Analo Interated Circuits Lecture 6: Noise Analysis ELC 60 Fall 03 Dr. Ahmed Nader Dr. Mohamed M. Aboudina anader@ieee.or maboudina@mail.com Department of Electronics and Communications Enineerin Faculty of Enineerin Cairo University
Noise Overview The phenomenon of noise and its effect on analo circuits. Noise characteristics in the frequency and time domains. Thermal noise Shot noise (in BJT) Flicker noise Methods of representin noise in circuits. Noise in sinle-stae and differential amplifiers.
Statistical Characteristics of Noise Noise is a random process, which means the value of noise cannot be predicted at any time. How can we incorporate noise in circuit analysis? This is accomplished by observin the noise for a lon time and usin the measured results to construct a statistical model for the noise. While the instantaneous amplitude of noise cannot be predicted, a statistical model provides knowlede about some other important properties of the noise that prove useful and adequate in circuit analysis.
Averae Power of Random Sinals Since the sinal are not periodic, the measurement must be carried out over a lon time: P av lim T T + T T / / x R ( t) where x(t) is a voltae quantity. L dt Low-power random sinal Hih-power random sinal
Averae Noise Power To simplify calculations, we write the definition of P av as + T/ P av lim x ( t) dt T T T/ where P av is expressed in rather than W. In analoy with deterministic sinals, we can also define a root-mean-square (rms) voltae for noise as P av.
Calculation of noise spectrum Noise Spectrum Power spectral density (PSD): The spectrum shows how much power the sinal carries at each frequency. More specifically, the PSD, S X (f), of a noise waveform x(t) is defined as the averae power carried by x(t) in a one-hertz bandwidth around f. S X (f) is expresses in /Hz. We can apply x(t) to a bandpass filter with center frequency f and -Hz bandwidth, square the output, and calculate the averae over a lon time to obtain S X (f ). Repeatin the procedure for different center In summary, the spectrum shows the power frequencies, we arrive at the overall shape of S X (f). carried in a small bandwidth at each frequency It is also common to take the square root of S X (f), revealin how fast the waveform is expected expressin the result in / Hz. to vary in the time domain.
Noise Shapin White spectrum (white noise) Noise shapin by a transfer function Example: Spectral shapin by telephone BW S ( f) S ( f) H( f) Y X
Spectrum Power Two-sided and one-sided noise spectra Since S X (f) is an even function of f for real x(t), the total power carried by x(t) in the frequency rane [f f ] is equal to P f, f f f X S + f + f ( f) df + S ( f) df S ( f) + f X + f X df Folded white spectrum
Amplitude Distribution Probability density function (PDF) Probability density function (PDF): By observin the noise waveform for a lon time, we can construct a distribution of the amplitude, indicatin how often each value occurs. The distribution of x(t) is defined as pdf(x)dx probability of x < X < x +dx, where X is the measured value of x(t) at some point in time. Gaussian PDF is defined as ( x m) pdf( x) exp, σ π σ where σ and m are the standard deviation and mean of the distribution, respectively.
Correlated and Uncorrelated Sources We add two noise waveforms and take averae of the resultin power: P av lim T T lim T T P av + P + T/ T/ + T/ T/ av [ x ( t) + x ( t) ] x + lim + T/ + T/ ( t) dt+ lim x ( t) dt+ lim x ( t) x ( t) T T T + T/ T/ T x dt T/ ( t) x ( t) correlation dt T T T/ dt Uncorrelated noise Correlated noise
Resistor Thermal Noise (/3) Thermal noise of a resistor The thermal noise of a resistor R can be modeled by a series voltae source, with the one-sided spectral density n S v (f) 4kTR, f 0, where k.38 0 3 J/K is the Boltzmann constant and S v (f) is expressed in /Hz.
Resistor Thermal Noise (/3) Example: low-pass filter out We compute the transfer function from R to out : ( s) out From the theorem, we have S ( f) S ( f) ( f) out R R R RCs + ktr 4π R C 4 f. + The total noise power at the output: 4kTR kt u df tan u 0 4π R C f + πc u 0 P n, out kt C ( )
Resistor Thermal Noise (3/3) Representation of resistor thermal noise by a current source n In 4kT (A /Hz) R R Example Since the two noise sources are uncorrelated, we add the powers: I n, tot I n + I n 4kT R + R The equivalent noise voltae is iven by ( R R ) kt( R ) n, tot In, tot 4 R
MOSFET Thermal Noise Thermal noise of a MOSFET For the MOS devices operatin in saturation, the channel noise can be modeled by a current source connected between the drain and source terminals with a spectral density: Output noise voltae I 4kTγ, where γ is equal to /3 for lon-channel transistors and may be a lare value () for submicron MOSFETs. n m n I n r o γ 4kT mro
Danlin bonds at the oxide-silicon interface As chare carriers move at the interface, some are randomly trapped and later released by such enery states, introducin flicker noise in the drain current. The flicker noise is modeled as a voltae source in series with the ate and rouhly iven by n K, where K is a process-dependent C WL f ox constant on the order of 0 5 F. The flicker noise is also called /f noise, and it does not depend on the bias current or the temperature. It is believed that PMOS devices exhibit less /f noise than NMOS transistors because the former carry the holes in a buried channel, i.e., at some distance from the oxide-silicon interface. Flicker Noise (/)
Concept of flicker noise corner frequency For /f noise, the drain noise current per unit bandwidth is I n,/ f K n,/ f m m. C WL f ox For the thermal noise, the drain noise current per unit bandwidth is I 4kT 3 n, th m. Thus, the /f noise corner, f C, of the output current is determined by f C device dimensions and bias current. Flicker Noise (/) K 3 m, which depends on C WL 8kT ox. / )
MOS Noise. / )
Lemma Assinment 4a: Prove this Lemma
Noise in Circuits (/) How to quantify the effect of noise? The natural approach would be to set the input to zero and calculate the total noise at the output due to various sources of noise in the circuit. Example n, out ( In, th+ In,/ f + In, R ) 3 K D 4kT m+ m+ RD CoxWL f R D R D 4kT
Noise in Circuits (/) Determination of input-referred noise voltae If the voltae ain is A v, then we have A n, out v n, in, that is, the input-referred noise voltae is iven by the output noise voltae divided by the ain. The input-referred noise indicates how much the input sinal is corrupted by the circuit s noise, i.e., how small an input the circuit can detect with acceptable SNR. The input-referred noise is a fictitious quantity in that it cannot be measured at the input of the circuit.
Common-Source Stae(/3) CS stae n, out 3 K 4kT 4kT m+ m+ RD CoxWL f RD n, out n, in 4 + + Av m mr D kt 3 K C WL ox f oltae Amplification Discussion Current Generation How can we reduce the input-referred noise voltae? It implies that the transconductance of M must be maximized. The transconductance must be maximized if the transistor is to amplify a voltae sinal applied to its ate [Fi.(a)] whereas it must be minimized if the transistor operates as a current source [Fi.(b)].
Example: calculate the input-referred thermal noise voltae of the amplifier Thermal noise: ( ), 3 3 4 o o m m out n r r kt + oltae ain: A v m (r o r o ) The total noise voltae referred to the ate of M is + +, 3 3 4 3 3 4 m m m m m m in n kt kt It reveals the dependence of n,in upon m and m, confirmin that m must be minimized because M serves as a current source. Common-Source Stae(/3)
Common-Source Stae(3/3) How to desin a CS stae for low-noise operation? n, out n, in 4 + + Av m mr D kt 3 K C WL ox f For thermal noise, we must maximize m by increasin the drain current or the device width. A hiher I D translates to reater power dissipation and limited output voltae swins while a wider device leads to larer input and output capacitance. We can also increase R D, but at the cost of limitin the voltae headroom and lowerin the speed. For /f noise, the primary approach is to increase the area of the transistor. If WL is increased while W/L remains constant, then the device m and its thermal noise do not chane but the device capacitances increase. These observations point to the trade-offs between noise, power dissipation, voltae headroom, and speed.
Since the input impedance of the source follower is quite hih, the input-referred noise current can usually be nelected for moderate drivin source impedance. Compute the input-referred thermal noise:, o o mb m n M out n r r I (thermal noise) and m o o mb o o mb v r r r r A + then + +,, 3 4 m m m v M out n n in n kt A Source Followers (/)
Source Followers (/) Compute the input-referred /f noise: and then K n, out ( m Rout) + C f C ox ( WL) ox K ( WL) R out ro ro Av m m, mb n, in n, out K m + A Coxf v R ( WL) ( WL) m f out ( R ) m out
Input-Referred Noise Representation of input-referred noise by voltae and current sources Calculation of input-referred noise voltae (source impedance 0) Calculation of input-referred noise current (source impedance )
Common Gate Stae (/) Since the input impedance of the source follower is quite low, the input-referred noise current cannot be nelected especially for hih drivin source impedance.
Common Gate Stae (/) Equate the output noise voltae when the input is short circuit and divide by Av Equate the output noise voltae when the input is open circuit and divide by output impednace
Cascode Stae M, R D : Since the noise currents of M and R D flow throuh R D, the noise contributed by these two devices is quantified as in a CS stae: n, in M, R 4kT 3 m + D R m D M : In Fi.(b), if the channel lenth modulation in M is nelected, then I n + I D 0, and hence M does not affect n,out. In Fi.(c), the voltae ain from n to the output is quite small if the impedance at node X is lare. At hih frequencies, the total capacitance at node X, C X, ives rise to a ain: n, out n RD / m+ / increasin the output noise. ( C s) X NoiseofM modeledby(b)acurrent source and(c) a voltae source At hih frequencies, noise of cascodedevices start to show up. Capacitor at sources of cascodedevices shorts this point to round Gain from cascodeto output increases at hih frequencies Effect of noise increases.
Differential Pairs (/) Differential pair circuit I SS contribute common-mode noise only Circuit includin input-referred noise source For low-frequency operation, the manitude of I n,in is typical neliible.
With the inputs shorted toether, we have I n, out R + I R n n M D D n, out M I nrd Differential Pairs (/) Calculation of input-referred noise of a differential pair (if R D R D R D ) Similarly, n, out M I nrd. nout ( I I ) R M, M n n D +, Takin into account the noise of R D and R D, the total output noise: n, out M, M ( I + I ) R + ( 4kTR ) n 8kT 3 n m R D D And, A v m R D, we have + R n, out n, in 8kT + n, in CS A 3 v m mr stae D D D
Noise Bandwidth Output noise spectrum of a circuit The total output noise: and n out, tot 0 n, out, df 0 0 B, df n nout Noise bandwidth (B n ): B n allows a fair comparison of circuits that exhibit the same low-frequency noise, 0, but different hih-frequency transfer functions.
Assinment 4b: Calculate the input referred thermal noise voltae (nelect both channel and body effects)
Assinment 4b (cont):