling model in [Mal 03] is employed.
We assume that SR = 70V/µs, GBW = 100MHz, R = 300Ω, OSR = 16, fB = 1MHz and CS = 2pF, and a 1MHz sinusoidal input signal is used. After performing FFT to the output data of the Σ∆ modulator, we obtain the simulated PSD (Power Spectrum Density) which is
shown in Fig. 5.3. It shows that HD3 is -112.5dB and HD5 is -117.5dB. The theoretical harmonic powers calculated from（5.8）and（5.9）are HD3 = -112.4dB and HD5 = -117.3dB.
The simulated and theoretical results are very close, and this confirms that our settling distortion model is reasonably precise.
Fig. 5.3 Output spectrum of a second-order Σ∆ modulator with harmonic distortion
In order to provide insight on how settling distortions are related to circuit and system parameters, we further analyze the 3rd and 5th harmonic powers as follows:
= 2 4
log 1 20 ) ( 3
3 3 VS settling
dB A
HD α
( )
[
log log 2 log4 2]
20 3 + 3−
= α AωT
095 . 30 log
60 log
20 3 − +
= α OSR （5.14） 15
. 48 log
100 log
20 ) (
5 dB = 5 − OSR+
HD settling α
From （5.14） we can see that OSR can effectively influence settling harmonic powers. The
（5.8）reveals that α3 and α5 are functions of T, GBW, R,CS and SR. Using the parameters designed in Chapter 8 with fS = 52MHz, R = 300ohm, CS= 1.7pF, and setting GBW and SR at medium values as GBW = 250MHz and SR = 250V/µs, we plot 20logα 3 vs. SR in Fig. 5.4 and
log 3
20 α vs. GBW in Fig. 5.5.
Fig. 5.4 log 3
20 α vs. SR
Fig. 5.5 20logα3 vs. GBW
In general, harmonic distortion less than -110dB can be ignored because it is below the noise floor of modulator output spectrum. From（5.14）, Fig. 5.4 and Fig. 5.5, we can obtain the minimum required SR and GBW w. r. t. a specific OSR. The results are summarized in Table 5.1.
OSR HD3(dB) SR
) / (V µs
GBW (MHz)
8 20logα3 -24 ≥500 ≥380
16 20logα3 -42 ≥200 ≥180
32 20logα3 -60 ≥120 ≥70
50 20logα3 -72 ≥110 ≥60
64 20logα3 -78 ≥100 ≥50
96 20logα3 -89 ≥90 ≥40
Table 5.1 Minimum SR and GBW required w. r. t. OSR
It is clear from Table 5.1 that as OSR decreases, SR and GBW have to increase dramatically so that the effect of settling distortion can be contained. This can be explained by（5.13）, since T increases when OSR decreases.