EE 505 Lecture 10 Statistical Circuit Modeling
mplifier Gain ccuracy eview from previous lecture: - +
eview from previous lecture: String DC Statistical Performance 1 1 k k k ILk j 1 j 1 k 1 OM j1 1 1 jk1 Since the resistors are identically distributed and the coefficients are not a function of the index i, it follows that 1 k k k 1 1 k 1 j1 1 jk1 1 OM ILk Since the index in the sum does not appear in the arguments, this simplifies to ILk 1k k 1 k 1 1 OM ote there is a nice closed-form expression for the IL k for a string DC!!
eview from previous lecture: String DC Statistical Performance IL k assumes a maximum variance at mid-code ILk max OM
String DC Statistical Performance How about statistics for the IL? eview from previous lecture: IL is not zero-mean and not Gaussian and closed form solutions do not exist
eview from previous lecture: Current Steering DC Statistical Characterization Unary weighted FF I0=0 I1 I I-1 s for the string DC, the maximum IL k occurs near mid-code at about k=/ thus XI n Binary to Thermometer Decoder S0 S1 S S-1 OUT IL kmx I I OM I j=i +Ij nd, as for the string DC, the IL is an order statistic and thus a closed-form solution does not exist
eview from previous lecture: Current Steering DC Statistical Characterization Binary Weighted IL b=<1000..0> 1 1 IGk 1 1 I LSBX FF XI n I1 S1 I S I3 S3 In Sn OUT ILMX ILb=<1,0,...0> I I G LSBX ote this is the same result as obtained for the unary DC But closed form expressions do not exist for the IL of this DC since the IL is an order statistic
Statistical Modeling of Current Sources I D I X GS Simple Square-Law MOSFET Model Usually dequate for static Statistical Modeling ssumption: Layout used to marginalize gradient effects, contact resistance and drain/source resistance neglected μc W I = - L OX D GS TH andom ariables: {µ, C OX, TH, W, L } From previous analysis, need: I I D D Thus I D is a random variable
Statistical Modeling of Current Sources μc W I = - L OX D GS TH I X GS I D andom ariables: {µ, C OX, TH, W, L } Thus I D is a random variable Will assume {µ, C OX, TH, W, L } are uncorrelated This is not true : T OX is a random variable that affects both TH and C OX This assumption is widely used and popularized by Pelgrom It is also implicit in the statistical model available in simulators such as SPECTE Statistical information about T OX often not available Drenen and Mcndrew (XP) published several papers that point out limitations Would be better to model physical parameters rather than model parameters but more complicated Statistical analysis tools at XP probably have this right but not widely available ssumption simplifies analysis considerably Error from neglecting correlation is usually quite small but don t know how small
Statistical Modeling of Current Sources Model parameters are position dependent μc W I = - L OX D GS TH W D µ(x,y), C OX (x,y), TH (x,y) L (x,y) S
Statistical Modeling of Current Sources Model parameters are position dependent μc W I = - L OX D GS TH ssume that model parameters can be modeled as a position-weighted integral C OX TH OX TH easonably good assumption if current density is constant x,y dxdy C x,y dxdy x,y dxdy L W (x,y) D S
Statistical Modeling of Current Sources ssume that model parameters can be modeled as a position-weighted integral s seen for resistors, this model is not good if current density is not constant D C W I L OX D GS TH1 L W TH1 ε W TH THEQ x,y dxdy TH TH1 TH If TH1 =1, TH = W X THEQ =1.5 S ote dramatically different current densities But reasonably good assumption if current density is constant
Statistical Modeling of Current Sources μcoxw I D= GS -TH L Model parameters characterized by following equations μ = μ μ = C = C TH TH TH OX OX OX L L L W W W C eglecting random part of W and L which are usually less important C μ μ COX OX W I D= GS -TH - L TH
Statistical Modeling of Current Sources C μ μ COX OX W I D= GS -TH - L This appears to be a highly nonlinear function of random variables!! Will now linearize the relationship between I D and the random variables Since the random variables are small, we can do a Taylor s series expansion and truncate after first-order terms to obtain μ C W C W μw μ C W I - + - + L L L L μ C - - OX OX OX D GS TH GS TH OX GS TH TH GS TH TH This is a linearization of I D in the random variables µ, C OX, and TH COX W μw μcox W I D μ GS - TH + OX GS -T H TH GS - L L L C C W μ W μ C W - - - OX OX I GS TH GS TH GS D μ L + L L COX TH ID ID ID ID Could easily include L and W but usually not important unless lots of perimeter TH TH
Statistical Modeling of Current Sources C W μ W μ C W - - - OX OX I GS TH GS TH GS D μ L + L L COX TH ID ID ID ID μ C W I = - L OX D GS TH COX W μw μcox W I GS -TH GS -TH GS -TH D μ L L L + COX TH I μ COXW μ COXW μ COXW D GS -TH I GS -TH GS -TH L L L TH Thus ID μ COX TH + I μ C - D OX GS TH 4 TH ID μ C I μ C GS TH D OX TH OX TH or I D μ C I μ C GS TH D OX TH OX
Statistical Modeling of Current Sources 4 TH ID μ C I μ C GS TH D OX TH OX TH or I D μ C I μ C GS TH D OX TH OX It will be assumed that μ μ C C OX OX T 0 TH WL Cox WL WL where µ, Cox, T0 are Pelgrom process parameters 1 4 ID Cox T 0 I WL EB D Define Cox Thus 1 4 ID T 0 I WL EB D Often only β is available
Statistical Modeling of Current Sources 1 4 ID T 0 I WL EB D Gate area: =WL Standard deviation decreases with Large EB reduces standard deviation Operating near cutoff results in large mismatch Often threshold voltage variations dominate mismatch ID T 0 I EB WL D
Statistical Modeling of Circuits The previous statistical analysis was somewhat tedious Will try to formalize the process ssume Y is a function of n random variables x 1, x n where the mean and variance of x i are small Y = f x, x,... x 1 n n f Y = f X + x + x x x X 0 x,,... j1 j X 0 x x X =... x 1 n j 1 n where x, x,... x 1 n small
Statistical Modeling of Circuits n f f X x X 0 x Y = + X 0 j1 j1 j n Y f X 1 f = + Y Y Y x Y X 0 X j 0 n 1 f = Y x j1 Y xj Sˆ f xj = 1 Y f x j X = ˆ S Y Y n j1 f xj x j 0 X j 0 j x j
Statistical Modeling of Circuits lternatively, if we define S f x j = x Y i f x j X 0 f S x j is the more standard sensitivity function we thus obtain n = f S Y xj x Y j1 x j i
Claim: μ where µ is the Pelgrom matching parameter and is the gate area rgument: ssume, μ eq x y dxdy Let S k be a square of area k in the channel μ eq i1 ki x, y dxdy y D k S Channel x where the channel has been partitioned into disjoint regions each of area ki For convenience, assume ki = kj = k for all i,j
Theorem: If the random part of two uncorrelated current sources I 1 and I are identically distributed with normalized variance, then the random I I variable ΔI=I -I 1 has a variance given by the equation I I I I Proof: I I I 1 I I I I I I 1 I I I I I I I I I I I I 1 1 I I I I I I I I
Theorem: f(y) and F(y) are any pdf/cdf pair and if X~ U[0,1], then y F x has a pdf of f(y). 1 Corollary: If X~ U[0,1] and then y=f -1 (x) is [0,1] x 1 h F h e dx CDF showing random variable mapping of x 1 from U(0,1) 1 F x 1 x F -1 (x 1 )
Theorem: If y~[0,1], then z = σy+µ is [µ,σ]
Some useful relationships: erf x = x π -t e dt 0 The CDF of the (0,1) random variable x is given by 1 x F x = 1+erf Excel @OM.DIST @OM.S.DIST @OM.I @OM.S.I Older Excel @OMI @OMSI @OMDIST @OMI x f x f 1 x 1 x x F F F F x where f: PDF F:CDF
Example: Determine the area required for the resistors for an n-bit -string DC to achieve a yield of P if the device is marketable provided 1 IL kmx LSB ssume the area of each resistor is WL OM WL
Since P is fixed, can solve for X
P F X 1 where F (X ) is the CDF on (0,1) 1 P 1 X F X 1 x 1 WL X 1 X WL WL X 1 P 1 WL F
1 P 1 WL F WL X Total area n n n n 3n TOT WL X X X 3 n TOT X or equivalently 3n 1 P 1 TOT F
End of Lecture 10