EE 505 Lecture 10 Statistical Circuit Modeling
Amplifier Gain Accuracy eview from previous lecture: - +
String DAC Statistical Performance eview from previous lecture: esistors are uncorrelated but identically distributed, typically zero mean Gaussian OUT k j1 0 k 0 k j1 j EF 1 k 1 j 1 FT k -1 j k j1 EF j1 j 0 k -1
eview from previous lecture: String DAC Statistical Performance 1 1 k k k Lk 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 Lk Since the index in the sum does not appear in the arguments, this simplifies to Lk 1k k 1 k 1 1 OM ote there is a nice closed-form expression for the L k for a string DAC!!
eview from previous lecture: String DAC Statistical Performance L k assumes a maximum variance at mid-code Lk max OM
String DAC Statistical Performance How about statistics for the L? eview from previous lecture: L is not zero-mean and not Gaussian and closed form solutions do not exist
eview from previous lecture: Current Steering DAC Statistical Characterization Unary weighted FF 0=0 1-1 The variance of Kk can be readily calculated X n Binary to Thermometer Decoder S0 S1 S S-1 OUT k1 1 k k 1 L i 1 1 1 i k k k k OM OM j= +j k k 1 L k 1 k k 1 1 k OM This simplifies to k 1 k 1 Lk k OM
eview from previous lecture: Current Steering DAC Statistical Characterization Unary weighted FF 0=0 1-1 As for the string DAC, the maximum L k occurs near mid-code at about k=/ thus X n Binary to Thermometer Decoder S0 S1 S S-1 OUT L k MAX OM j= +j And, as for the string DAC, the L is an order statistic and thus a closed-form solution does not exist
eview from previous lecture: Current Steering DAC Statistical Characterization Binary Weighted This can be expressed as FF 1 3 n L b i k n 1 b Gk bi i 1 i 1 k 1 LSBX X n S1 S S3 Sn OUT This is now a sum of uncorrelated random variables, thus i n 1 k b L bi b i 1 i 1 k 1 Gk LSBX This reduces to n i 1 k b L bi b i 1 1 Gk LSBX
eview from previous lecture: Current Steering DAC Statistical Characterization Binary Weighted L b=<1000..0> 1 1 Gk 1 1 LSBX FF X n 1 S1 S 3 S3 n Sn OUT LMAX Lb=<1,0,...0> G LSBX ote this is the same result as obtained for the unary DAC But closed form expressions do not exist for the L of this DAC since the L is an order statistic
Statistical Modeling of Current Sources D X GS Simple Square-Law MOSFET Model Usually Adequate for static Statistical Modeling Assumption: Layout used to eliminate gradient effects, contact resistance and drain/source resistance neglected μc W = - L OX D GS TH andom ariables: µ, C OX, TH, W,L eed: D D Thus D is a random variable
Statistical Modeling of Current Sources μc W = - L OX D GS TH X GS D andom ariables: µ, C OX, TH, W,L Thus 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 t is also implicit in the statistical model available in simulators such as SPECTE Statistical information about T OX often not available Drenen and McAndrew (XP) published several papers that point out limitations Would be better to model physical paramaters rather than model parameters but more complicated Statistical analysis tools at XP probably have this right but not widely available Assumption 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 = - 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 = - L OX D GS TH Assume that model parameters can be modeled as a positionweighted integral easonably good assumption if current density is constant
Statistical Modeling of Current Sources Assume that model parameters can be modeled as a position-weighted integral As seen for resistors, this model is not good if current density is not constant
Statistical Modeling of Current Sources μc W = - L OX D GS TH Model parameters characterized by 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 D= GS -TH - L TH
Statistical Modeling of Current Sources C μ μ COX OX W D= GS -TH - L This appears to be a highly nonlinear function of random variables!! TH But 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 - + - + L L L L μ C - - OX OX OX D GS TH GS TH OX GS TH TH GS TH This is a linearization of D in the random variables µ, C OX, and TH COX W μw μcox W D μ GS - TH + OX GS -T H TH GS - L L L C TH C W μw μ C W - - - OX OX GS TH GS TH GS D μ L + L L COX TH D D D D TH
Statistical Modeling of Current Sources C W μw μ C W - - - OX OX GS TH GS TH GS D μ L + L L COX TH D D D D μ C W = - L OX D GS TH COX W μw μcox W GS -TH GS -TH GS -TH D μ L L L + COX TH μ COXW μ COXW μ COXW D GS -TH GS -TH GS -TH L L L TH Thus D μ COX TH + μ C - D OX GS TH 4 TH D μ C μ C GS TH D OX TH OX TH or D μ C μ C GS TH D OX TH OX
Statistical Modeling of Current Sources 4 TH D μ C μ C GS TH D OX TH OX TH or D μ C μ C GS TH D OX TH OX t will be argued that μ μ C C OX OX T 0 TH A WL ACox WL A WL A µ,a Cox,A T0 are process parameters 1 4 A A A D Cox T 0 WL EB D Define A A A Cox Thus 1 4 A D T 0 WL EB D A Often only A β is available
Statistical Modeling of Current Sources 1 4 A D T 0 WL EB D A Standard Deviation Decreases with Sqrt of A Large EB reduces standard deviation Operating near cutoff results in large mismatch Often threshold voltage variations dominate mismatch D T 0 EB WL D A
Statistical Modeling of Circuits The previous statistical analysis was somewhat tedious Will try to formalize the process Assume 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 0 j X j 0 j x j
Statistical Modeling of Circuits Alternatively, if we define S f xj = x Y i f x j X 0 we thus obtain n = f S Y xj x Y j1 x j i
Theorem: f the random part of two uncorrelated current sources 1 and are identically distributed with normalized variance, then the random variable Δ= - 1 has a variance given by the equation Proof: 1 1 1 1
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: f 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: f y~[0,1], then z = σy+µ is [µ,σ]
End of Lecture 10