ECE594I notes, M. Rodwell, copyrighted
ECE594I Notes set 13: Two-port Noise Parameters
Mark Rodwell University of California, Santa Barbara
[email protected] 805-893-3244, 805-893-3262 fax ECE594I notes, M. Rodwell, copyrighted References and Citations:
Sources / Citations : Kittel and Kroemer : Thermal Physics Van der Ziel : Noise in Solid - State Devices Papoulis : Probability and Random Variables (hard, comprehens ive) Peyton Z. Peebles : Probability, Random Variables, Random Signal Principles (introductory) Wozencraft & Jacobs : Principles of Communications Engineering. Motchenbak er : Low Noise Electronic Design Information theory lecture notes : Thomas Cover, Stanford, circa 1982 Probability lecture notes : Martin Hellman, Stanford, circa 1982 National Semiconductor Linear Applications Notes : Noise in circuits.
StdSuggested references for stdtudy. Van der Ziel, Wozencraft & Jacobs, Peebles, Kittel and Kroemer Papers by Fukui (device noise), Smith & Personik (optical receiver design) National Semi. App. Notes (!) Cover and Williams : Elements of Information Theory ECE594I notes, M. Rodwell, copyrighted Two-Port Noise Description
Through the methods of circuit analysis, the internal noise generators of a circuit can be summed and represente d by two noise generators En and In .
The spectral densities of En and In must be calculated and specified. The cross spectltral ditdensity must also be calcul ltdated and specified . ECE594I notes, M. Rodwell, copyrighted Calculating Total Noise
If the generator just has thermal noise, ~ S = 4kTR EN ,gen gen
Represent the combination of amplifier voltage and current noise by a single source
ETotal = EN + I N ⋅ Z gen
We can now calculate the spectral density of this total noise : ~ ~ ~ S = || Z ||2 S + 2 Re{S Z * } En ,total ,amplifier g In En In g ~ ~ = || Z ||2 S + 2 Re{S ()R − jX } g In En In gen gen ECE594I notes, M. Rodwell, copyrighted Signal / Noise Ratio of Generator
Vsignal , En,total and En,gen are in series and see the same ldload idimpedance. The ratios of powers delivered by these will not depend upon the load. Therefore consider t he aviable noise powers.
2 The signal power available from the generator is Psignal ,available = Vsignal ,RMS / 4Rgen
If we consider a narrow bandwidth between ( f signal − Δf / 2) and ( fsignal + Δf / 2),
then the availabl e noise power from EN ,gen is 2 ~ Pnoise,available ,generator = E[En,gen ] = S ( jf ) ⋅ Δf / 4Rgen
The signal/noise ratio of the generator is then 2 2 Psignal ,available Vsignal ,RMS / 4Rgen Vsignal ,RMS / 4Rgen SNR = = ~ = P S ( jf ) ⋅ Δf / 4R kT ⋅ Δf noise,available ,generator En ,gen gen ECE594I notes, M. Rodwell, copyrighted Signal / Noise Ratio of Generator+Amplifier
2 Signal power available from the generator : Psignal ,available = Vsignal ,RMS / 4Rgen ~ Noise power available from generator : Pnoise,av,gen = SV ⋅ Δf / 4Rgen = kT ⋅ Δf ~ Noise power available from amplifier : P = S ⋅ Δf / 4R noise,av,Amp En ,total ,amplifier gen
Signal/noise ratio including amplifier noise : 2 Psignal ,available Vsignal ,RMS / 4Rgen SNR = = ~ ~ P + P S ⋅ Δf / 4R + S ⋅ Δf / 4R noise,avail ,gen noise,avail ,amp Etotal gen En ,gen gen 2 Vsignal ,RMS / 4Rgen = ~ S ⋅ Δf / 4R + kT ⋅ Δf Etotal gen ECE594I notes, M. Rodwell, copyrighted Noise Figure: Signal / Noise Ratio Degradation
signal/noise ratio before adding amplifier Noise figure = signal/noise ratio before adding amplifier
V 2 / 4R Signal/noise ratio before adding amplifier : SNR = signal ,RMS gen kT ⋅ Δf
2 Vsignal ,RMS / 4Rgen Signal/noise ratio after adding amplifier : SNR = ~ S ⋅ Δf / 4R + kT ⋅ Δf Etotal gen ~ S ⋅ Δf / 4R + kT ⋅ Δf Noise figure = F = Etotal gen kT ⋅ Δf
~ S / 4R amplifier available input noise power Noise figure = 1 + Etotal gen = 1+ kT kT ECE594I notes, M. Rodwell, copyrighted Calculating Noise Figure ~ S / 4R Noise figure = 1 + Etotal gen kT
We also know that : ~ ~ ~ S = || Z ||2 S + 2 Re{}S Z * En ,total ,amplifier g In En In g
We can calculate from this an expression for noise figure :
S + Z 2 S + 2 ⋅ Re(Z *S ) F = 1+ En s In s En In 4kTRgen ECE594I notes, M. Rodwell, copyrighted Minimum Noise Figure
Noise figure varies as a function of Z gen = Rgen + jX gen :
~ 2 ~ ~ S + Z S + 2 ⋅ Re(Z *S ) F = 1 + En s In s En In 4kTRgen After some calculus, we can find a mimimum noise figure and a generator impedance which gives us this minimum :
1 ~ ~ ~ 2 ~ F = 1+ ⎡2 S S − (Im[S ]) + 2 Re[S ]⎤ min 4kT ⎣⎢ En In En In En In ⎦⎥ ~ ~ 2 ~ S ⎛ Im[S ]⎞ Im[S ] En ⎜ En In ⎟ En In Zopt = Ropt + jX opt = ~ − ~ − j ~ S ⎜ S ⎟ S In ⎝ In ⎠ In
PitPoints to remember : ()(a) F varies with Z gen , (b) hence there is an optimum Z gen which gives a minimum F(c) . ECE594I notes, M. Rodwell, copyrighted Noise Figure in Wave Notation
Written instead in terms of wave parameters , 2 4rn ⋅ Γs − Γopt F = Fmin + 2 2 [1− Γs ]⋅[1− Γopt ]
These describe contous in the Γs − plane of constant noise figure :"noise figure circles" , i.e. a description of the variation of noise figure with source reflection coefficien t. ECE594I notes, M. Rodwell, copyrighted Low-Noise Amplifier Design
Design steps are 1) in-band stabilization: this is best done at output port to avoid degrading noise
2) input tuning for Fmin 3) output tuning (match) 4) out-of-band stabilization
Note that tuning for minimum noise figure requires a *mismatch* on the amplifier input; amplifier gain therefore must lie below the transistor MAG/MSG.
Note that tuning for minimum noise figure implies that amplifier input is mismatched: input reflection coefficient is therefore not zero !
Discrepancy in input noise-match & gain-match can be reduced by adding source inductance
Zs Zs=Zo noise match
Vgen ECE594I notes, M. Rodwell, copyrighted Example LNA Design: 60 GHz, 130 nm SiGe BJT
gain & noise circles after input matching note compromise between gain & noise tuning ECE594I notes, M. Rodwell, copyrighted Friis Formula for Noise Figure
Available gain : power gain of then amplifier with the * output * matched to the load
PAVA power available from the amplifier output GA = = PAVG power availabl e from the generator
Noise figure of a cascade of amplifiers
F2 −1 F3 −1 Ftotal = F1 + + + ... GA1 GA1GA2 Here the noise figures and available gains of each amplifier are calculated given using a source impedance equal to the output impedance of the prior stage.
The Friis expression will not be proven here due to time limits. ECE594I notes, M. Rodwell, copyrighted Noise Measure
One peculiarity of noise figure is that any active device has poorer noise figure than a simple wire connecting input and output. We need to amplify a signal to use it, and that comes at the cost of increased noise relative to the signal. Clearly F is not a the best figure - of - merit for a low - noise amplfier !
Define F∞ as the noise figure of an infinite cascade of identical amplifiers : F −1 F −1 F −1 F∞ = F + + 2 + 3 + ... GA GA GA
The * noise measure * is then defined as so :
M = F∞ −1
Mason proves that M is a network invariant, i.e. is invariant with respect to embedding the device in a lossless reciprocal network. This implies in particular that M is the same for a FET in common - source / common - gate and common - didrain configurations.