In Mixed-Signal Circuits, Part 1: From s to z - The Discrete Complex-Frequency Domain, discrete-time analog circuit analysis introduced the z-domain, a simple shortcut from discrete time-domain equations to the sampled s, or s*-domain. In this part, we push forth to transfer functions in s and z of ADCs and DACs, then consider briefly how to design feedback loops that span the analog-digital boundary.
Part 1 left us hanging as to how impulse trains relate to discrete-time waveforms. The development continues, and in the concrete context of ADCs, so that the abstraction of impulses can have a clearer meaning.
ADC Dynamics in z
To demonstrate how the z-domain can be applied in circuit engineering, a 12-bit ADC transfer function is
where vADC is the input voltage, VR is the full-scale 3.3 V ADC reference voltage, and wADC is the 12-bit ADC output. This ADC transfer function is quasistatic; it applies at 0+ Hz. The ADC dynamics of the clock-sampled zero-order hold (ZOH) output accounts for the sampling behavior of the ADC. The sampler, which produces the impulse trains from Part 1, is followed conceptually by a hold function,
The right-side expression can be interpreted as the gain-normalized integration (1/s x Ts) of an impulse train in the time domain, resulting in a sequence of step functions, each turned off Ts later (− exp(−s x Ts)) so that their effect only occurs over one sample interval. At each sample time, a new value of ADC output is acquired and held. Plotted against time, this waveform has the stair-step look of a sample-and-hold waveform described by TZOH(s), as shown below.
The acquired continuous voltage waveform, vi(t), is plotted along with its piecewise-continuous or “stepped” digital output, approximated as the continuous vo(t), shown as a dotted curve. This approximation of the acquired waveform is reconstructed by shifting the continuous waveform, vi, by Ts/2 to the right, a delay of a half sampling period. It passes through the centers of the vertical steps of the acquired ZOH waveform.
The ADC sample-hold frequency response is the magnitude and phase of TZOH(j
The main step in this derivation is to multiply numerator and denominator by e-jω Ts/2, then apply (from Euler’s equation, ej
The amplitude of TZOH(j
) is thus
This function begins with a value of one at zero frequency (
= 0 s–1) and decreases to zero at the sampling frequency. Then it “bounces” (because of the absolute value) between zero values at harmonics of fs, outside the Nyquist interval. The rolloff of magnitude with frequency can be compensated with an inverse ZOH (ZOH-1) function and can be included following the ADC block as code in the
C. It can be omitted if the ADC sample rate is enough above the analog-circuit bandwidth to make amplitude roll-off and phase delay insignificant so that the sampled result is approximately continuous.
The phase is
In degrees, the phase is −180o(f/fs), where fs = 1/Ts. The Ts/2 delay in the phase is the half-step of delay observed graphically in the plot above of vo delayed from vi.
At the Nyquist frequency, fs/2, the magnitude will have decreased to (sin(
/2) = 2/
0.637 with a phase delay of −180o/2 = −90o. A sampling rate of 10 times fs results in a normalized amplitude of 0.9836, with about 6 bits of accuracy. Some values are given in the following table.
We arrive at the final step, the ZOH function, which completes the transformation from z to s:
Thus, to transform from z to s, substitute z(s) = exp(sTs) for z, then multiply the resulting impulse-train or sampled expression in s by the ZOH function.