A probabilistic electrical simulator

Jonas Bodingbauer

February 01, 2024

Electrical Circuit: Voltage divider

Voltage divider circuit

\[U_{out} = U \frac{R_2}{R_1 + R_2}\]

Idea

  • Components in a circuit have an expected value and a tolerance.
  • Circuit gets input and produces a response.
  • The response is a random variable described by a probability distribution.
  • Priors and Observed are linked by the simulator model.
  • Additional randomness: device noise, instrument noise, offsets, ...

Electrical Simulations

  • DC Analysis
  • AC Analysis (treat everything as a linear component)
  • DC Analysis with nonlinearity (e.g. diodes)
  • Transient Analysis
  • Lots more...

Finally, some modelling!

\[ \underbrace{\begin{bmatrix} \mathbf{Y'} & \mathbf{B}\\ \mathbf{C} & \mathbf{D} \end{bmatrix}}_{Y} \cdot \underbrace{\begin{bmatrix} \vec{\varphi}\\ \vec{I} \end{bmatrix}}_{x} = \underbrace{\begin{bmatrix} \vec{J}\\ \vec{E} \end{bmatrix}}_{F} \]

AC-Case: Domain changes to complex numbers (Fourier transform)

Nonlinear case: Newton-Raphson fixed-point iteration

\[ \mathbf{Y}(\vec{x}_i) \cdot \vec{x}_{i+1} = -\mathbf{F}(\vec{x}_i) + \mathbf{Y}(\vec{x}_i) \cdot \vec{x}_i \]

Program flow

  1. Read SPICE-like file

voltagediv.cir


V1 0 in DC 2
R3 in 2 11.72
R1 2 3 1.2k
R2 3 0 1.8k

voltagediv.tol


R3 0.1%
R1 5%
R2 5%
  1. Determine the matrix size
  2. Read observed data from file(s)
  3. Run inference using the simulator as a model

Measurements

Circuit setup Oscilloscope setup

Measured values

First example extended

Extended first example circuit
  • \(R_m\) is an additional precision resistor for measuring current.
  • \(\varphi_1\) and \(\varphi_2\) are potentials.
  • Priors:
    \(R_1\), \(R_2\) normally distributed around nominal value.
    \(U_{offset_1}\), \(U_{offset_2}\) normally distributed around 0.
    \(U_{noise_1}\), \(U_{noise_2}\) uniformly distributed in \([0, 0.1]V\).
  • Observed: \(\varphi_1\), \(\varphi_2\)

Results

Divider data results

Glorified linear regression?

Divider sampling

Glorified linear regression?

Divider sampling more

Glorified linear regression?

Divider sampling hexbin

Glorified linear regression?

Component Nominal Value "True" Value MAP-Estimate
\(R_1\)\(1.2\,\text{k}\Omega\)\(1.176\,\text{k}\Omega\)\(1.186\,\text{k}\Omega\)
\(R_2\)\(1.8\,\text{k}\Omega\)\(1.784\,\text{k}\Omega\)\(1.776\,\text{k}\Omega\)
\(R_3\)\(11.72\Omega\)\(11.72\Omega\)\(11.72\Omega\)
\(U_{offset_1}\)??\(-0.059\text{V}\)
\(U_{offset_2}\)??\(-0.026\text{V}\)
\(U_{noise_1}\)??\(0.2\,m\text{V}\)
\(U_{noise_2}\)??\(10^{-5}\text{V}\)

A more complex circuit

Sallen key circuit

Frequency Response

Sallen key frequency response
Posterior pair plot
Posterior vs prior pair plot

Results

Obtained with three measured potentials: \(\varphi_1\), \(\varphi_2\), \(\varphi_5\).

Component Nominal Value "True" Value MAP-Estimate
\(R_1\)\(100\,\text{k}\Omega\)\(100.09\,\text{k}\Omega\)\(100.08\,\text{k}\Omega\)
\(R_2\)\(100\,\text{k}\Omega\)\(99.97\,\text{k}\Omega\)\(98.6\,\text{k}\Omega\)
\(R_3\)\(150\,\text{k}\Omega\)\(149.5\,\text{k}\Omega\)\(151.2\,\text{k}\Omega\)
\(R_4\)\(100\,\text{k}\Omega\)\(100.06\,\text{k}\Omega\)\(99.19\Omega\)
\(C_1\)\(3.3\,\text{nF}\)\(2.75\,\text{nF}\)\(2.73\,\text{nF}\)
\(C_2\)\(3.3\,\text{nF}\)\(3.33\,\text{nF}\)\(3.32\,\text{nF}\)

Inference

  • Turing NUTS sampler performs really well.
  • Also tried other samplers (SMC, PG, Gibbs), but they did not work reasonably well without further tuning.
  • ADVI works wonders for this application (MAP estimate in seconds).
  • Choice of priors is important: output often depends on a ratio of variables, so scaled variables may be equally likely.

Normal distributed priors

Normal prior result

Uniform priors

Uniform prior result

Challenges

  • Getting Automatic Differentiation to work was quite a hassle (types...).
  • Big range of values (e.g. \(10^{-12} - 10^{6}\)) does not play nicely with NUTS.
  • Fixed by scaling variables and using logarithmic frequency response.
  • Hard to know if results are actually reasonable if you do not know the true values.

Conclusion

  • It works!
  • Modelling needs to account for many effects to be realistic (measurement offsets, noise, parasitics, ...).
  • The nonlinear system suffers from convergence issues with some parameters explored by inference.
  • More components: transistors (BJT, MOSFETs, ...).
  • Transient analysis.

Thank you for your attention!
Questions?

SBICE QR code

git: https://github.com/Caesiumhydroxid/SBICE

Nonlinear Diode

Diode inference

DC Analysis Model


@model function dcanalysis(...)
  for e in keys(tolerances) ...
      elementsR[e].R ~ Normal(elements[e].R,
                  elements[e].R * tolerances[e])
  end
  offsets ~ MvNormal(zeros(observedNodeIndicesLength),
                    diagm(0.01 .* ones(observedNodeIndicesLength)))
  for i in 1:observedNodeIndicesLength
      noise[i] ~ Uniform(0, 0.01)
  end
  A,RHS = assembleMatrixAndRhs(elementsR,nodes,type)
  Ainv = inv(A)
  for i in 1:amountOfMeasurements
      RHS[nodes[elementsR["V1"].name]-1] = -voltages[i]
      res = Ainv * RHS
      results[i,:] = res[observedNodeIndices]
  end
  for (i,col) in enumerate(eachcol(results))
      measurementMatrix[:,i] ~ MvNormal(col .+ offsets[i],
              diagm(noise[i] * ones(amountOfMeasurements)))
  end
end

Sallen Key Input


V1 0 in AC 1
R5 in 1 1.484k
R1 1 2 100k
R2 2 3 100k
C1 2 out 3.3n
C2 3 0 3.3n
R3 out 4 150k
R4 4 0 100k
O1 3 4 out 0 1e5
            

R5 0.001%
R1 0.5%
R2 0.5%
R3 0.5%
R4 0.5%
C1 40%
C2 40%

Sallen Key Measurements


in Sa,Frequency in Hz,Gain in dB,Phase in deg,Amplitude in Vpp
1.00E+00,1.000E+01,-2.426E-02,2.693E-03,2.500E+00
2.00E+00,1.023E+01,-2.448E-02,3.685E-03,2.500E+00
3.00E+00,1.047E+01,-2.481E-02,6.256E-04,2.500E+00
4.00E+00,1.072E+01,-2.359E-02,1.049E-02,2.500E+00
5.00E+00,1.096E+01,-2.475E-02,1.010E-02,2.500E+00
6.00E+00,1.122E+01,-2.326E-02,4.135E-03,2.500E+00
7.00E+00,1.148E+01,-2.272E-02,1.164E-02,2.500E+00
8.00E+00,1.175E+01,-2.387E-02,-7.126E-03,2.500E+00
9.00E+00,1.202E+01,-2.171E-02,1.350E-03,2.500E+00
...