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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,347 @@ | ||
| --- | ||
| title: NAND Gate Differential-Algebraic Equation (DAE) Work-Precision Diagrams | ||
| author: Jayant Pranjal | ||
| --- | ||
|
|
||
| ```julia | ||
| using OrdinaryDiffEq, DiffEqDevTools, ModelingToolkit, ODEInterfaceDiffEq, | ||
| Plots | ||
| using LinearAlgebra | ||
| using ModelingToolkit: t_nounits as t, D_nounits as D | ||
| ``` | ||
|
|
||
| ## Problem Parameters | ||
|
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||
| ```julia | ||
| const RGS = 4.0 | ||
| const RGD = 4.0 | ||
| const RBS = 10.0 | ||
| const RBD = 10.0 | ||
| const CGS = 6e-5 | ||
| const CGD = 6e-5 | ||
| const CBD = 2.4e-5 | ||
| const CBS = 2.4e-5 | ||
| const C9 = 5e-5 | ||
| const DELTA = 0.02 | ||
| const CURIS = 1e-14 | ||
| const VTH = 25.85 | ||
| const VDD = 5.0 | ||
| const VBB = -2.5 | ||
| const VT0_DEPL = -2.43 | ||
| const CGAMMA_DEPL = 0.2 | ||
| const PHI_DEPL = 1.28 | ||
| const BETA_DEPL = 5.35e-4 | ||
| const VT0_ENH = 0.2 | ||
| const CGAMMA_ENH = 0.035 | ||
| const PHI_ENH = 1.01 | ||
| const BETA_ENH = 1.748e-3 | ||
| ``` | ||
|
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| ## Input Signal Functions | ||
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| ```julia | ||
| function pulse(t, t_start, v_low, t_rise, v_high, t_high, t_fall, t_period) | ||
| t_mod = mod(t, t_period) | ||
|
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| if t_mod < t_start | ||
| return v_low | ||
| elseif t_mod < t_start + t_rise | ||
| return v_low + (v_high - v_low) * (t_mod - t_start) / t_rise | ||
| elseif t_mod < t_start + t_rise + t_high | ||
| return v_high | ||
| elseif t_mod < t_start + t_rise + t_high + t_fall | ||
| return v_high - (v_high - v_low) * (t_mod - t_start - t_rise - t_high) / t_fall | ||
| else | ||
| return v_low | ||
| end | ||
| end | ||
|
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| V1(t) = pulse(t, 0.0, 0.0, 5.0, 5.0, 5.0, 5.0, 20.0) | ||
| V2(t) = pulse(t, 0.0, 0.0, 15.0, 5.0, 15.0, 5.0, 40.0) | ||
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| function V1_derivative(t) | ||
| t_mod = mod(t, 20.0) | ||
| if 0.0 < t_mod < 5.0 | ||
| return 1.0 | ||
| elseif 10.0 < t_mod < 15.0 | ||
| return -1.0 | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
| function V2_derivative(t) | ||
| t_mod = mod(t, 40.0) | ||
| if 0.0 < t_mod < 15.0 | ||
| return 1.0/15.0 | ||
| elseif 20.0 < t_mod < 35.0 | ||
| return -1.0/15.0 | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
| ``` | ||
|
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| ## MOSFET Model Functions | ||
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| ```julia | ||
| function gdsp(ned, vds, vgs, vbs) | ||
| if ned == 1 | ||
| vt0, cgamma, phi, beta = VT0_DEPL, CGAMMA_DEPL, PHI_DEPL, BETA_DEPL | ||
| else | ||
| vt0, cgamma, phi, beta = VT0_ENH, CGAMMA_ENH, PHI_ENH, BETA_ENH | ||
| end | ||
| phi_vbs = max(phi - vbs, 1e-12) | ||
| phi_safe = max(phi, 1e-12) | ||
| vte = vt0 + cgamma * (sqrt(phi_vbs) - sqrt(phi_safe)) | ||
| if vgs - vte <= 0.0 | ||
| return 0.0 | ||
| elseif 0.0 < vgs - vte <= vds | ||
| return -beta * (vgs - vte)^2 * (1.0 + DELTA * vds) | ||
| elseif 0.0 < vds < vgs - vte | ||
| return -beta * vds * (2.0 * (vgs - vte) - vds) * (1.0 + DELTA * vds) | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
|
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| function gdsm(ned, vds, vgd, vbd) | ||
| if ned == 1 | ||
| vt0, cgamma, phi, beta = VT0_DEPL, CGAMMA_DEPL, PHI_DEPL, BETA_DEPL | ||
| else | ||
| vt0, cgamma, phi, beta = VT0_ENH, CGAMMA_ENH, PHI_ENH, BETA_ENH | ||
| end | ||
| phi_vbd = max(phi - vbd, 1e-12) | ||
| phi_safe = max(phi, 1e-12) | ||
| vte = vt0 + cgamma * (sqrt(phi_vbd) - sqrt(phi_safe)) | ||
| if vgd - vte <= 0.0 | ||
| return 0.0 | ||
| elseif 0.0 < vgd - vte <= -vds | ||
| return beta * (vgd - vte)^2 * (1.0 - DELTA * vds) | ||
| elseif 0.0 < -vds < vgd - vte | ||
| return -beta * vds * (2.0 * (vgd - vte) + vds) * (1.0 - DELTA * vds) | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
|
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| function ids(ned, vds, vgs, vbs, vgd, vbd) | ||
| if vds > 0.0 | ||
| return gdsp(ned, vds, vgs, vbs) | ||
| elseif vds == 0.0 | ||
| return 0.0 | ||
| else | ||
| return gdsm(ned, vds, vgd, vbd) | ||
| end | ||
| end | ||
|
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| function ibs(vbs) | ||
| if vbs <= 0.0 | ||
| return -CURIS * (exp(vbs / VTH) - 1.0) | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
|
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| function ibd(vbd) | ||
| if vbd <= 0.0 | ||
| return -CURIS * (exp(vbd / VTH) - 1.0) | ||
| else | ||
| return 0.0 | ||
| end | ||
| end | ||
| ``` | ||
|
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| ## Capacitance Matrix Function | ||
|
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| ```julia | ||
| function capacitance_matrix!(C, y, t) | ||
| fill!(C, 0.0) | ||
|
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| C[1,1] = CGS | ||
| C[2,2] = CGD | ||
| C[3,3] = CBS | ||
| C[4,4] = CBD | ||
| C[5,5] = 0.0 | ||
| C[6,6] = CGS | ||
| C[7,7] = CGD | ||
| C[8,8] = CBS | ||
| C[9,9] = CBD | ||
| C[10,10] = 0.0 | ||
| C[11,11] = CGS | ||
| C[12,12] = CGD | ||
| C[13,13] = CBS | ||
| C[14,14] = CBD | ||
|
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| return C | ||
| end | ||
| ``` | ||
|
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| ## DAE System Definition | ||
|
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| ```julia | ||
| function nand_rhs!(f, y, p, t) | ||
| v1 = V1(t) | ||
| v2 = V2(t) | ||
| v1d = V1_derivative(t) | ||
| v2d = V2_derivative(t) | ||
|
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||
| y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11, y12, y13, y14 = y | ||
|
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||
| f[1] = -(y1 - y5) / RGS - ids(1, y2 - y1, y5 - y1, y3 - y5, y5 - y2, y4 - VDD) | ||
| f[2] = -(y2 - VDD) / RGD + ids(1, y2 - y1, y5 - y1, y3 - y5, y5 - y2, y4 - VDD) | ||
| f[3] = -(y3 - VBB) / RBS + ibs(y3 - y5) | ||
| f[4] = -(y4 - VBB) / RBD + ibd(y4 - VDD) | ||
| f[5] = -(y5 - y1) / RGS - ibs(y3 - y5) - (y5 - y7) / RGD - ibd(y9 - y5) | ||
|
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| f[6] = CGS * v1d - (y6 - y10) / RGS - ids(2, y7 - y6, v1 - y6, y8 - y10, v1 - y7, y9 - y5) | ||
| f[7] = CGD * v1d - (y7 - y5) / RGD + ids(2, y7 - y6, v1 - y6, y8 - y10, v1 - y7, y9 - y5) | ||
| f[8] = -(y8 - VBB) / RBS + ibs(y8 - y10) | ||
| f[9] = -(y9 - VBB) / RBD + ibd(y9 - y5) | ||
| f[10] = -(y10 - y6) / RGS - ibs(y8 - y10) - (y10 - y12) / RGD - ibd(y14 - y10) | ||
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| f[11] = CGS * v2d - y11 / RGS - ids(2, y12 - y11, v2 - y11, y13, v2 - y12, y14 - y10) | ||
| f[12] = CGD * v2d - (y12 - y10) / RGD + ids(2, y12 - y11, v2 - y11, y13, v2 - y12, y14 - y10) | ||
| f[13] = -(y13 - VBB) / RBS + ibs(y13) | ||
| f[14] = -(y14 - VBB) / RBD + ibd(y14 - y10) | ||
|
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| return nothing | ||
| end | ||
|
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| function nand_mass_matrix!(M, y, p, t) | ||
| capacitance_matrix!(M, y, t) | ||
| return nothing | ||
| end | ||
|
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||
| function create_nand_problem() | ||
| y0 = [5.0, 5.0, VBB, VBB, 5.0, 3.62385, 5.0, VBB, VBB, 3.62385, 0.0, 3.62385, VBB, VBB] | ||
| dy0 = zeros(14) | ||
| tspan = (0.0, 80.0) | ||
| M = zeros(14, 14) | ||
| capacitance_matrix!(M, y0, 0.0) | ||
| f = ODEFunction(nand_rhs!, mass_matrix=M) | ||
| prob = ODEProblem(f, y0, tspan) | ||
| return prob | ||
| end | ||
| nand_prob = create_nand_problem() | ||
| ``` | ||
|
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| ## Generate Reference Solution | ||
|
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| ```julia | ||
| ref_sol = solve(nand_prob, Rodas5P(), abstol=1e-12, reltol=1e-12, | ||
| tstops=0.0:5.0:80.0) # Include discontinuity points | ||
|
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| plot(ref_sol, title="NAND Gate Circuit - Node Potentials", | ||
| xlabel="Time", ylabel="Voltage (V)", legend=:outertopright) | ||
| ``` | ||
|
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| ## Work-Precision Benchmarks | ||
|
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| ### High Tolerances | ||
|
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| ```julia | ||
| abstols = 1.0 ./ 10.0 .^ (4:7) | ||
| reltols = 1.0 ./ 10.0 .^ (1:4) | ||
|
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| setups = [ | ||
| Dict(:alg=>Rosenbrock23()), | ||
| Dict(:alg=>Rodas4()), | ||
| Dict(:alg=>Rodas5P()), | ||
| Dict(:alg=>FBDF()), | ||
| Dict(:alg=>QNDF()), | ||
|
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|
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| ] | ||
|
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| wp = WorkPrecisionSet([nand_prob], abstols, reltols, setups; | ||
| save_everystep=false, appxsol=[ref_sol], | ||
| maxiters=Int(1e6), numruns=5, | ||
| tstops=0.0:5.0:80.0) | ||
| plot(wp, title="NAND Gate DAE - Work-Precision (High Tolerances)") | ||
| ``` | ||
|
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| ### Medium Tolerances | ||
|
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| ```julia | ||
| abstols = 1.0 ./ 10.0 .^ (6:9) | ||
| reltols = 1.0 ./ 10.0 .^ (3:6) | ||
|
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| setups = [ | ||
| Dict(:alg=>Rodas4()), | ||
| Dict(:alg=>Rodas5P()), | ||
| Dict(:alg=>FBDF()), | ||
| Dict(:alg=>QNDF()), | ||
|
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|
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| ] | ||
|
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| wp = WorkPrecisionSet([nand_prob], abstols, reltols, setups; | ||
| save_everystep=false, appxsol=[ref_sol], | ||
| maxiters=Int(1e6), numruns=5, | ||
| tstops=0.0:5.0:80.0) | ||
| plot(wp, title="NAND Gate DAE - Work-Precision (Medium Tolerances)") | ||
| ``` | ||
|
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| ### Low Tolerances (High Accuracy) | ||
|
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| ```julia | ||
| abstols = 1.0 ./ 10.0 .^ (8:11) | ||
| reltols = 1.0 ./ 10.0 .^ (5:8) | ||
|
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| setups = [ | ||
| Dict(:alg=>Rodas5P()), | ||
| Dict(:alg=>FBDF()), | ||
| Dict(:alg=>QNDF()), | ||
|
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| ] | ||
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| wp = WorkPrecisionSet([nand_prob], abstols, reltols, setups; | ||
| save_everystep=false, appxsol=[ref_sol], | ||
| maxiters=Int(1e6), numruns=5, | ||
| tstops=0.0:5.0:80.0) | ||
| plot(wp, title="NAND Gate DAE - Work-Precision (Low Tolerances)") | ||
| ``` | ||
|
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| ### Timeseries Error Analysis | ||
|
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| ```julia | ||
| abstols = 1.0 ./ 10.0 .^ (5:8) | ||
| reltols = 1.0 ./ 10.0 .^ (2:5) | ||
|
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| setups = [ | ||
| Dict(:alg=>Rodas4()), | ||
| Dict(:alg=>Rodas5P()), | ||
| Dict(:alg=>FBDF()), | ||
|
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|
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| ] | ||
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| wp = WorkPrecisionSet([nand_prob], abstols, reltols, setups; | ||
| error_estimate=:l2, save_everystep=false, | ||
| appxsol=[ref_sol], maxiters=Int(1e6), numruns=5, | ||
| tstops=0.0:5.0:80.0) | ||
| plot(wp, title="NAND Gate DAE - Timeseries Error Analysis") | ||
| ``` | ||
|
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| ## Analysis of Key Nodes | ||
|
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| ```julia | ||
| key_nodes = [1, 5, 6, 10, 11, 12] # Representative nodes from different parts of the circuit | ||
| node_names = ["Node 1", "Node 5", "Node 6", "Node 10", "Node 11", "Node 12"] | ||
|
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| p_nodes = plot() | ||
| for (i, node) in enumerate(key_nodes) | ||
| plot!(ref_sol.t, [u[node] for u in ref_sol.u], | ||
| label=node_names[i], linewidth=2) | ||
| end | ||
| plot!(p_nodes, title="NAND Gate - Key Node Potentials", | ||
| xlabel="Time (s)", ylabel="Voltage (V)", legend=:outertopright) | ||
| ``` | ||
|
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| ### Conclusion | ||
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| ```julia, echo = false | ||
| using SciMLBenchmarks | ||
| SciMLBenchmarks.bench_footer(WEAVE_ARGS[:folder],WEAVE_ARGS[:file]) | ||
| ``` | ||
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Needs a DAEProblem formulation as well to test IDA, and should have an MTK version. See the other DAE benchmarks.
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Note that the 3 can all be generated from the same form so it shouldn't be too much more code, and the static array version isn't necessary here.