Outline. Thermal noise, noise power and noise temperature. Noise in RLC single-ports. Noise in diodes and photodiodes

Transcription

1 3. Noise 1

2 Outline Thermal noise, noise power and noise temperature Noise in RLC single-ports Noise in diodes and photodiodes -port and multi-port noise parameters Noise temperature and noise parameter measurements

3 Thermal noise, noise power, noise temp definitions of T n T n T of resistor for given p n T n p n k 3 definitions of noise power density, p n Planck, [ hf ] p Planck kt n =kt hf e kt 1 R Rayleigh-Jeans (from Planck), p J n =kt [ Callen-Welton p C W n = p Planck n p vac n =kt e hf kt hf kt 1 ] hf 3

4 Difference between T n and T for a resistor R T J n =T [ hf ] T Planck kt n =T hf e kt 1 T n C W =T [ hf kt hf e 1] hf k kt A.R. Kerr and J. Randa, Thermal Noise and Noise Measurements A 010 Update, IEEE Microwave Magazine, pp.40-5, October

5 Lossy Single-Port Noise: Resistor v n =4 k T f R R= 1 G k = JK 1 i n =4 k T f G v n f = R 1 k 4.06 nv Hz 5

6 Lossy Single-Port Resistor + Inductor v n =4kT f R R= 1 G v n i 4k T f G n = R = L 1 L G 6

7 Lossy Single-Port Noise: Resistor Capacitor i n =4k T f G R= 1 G i n v 4k T f R n = G C = 1 C R 7

8 Example: series L-R series + shunt C-R 8

9 Diode Noise v n s v ns =4 k T f R s Intrinsic noise sources i nj, v ns R S C J i n j Equivalent input thermal noise sources: v n or i n R J i n j = q f I d K 1 f I d f Equivalent noise conductance v n R S R S G e i n 4 kt f q I d k T = 1 R j C J R J C J R J i n 9

10 Photodetectors PIN diodes: M = 1, F(M) = 1 APDs: M = , F(M) = where: M is the multiplication factor and F(M) is the noise factor of the photo-diode I p h =RM P; R= diode responsivity and P= optical power i n, p h = q I f = q f RM P M I dark MF M G seq = RP I dark M F M kt q P m i n = 1 R q f I dark F M 10

11 Noise source T e and ENR White noise T N e source o R N o e R R e e T e = equivalent (thermal) noise temperature of a white noise source ENR = excess noise ratio ENR db =10 log 10 T e T 0 T 0 T 0 = reference temperature, typically 90 o K. 11

12 Multi-port noise representations: Y-params One noise current source exiting each port Y correlation matrix [I1 11 Y 1.. Y 1n V 1 I Y 1 Y.. Y n V I n]=[y n] [ i n1 i n.. V. i nn] Y n1 Y n.. Y nn][ Noisy n-port network [Y] V 1 V I 1 I Noiseless n-port network [Y] i n1 i n I 1 V 1 I V, i n1 i n1, i n.. i n1, i nn n, i n1 i n, i n.. i n, i nn ] [C y ]=[ in i nn, i n1 i nn, i n.. i nn, i nn V n I n i nn I n V n For passive network: [C y ] = 4 kt f Re{[Y]} 1

13 Multi-port noise representations: Z-params One noise voltage source at each port Z correlation matrix [V 1 V n]=[ Z 11 Z 1.. Z 1n Z 1 Z.. Z n V nn][i1 n] [ Z n1 Z n.. Z I v I.. n1 v n...v nn] Noisy n-port network [Z] V 1 V I 1 I Noiseless n-port network [Z] v n1 v n I 1 V 1 I V, v n1 v n1, v n.. v n1, v nn n, v n1 v n, v n.. v n, v nn ] [C z ]=[ vn v nn, v n1 v nn, v n.. v nn, v nn V n I n v nn V n I n For passive network: [C z ] = 4 kt f Re{[Z]} 13

14 Multi-port noise representations: noise wave One noise wave emanating from each port wave correlation matrix [b1 1 S1.. S1 n b n 1 b S 1 S.. S n a b n b n]=[s1 n]. b n 1, bn 1 b n 1, b n.. b n 1, b n n n, b n 1 b n, b n.. b n, b n n ] [C S ]=[ bn b n n, b n 1 b n n, b n.. b n n, b n n n][a1 n] [ S n 1 S n.. S n a Noisy n-port network [S] a 1 b 1 Z 0 a b a n b n Noiseless n-port network [S] b n1 a 1 b 1 b n Z 0 a b a n b nn T e 1 T e T b e n n For passive network: [C S ]=k T f [U ] [ S ][ S ] + 14

15 Other Noisy -Port (only) Correlation Matrices I 1 V 1 Noisy -port Y,Z,G,H,S or ABCD matrix V I Noisy two-port with internal noise sources which can be represented as: I 1 I V Noisy -port 1 V H-matrix v n1 Noiseless H-matrix -port i n [ V 1 I ] = [ H 11 H 1 H 1 H ][ I 1 V ] [ v n1 i n ] [C h ]=[ v, v n1 n1 v n1, i n ] i n, v n1 i n, i n 15

16 Two-port Correlation Matrices: G I 1 I V Noisy -port 1 V G-matrix i n1 Noiseless G-matrix -port v n [ I 1 V ] = [ G 11 G 1 G 1 G ][ V 1 I ] [ i n1 v n] [C g ]=[ i n1 i n1 v n i n1 i n1 v n v n v n ] 16

17 Correlation Matrices: ABCD I 1 I V Noisy -port 1 V ABCD-matrix v n i n Noiseless ABCD-mat -port [ V 1 I 1 ] = [ A B D][ V ] C I [ v n i n ] [C A ]=[ v n v n v n i n ] i n v n i n i n 17

18 Noise Correlation Matrix Transformations [C z ]=[ 1 Z 11 0 Z ] [ C A ] [ 1 0 Z 11 ] ]=[ Z ; [C y Y 11 1 Y 0] [ C A ] [ Y 11 Y ] 1 0 [C z ]=[ Z ] [C y ] [ Z ] + ; [C z ]= [U] [ Z ] [C S ] [U] [Z ] + [C y ]=[Y ] [C z ] [Y ] + ; [C y ]= [U ] [Y ] [C S ] [U ] [Y ] + [C S ]= 1 4 [U ] [S ] [C y] [U ] [S ] + ; [C S ]= 1 4 [U ] [S ] [C Z ] [U ] [S ] + Im c nn =Im v nn =Im i nn =Im b nn =0 c ki =c ik 18

19 Method to form the noise admittance correlation matrix of an n-port C y [i,i] = sum of all noise currents connected to node i C y [i,j] = negative of sum of all noise currents between i and j If j = GND, the noise current is only added to C y [i,i] 19

20 Example: C C R P 1 P R P 1 P R 1 C 3 i n1 R 1 i n C 3 [C Y ]=[ i n1 i n i n i n ] i n =4kT f [ G 1 G G G G ] 0

21 Noise Factor of a -port driven by a signal source -port with input equivalent noise sources (sign is not important) Noise of signal source is typically thermal: i sn = 4kT f Re{Y s } Signal source may be a photodiode: i sn = 4kT fg seq ; Y s = j C D G seq is due to dark current and optical power. 1

22 Deriving the Noise Factor of the -Port Y in, g m Find input shortcircuit current i due to i, i, v sc by superposition n sn n n Input short circuit current i scs n due to i sn Determine F as the ratio: F= i scn i scsn

23 Two-Port Noise Parameters (1) Noise admittance formalism Noise resistance formalism v n =v n ; i n =i u i c =i u Y cor v n i n =i n ; v n =v u v c =v u Z cor i n v n R n = 4kT f G u = i u 4kT f v u R u = 4kT f G n = i n 4kT f Y cor = i v x n n v =G cor j B cor n Z cor = v i x n n i =R cor j X cor n 3

24 Two-Port Noise Parameters () F=1 v n Y cor Y s i sn i u i sn F=1 i n ' Z cor Z s v sn v u v sn F=1 R n G s Y cor Y s G u G s F=1 G n R s Z cor Z s R u R s F=F MIN R n G s Y s Y sopt F=F MIN G n R s Z s Z sopt Constant Noise Circles on Smith Chart 4

25 Two-Port Noise Parameters (3) Y sop =G sopt jb sopt Z sopt =R sopt jx sopt F min =1 R n G cor G sopt F min =1 G n R cor R sopt G sopt= G cor G u ;B R sopt = B cor n R sopt= R cor R u ;X G sopt = X cor n Y sop = 1 Z sopt Y cor x = Z cor Z sopt R n =G n Z sopt G u = R u Z sopt 5

26 Two-Port Noise Circles F s =F min 4 R n Z 0 s sopt 1 S 1 sopt N r i = i N i 1 sopt 1 N i C i = sopt 1 N i N i = F i F min 4 r n 1 sopt Constant Noise Circles on Smith Chart 6

27 Correlation Matrix and -Port Noise Parameters [C A ]=[ v v n1 n1 v n1 i n1 i n1 v n1 i n1 i n1 ] = [ C uu C ui C iu C ii ] =[ F MIN 1 F MIN 1 R n Y opt ] R n Y opt R n Y opt R n = C Y opt ii C uu [ I C ui C uu ] j I C ui C uu F MIN =1 C C Y ui uu opt kt R n = C uu kt From Stephen Maas Noise 7

28 Correlation Matrix and -Port Noise Parameters [C y ]=[ i i n1 n1 i n i n1 i n1 i n i n i n ] [ = G R u n Y Y 11 cor R n Y 1 R n Y 1 Y 11 Y 11 Y cor ] Y cor R n Y 1 R n = C y Y 1 Y cor =Y 11 C y1 C y Y 1 G u =C y11 R n Y 11 Y cor From Stephen Maas Noise 8

29 Noise in Passive -ports A lossless passive -port has NF MIN =0 db In a lossy passive -port the noise figure is equal to the insertion loss. If a noisy -port is cascaded with a lossless passive -port the minimum noise figure is preserved. Hence the popularity of reactive matching networks in tuned LNA design 9

30 Two-Port Noise Temperature Measurements R, T 1 (hot) T a N 1 G, f N R, T (cold) N 1 =GkT 1 f GkT a f ; N =GkT f GkT a f Y= N 1 = T T 1 a ; T N T T a = T YT 1 a Y 1 Also know as the Y-factor measurement Measures output noise for two loads at significantly different temperatures connected at the input of the two-port 30

31 Two-Port Noise Parameter Measurements F k =F MIN R n G sk Y sk Y sopt Want small R n for insensitivity to source noise impedance mismatch 31

32 Two-Port Noise Parameter Measurements () The 4 noise parameters are measured indirectly using a Noise Figure Meter Measure noise figure for 4 or more different source impedances and solve for the noise parameters: F k =F MIN R n G sk Y sk Y sopt k = Choice of Y sk is critical to meas. accuracy 3

33 Summary NF is a function of signal source impedance and of the 4 noise parameters of the -port For a -port there is an optimum signal source impedance which minimizes NF To achieve NF MIN the -port must be "noise-matched" Noise matching can be achieved using negative feedback or (lossless) impedance transformation techniques The noise figure of a passive -port is equal to its insertion loss. 33