Gábor C. Temes School of Electrical Engineering and Computer Science Oregon State University temes@ece.orst.edu 1/25
Noise Intrinsic (inherent) noise: generated by random physical effects in the devices. Interference (environmental) noise: coupled from outside into the circuit considered. Switching noise: charge injection, clock feedthrough, digital noise. Mismatch effects: offset, gain, nonuniformity, ADC/DAC nonlinearity errors. Quantization (truncation) noise : in internal ADCs, DSP operations. temes@ece.orst.edu 2/25
Topics Covered Types of Noise in Analog Integrated Circuits The characterization of Continuous and Sampled Noise Thermal Noise in OpAmps Thermal Noise in Feedback Amplifiers Noise in an SC Branch Noise Calculation in Simple SC Stages Sampled Noise in OpAmps temes@ece.orst.edu 3/25
Characterization of Continuous-Time Random Noise (1) Noise x(t) e.g., voltage. Must be stationary as well as mean and variance ergodic. Average power defined as mean square value of x(t): For uncorrelated zero-mean noises Power spectral density S x (f) of x(t): P av contained in a 1Hz BW at f. Measured in V 2 /Hz. Even function of f. Filtered noise: if the filter has a voltage transfer function H(s), the output noise PSD is H(j2πf) 2 S x (f). temes@ece.orst.edu 4/25
Characterization of Continuous-Time Random Noise (2) Average power in f 1 < f < f 2 : if S x is regarded as a one-sided PSD. Hence, P 0, = P av. White noise has infinite power!? Amplitude distribution: probability density function(psd) p x (x). p x (x 1 ) Δx: probability of x 1 < x < x 1 +Δx occurring. E.g., p x (q)=1/lsb for quantization noise q(t) if q < LSB/2, 0 otherwise. temes@ece.orst.edu 5/25
Characterization of Sampled-Data Random Noise Noise x(n) e.g., voltage samples. Ave. power: P av is the mean square value of x(n). For sampled noise, P av remains invariant! Autocorrelation sequence of x(n) PSD of x(n): S x (f) = F [r x (k)]. Periodic even function of f with a period f c. Real-valued, non-negative. White noise has finite power! Power in f 1 < f < f 2 : PDF of x(n): p x (x), defined as before. white noise has finite power! temes@ece.orst.edu 6/25
Thermal Noise (1) Due to the random motion of carriers with the MS velocity T. Dominates over shot noise for high carrier density but low drift velocity, occurring, e.g., in a MOSFET channel. Mean value of velocity, V, I is 0. The power spectral density of thermal noise is PSD = kt. In a resistive voltage source the maximum available noise power is hence giving temes@ece.orst.edu 7/25
Thermal Noise (2) The probability density function of the noise amplitude follows a Gaussian distribution Here, is the MS value of E. In a MOSFET, if it operates in the triode region, R=r can be used. In the active region, averaging over the tapered channel, R=(3/2)/g m results. The equivalent circuit is temes@ece.orst.edu 8/25
Noise Bandwidth Let a white noise x(t) with a PSD S x enter an LPF with a transfer function: where G 0 is the dc gain, and ω 3-dB is the 3-dB BW of the filter. The MS value of the output noise will be the integral of H 2. S x, which gives Assume now that x(t) is entered into an ideal LPF with the gain function: H =G 0 if f < f n and 0 if f > f n. The MS value of the output will then be: Equating the RHSs reveals that the two filter will have equivalent noise transfer properties if f n is the noise bandwidth of the LPF. temes@ece.orst.edu 9/25
Analysis of 1/f Noise in Switched MOSFET Circuits 1/f noise can be represented as threshold voltage variation. If the switch is part of an SC branch, it is unimportant. In a chopper or modulator circuit, it may be very important. See the TCAS paper shown. temes@ece.orst.edu 10/25
Thermal Op-Amp Noise (1) Simple op-amp input stage [3]: With device noise PSD: Equivalent input noise PSD: For g m1 >> g m3. Hence, it can be represented by a noisy resistor R N = (8/3)kT/g m1 at one input terminal. Choose g m1 as large as practical! temes@ece.orst.edu 11/25
Thermal Op-Amp Noise (2) All devices in active region, [id(f)] 2 =(8/3)kT g m. Consider the short-circuit output current I 0,sh of the opamp. The output voltage is I o,sh R o. If the ith device PSD is considered, its contribution to the PSD of I 0,sh is proportional to g mi. Referring it to the input voltage, it needs to be divided by the square of the input device g m, i.e, by g m12. Hence, the input-referred noise PSD is proportional to g m /g m12. For the noise of the input device, this factor becomes 1/g m1. Conclusions: Choose 1/g m1 as large as possible. For all noninput devices (loads, current sources, current mirrors, cascade devices) choose 1/g m3 as small as possible! temes@ece.orst.edu 12/25
Noisy Op-Amp in Unity-Gain Feedback Consider an op-amp with thermal input noise PSD P ni = 16kT/3g m1, where g m1 is the transconductance of the input devices. In a unity-gain feedback configuration: We shall assume a single-pole model for the op-amp, with a voltage gain A(s) = A 0 ω 3-dB /(s+ω 3-dB ), where A 0 is the DC gain, ω 3-dB is the 3-dB BW (pole frequency), and ω u =A 0 ω 3-dB is the unity-gain BW of the op-amp. For folded-cascode telescopic and 2- stage OTAs, usually ω u =g m1 /C, where C is the compensation capacitor and g m1 is again the transconductance of the input devices. Then the open-loop noise BW of the Op-Amp is f n = g m1 /4A 0 C, and the open-loop noise gain at DC is A 0. Hence, the openloop output noise power is If the op-amp is in a unity-gain configuration, then (for A 0 >>1) the noise bandwidth of the stage becomes A 0 f n, and the DC noise gain is 1. Hence, the output (and input) thermal noise power is This result is very similar to the kt/c noise power formula of the simple R-C circuit. temes@ece.orst.edu 13/25
Noisy Op-Amp in a Gain Stage A more general feedback stage is shown below: Let the ideal stage gain G i =Y 1 /Y 2 be constant. Then the noise voltage gain is the single-pole function where is the 3-dB frequency of A n (jω). The DC noise gain is, and the noise BW of the stage is. Hence, the output thermal noise power is and the input-referred thermal noise power is Note that P ni is smaller for a higher gain G i, so a higher SNR is possible for higher stage gains. temes@ece.orst.edu 14/25
Switched-Capacitor Noise (1) Two situations; example: Situation 1: only the sampled values of the output waveform matter; the output spectrum may be limited by the DSP, and hence V RMS,n reduced. Find V RMS from charges; adjust for DSP effects. Situation 2: the complete output waveform affects the SNR, including the S/H and direct noise components. Usually the S/H dominates. Reduced by the reconstruction filter. temes@ece.orst.edu 15/25
Switched-Capacitor Noise (2) temes@ece.orst.edu 16/25
Switched-Capacitor Noise (3) Thermal noise in a switched-capacitor branch: (a) circuit diagram; (b) clock signal; (c) output noise; (d) direct noise component; (e) sampled-and-held noise component. The noise power is kt/c in every time segment. temes@ece.orst.edu 17/25
Switched-Capacitor Noise Spectra (one-sided) - (4) For f << f c, S S/H >> S D! temes@ece.orst.edu 18/25
Switched-Capacitor Noise (5) The MS value of samples in V cn S/H is unchanged: Regarding it as a continuous-time signal, at low frequency its one-sided PSD is while that of the direct noise is Since we must have f sw /f c > 2/m, usually S S/H >> S d at low frequencies. (See also the waveform and spectra.) See Gregorian-Temes book, pp. 505-510 for derivation. temes@ece.orst.edu 19/25
Calculation of SC Noise (1) In the switch-capacitor branch, when the switch is on, the capacitor charge noise is lowpass-filtered by R on and C. The resulting charge noise power in C is ktc. It is a colored noise, with a noise-bandwidth f n = 1/(4 R on C). The low-frequency PSD is 4kTR on. When the switch operates at a rate f c <<f n, the samples of the charge noise still have the same power ktc, but the spectrum is now white, with a PSD = 2kTC/f c. For the situation when only discrete samples of the signal and noise are used, this is all that we need to know. For continuous-time analysis, we need to find the powers and spectra of the direct and S/H components when the switch is active. The direct noise is obtained by windowing the filtered charge noise stored in C with a periodic window containing unit pulses of length m/f c. This operation (to a good approximation) simply scales the PSD, and hence the noise power, by m. The low-frequency PSD is thus 4mkTR on. temes@ece.orst.edu 20/25
Calculation of SC Noise (summary) (2) To find the PSD of the S/H noise, let the noise charge in C be sampledand-held at fc, and then windowed by a rectangular periodic window w(t)=0 for n/f c < t< n/f c +m/f c w(t)=1 for n/f c +m/f c < t < (n+1)/f c n= 0, 1, 2, Note that this windowing reduces the noise power by (1 - m) squared(!), since the S/H noise is not random within each period. Usually, at low frequencies the S/H noise dominates, since it has approximately the same average power as the direct noise, but its PSD spectrum is concentrated at low frequencies. As a first estimate, its PSD can be estimated at 2(1-m) 2 kt/f c C for frequencies up to f c /2. temes@ece.orst.edu 21/25
Circuit Example 1: ΔΣ ADC Input Stage Two sampling operations result in q2 having a RMS noise component. Hence, the input-referred noise of the SC stage is. The onesided PSD=4kT/f c C 1. Including the LPF, the input-referred RMS noise voltage in the baseband becomes: Where OSR f c /2f 0. Independent of R on ; may be used to set C 1min,OSR min,t max. temes@ece.orst.edu 22/25
Circuit Example 2: Lossy Integrator with Ideal Op-Amp[4] RMS noise charge delivered into C 3 as φ 2 0, assuming OTA: From C 1 : Form C 2 : φ ia : advanced cutoff phase RMS noise charges acquired by C i during φ j = 1: Total: Input-referred RMS noise voltage: with V in set to 0. V in,n and V in are both low-pass filtered by the stage. temes@ece.orst.edu 23/25
Sampled Op-Amp Noise Example [4] φ 1 =1 Direct noise output voltage = V neq φ 2 =1 Charges delivered by C 1 and C 2 : -C 1 (V neq +V in )+C 2 (V 0 -V neq ). Charge error (C 1 +C 2 )V neq. Input-referred error voltage V neq (1+C 2 /C 1 ). temes@ece.orst.edu 24/25
Reference temes@ece.orst.edu 25/25