4.4 Application of PCE to Simple Expressions
In this section, PCE in terms of SGM will be applied to simple analytical formulas. Before looking at the general application of PCE to link models and physics-based approaches in general, the idea is to apply PCE to simple formulas where the computations can be performed analytically. First, an analytical expression for a parallel connection of a deterministic and a stochastic impedance is derived. Next, the corner frequency for the case of a deterministic resistor parallel to a stochastic capacitor is derived.
4.4.1 A Deterministic Impedance Parallel to a Stochastic One
Consider a stochastic impedance Z1(ξ) depending on the stochastic variable ξ in parallel to a deterministic impedance Z2. The total impedance is given by
Zp(ξ) =Z1(ξ)·Z2·(Z1(ξ) +Z2)−1. (4.59) Due to its stochastic nature, this equation cannot be evaluated in a straight forward fashion.
The stochastic impedance is Gaussian distributed and can be written asZ1(ξ) =µZ1 +ξσZ1. Projection onto a basis ofHermite polynomial leads to the expansion coefficients
z0(1) =µZ1, z1(1) =σZ1. (4.60) As this expansion is exact, the degree of approximation P = 1 is sufficient [175]. Applying PCE, (4.59) can be evaluated on the basis of expansion coefficients by using (4.40), (4.41), and (4.54). As there are only two basis polynomials, the operations can be conducted analytically. First, the sum in (4.59) is evaluated using (4.40). Next, the expansion coefficients of the inverse y0 and y1 are computed by inverting the corresponding matrix
"
µZ1 +Z2 σZ1 σZ1 µZ1 +Z2
#−1
(4.61) and are found to be
y0 = µZ1 +Z2
µZ12+ 2µZ1Z2+Z22−σZ12, (4.62) y1 = −σZ1
µZ12+ 2µZ1Z2+Z22−σZ12. (4.63)
These coefficients are then multiplied with the coefficients of Z1(ξ) and Z2 using (4.41).
The expansion coefficients of the parallel connection are found to be z(p)0 =Z2 µZ12 +µZ1Z2−σZ12
µZ12+ 2µZ1Z2+Z22−σZ12, (4.64) z(p)1 =Z2 Z2σZ1
µZ12+ 2µZ1Z2+Z22−σZ12. (4.65) The mean and the variance can be written directly in the form of expansion coefficients as
µZp =z0(p) =Z2 µZ12+µZ1Z2−σZ12
An alternative approach to obtain the expansion coefficients of the parallel connection is to write (4.59) in terms of augmented matrices. This way, the equation for P = 1 reads
Zˆ(p)=
It can be seen that the elements in the first column are equal to the expansion coefficients derived with the formulas based on the expansion coefficients.
Even though the expansion of Z1(ξ) is exact with P = 1, (4.66), (4.67), and (4.68) are not exact as the fact that higher order coefficients are zero is not considered in the inversion.
Figure 4.1 shows the error of the formula when compared to MCS for the mean and standard deviation. The error of the variance and mean are small, especially if the standard deviation of Z1(ξ),σZ1, is small compared to the meanµZ1. The observed accuracy is sufficient for practical applications.
For the sake of illustration we look at two specific cases. First assume, the standard deviation of Z1(ξ), σZ1, to be zero. In this case (4.66) and (4.67) can be simplified to
µZp = µZ1Z2
µZ1 +Z2, σZ2p = 0, (4.69) which equals the deterministic case and further validates the formulas. Next, assume the mean of Z1(ξ) to be equal to the second impedance µZ1 =Z2. In this case, the formulas
4.4 Application of PCE to Simple Expressions
0 50 100 150 200 µZ1 (Ω)
12%
9%
6%
3%
0%
σZ1/µZ1
Mean
0 50 100 150 200 µZ1 (Ω) Variance
10−6 10−5 10−4 10−3 10−2 10−1
Figure 4.1: Relative error of the mean µZ1 and variance σZ1 given by (4.66) and (4.67) when compared to MCS with 106 samples. Z2 is given with 50 Ω, µZ1 is varied from 1 mΩ to 200 Ω, and the standard deviationσZ1 is varied from 0% to 12% of the mean.
can be simplified to
µZp =µZ12−(σZ1/µZ1)2
4−(σZ1/µZ1)2, (4.70)
σZ2
p = µZ1 σZ1/µZ1 4−(σZ1/µZ1)2
!2
. (4.71)
The mean of the parallel connection depends on the standard deviation σZ1 of Z1(ξ). It decreases if σZ1 increases. As a practical conclusion: the parallel connection of two equal impedances is expected to have a lower impedance if the values are not exactly known compared to the deterministic case. To evaluate how the variability of Z1(ξ) affects the uncertainty of the parallel connection, the relative variance of the parallel connection is written as a function of the relative variance of Z1(ξ)
σZp µZp
= σZ1/µZ1
2−(σZ1/µZ1)2 ≈ 1
2σZ1/µZ1. (4.72)
Hence, the uncertainty of the parallel connections is approximately half the one of the single variable.
4.4.2 Stochastic Corner Frequency
Now, assume thatZ2 is a resistorR andZ1(ξ) is a capacitorZ1(ξ) =C(ξ) =µC+σCξ. In order to illustrate that the proposed method is applicable for various applications beyond the concatenation and combination of circuits, the stochastic corner frequency is analyzed.
At the corner frequency fc the transmission drops to−3 dB and the phase shift is 45◦. The parallel connection of a deterministic resistor and a capacitor, see Figure 4.2a, forms a low pass filter with the corner frequency
fc = 1
2πRC. (4.73)
For the given case of a stochastic capacitor and assuming time-invariant uncertainty, a similar equation for the corner frequency is found. As the capacitor is stochastic, the corner frequency is stochastic, too
fc(ξ) = (2πRC(ξ))−1, (4.74) which, again, cannot be evaluated in a straightforward fashion. By application of the proposed formulas, (4.74) can be approximated in the following way.
The stochastic capacitance is written in terms of expansion coefficients which are c0 = µC
and c1 = σC; all other coefficients are zero. Hence, PCE with P = 1 represents the capacitance accurately [175]. By applying the multiplication formula (4.41) and performing the inverse, (4.74) can be expressed in terms of expansion coefficients. This yields the mean µfcand variance σf2
cof the stochastic corner frequency µfc = µC
2πR(µ2C −σC2), (4.75)
σ2f
c = σC
2πR(µ2C−σC2)
!2
. (4.76)
These results are a first order approximation of the stochastic measures. Again, the capacitance is represented accurately with P = 1, however, this is not necessarily the case for its inverse. This can be seen by considering more coefficients and setting up a larger matrix for the computation of the inverse. Even though the additional coefficients are zero, higher order coefficients of the inverse are non-zero. Figure 4.2b shows the relative error of the mean and the variance of the proposed formulas in comparison to MCS. It can be seen that the relative error of the mean remains below 10−4 in all cases. The relative error of the variance is below 10−1. The accuracy of the formula increases if the variance of the capacitance is substantially smaller than the mean.