# Passivity: Test, visualize, and enforce passivity of rationalfit output

This example shows how to test, visualize, and enforce the passivity of output from the rationalfit function.

### Section 1 S-parameter data passivity

Time-domain analysis and simulation depends critically on being able to convert frequency-domain S-parameter data into causal, stable, and passive time-domain representations. Because the rationalfit function guarantees that all poles are in the left half plane, rationalfit output is both stable and causal by construction. The problem is passivity.

N-port S-parameter data represents a frequency-dependent transfer function H(f). An sparameter object can be created in RF Toolbox by reading a Touchstone file, such as passive.s2p, into the sparameter function. The ispassive function can be used to check the passivity of the S-parameter data, and the passivity function can be used to plot the 2-norm of the NxN matrices H(f) at each data frequency.

S = sparameters('passive.s2p');

ispassive(S)
ans = logical
1

passivity(S)

### Section 2 Testing and visualizing rationalfit output passivity

The rationalfit function converts N-port sparameter data S into an NxN matrix of rfmodel.rational objects.

Using the ispassive function on the NxN fit output reports that despite the fact that the input data S is passive, the output fit is not passive. In other words, the norm H(f) is greater than one at some frequency in the range [0,Inf].

The passivity function takes an NxN fit as input and plots its passivity. This is a plot of the upper bound of the norm(H(f)) on [0,Inf], also known as the H-infinity norm.

fit = rationalfit(S);

ispassive(fit)
ans = logical
0

passivity(fit)

### Section 3 Testing and visualizing rationalfit output passivity

The makepassive function takes as input an NxN array of fit objects and also the original S-parameter data, and produces a passive fit by using convex optimization techniques to optimally match the data of the S-parameter input S while satisfying passivity constraints. The residues C and feedthrough matrix D of the output pfit are modified, but the poles A of the output pfit are identical to the poles A of the input fit.

pfit = makepassive(fit,S,'Display','on');
ITER	 H-INFTY NORM	FREQUENCY		ERRDB		CONSTRAINTS
0		1 + 1.791e-02	17.6816  GHz	-40.4702
1		1 + 2.670e-04	283.089  MHz	-40.9161	5
2		1 + 8.711e-05	368.778  MHz	-40.9084	8
3		1 - 3.678e-07	368.979  MHz	-40.9053	10
ispassive(pfit)
ans = logical
1

passivity(pfit)

all(vertcat(pfit(:).A) == vertcat(fit(:).A))
ans = logical
1

### Section 4 Starting makepassive with prescribed poles and zero C and D

To demonstrate that only C and D are modified by makepassive, one can zero out C and D and re-run makepassive. The output pfit still has the same poles as the input fit. The differences between pfit and pfit2 arise because of the different starting points of the convex optimizations.

One can use this feature of the makepassive function to produce a passive fit from a prescribed set of poles without any idea of starting C and D.

for k = 1:numel(fit)
fit(k).C(:) = 0;
fit(k).D(:) = 0;
end
pfit2 = makepassive(fit,S);
passivity(pfit2)

all(vertcat(pfit2(:).A) == vertcat(fit(:).A))
ans = logical
1

### Section 5 Generating an equivalent SPICE circuit from a passive fit

The generateSPICE function takes a passive fit and generates an equivalent circuit as a SPICE subckt file. The input fit is an NxN array of rfmodel.rational objects as returned by rationalfit with an sparameters object as input. The generated file is a SPICE model constructed solely of passive R, L, C elements and controlled source elements E, F, G, and H.

generateSPICE(pfit2,'mypassive.ckt')
type mypassive.ckt
* Equivalent circuit model for mypassive.ckt
.SUBCKT mypassive po1 po2
Vsp1 po1 p1 0
Vsr1 p1 pr1 0
Rp1 pr1 0 50
Ru1 u1 0 50
Fr1 u1 0 Vsr1 -1
Fu1 u1 0 Vsp1 -1
Ry1 y1 0 1
Gy1 p1 0 y1 0 -0.02
Vsp2 po2 p2 0
Vsr2 p2 pr2 0
Rp2 pr2 0 50
Ru2 u2 0 50
Fr2 u2 0 Vsr2 -1
Fu2 u2 0 Vsp2 -1
Ry2 y2 0 1
Gy2 p2 0 y2 0 -0.02
Rx1 x1 0 1
Cx1 x1 0 2.73023882777141e-12
Gx1_1 x1 0 u1 0 -2.06032427808181
Rx2 x2 0 1
Cx2 x2 0 7.77758881964484e-12
Gx2_1 x2 0 u1 0 -2.91716577793217
Rx3 x3 0 1
Cx3 x3 0 2.29141630537012e-11
Gx3_1 x3 0 u1 0 -0.544107578771808
Rx4 x4 0 1
Cx4 x4 0 9.31845201916597e-11
Gx4_1 x4 0 u1 0 -0.654511656537136
Rx5 x5 0 1
Cx5 x5 0 4.89917763955558e-10
Gx5_1 x5 0 u1 0 -0.0811500692400149
Rx6 x6 0 1
Fxc6_7 x6 0 Vx7 18.739034998054
Cx6 x6 xm6 3.95175907081288e-09
Vx6 xm6 0 0
Gx6_1 x6 0 u1 0 -0.092217835782869
Rx7 x7 0 1
Fxc7_6 x7 0 Vx6 -0.0838076071904866
Cx7 x7 xm7 3.95175907081288e-09
Vx7 xm7 0 0
Gx7_1 x7 0 u1 0 0.00772855615724749
Rx8 x8 0 1
Cx8 x8 0 1.25490425523105e-08
Gx8_1 x8 0 u1 0 -0.947644374046793
Rx9 x9 0 1
Cx9 x9 0 2.73023882777141e-12
Gx9_2 x9 0 u2 0 -2.08383632474217
Rx10 x10 0 1
Cx10 x10 0 7.77758881964484e-12
Gx10_2 x10 0 u2 0 -2.92724232900967
Rx11 x11 0 1
Cx11 x11 0 2.29141630537012e-11
Gx11_2 x11 0 u2 0 -0.607553465753134
Rx12 x12 0 1
Cx12 x12 0 9.31845201916597e-11
Gx12_2 x12 0 u2 0 -0.692661089672121
Rx13 x13 0 1
Cx13 x13 0 4.89917763955558e-10
Gx13_2 x13 0 u2 0 -0.0860907831130595
Rx14 x14 0 1
Fxc14_15 x14 0 Vx15 18.3723640827824
Cx14 x14 xm14 3.95175907081288e-09
Vx14 xm14 0 0
Gx14_2 x14 0 u2 0 -0.0931984210726591
Rx15 x15 0 1
Fxc15_14 x15 0 Vx14 -0.0854802178516294
Cx15 x15 xm15 3.95175907081288e-09
Vx15 xm15 0 0
Gx15_2 x15 0 u2 0 0.00796662133671878
Rx16 x16 0 1
Cx16 x16 0 1.25490425523105e-08
Gx16_2 x16 0 u2 0 -0.948032067253446
Gyc1_1 y1 0 x1 0 -0.139137822315353
Gyc1_2 y1 0 x2 0 -0.0228134592644853
Gyc1_3 y1 0 x3 0 -1
Gyc1_4 y1 0 x4 0 -1
Gyc1_5 y1 0 x5 0 1
Gyc1_6 y1 0 x6 0 -1
Gyc1_7 y1 0 x7 0 -1
Gyc1_8 y1 0 x8 0 0.999803849072635
Gyc1_9 y1 0 x9 0 1
Gyc1_10 y1 0 x10 0 -1
Gyc1_11 y1 0 x11 0 0.809853570219065
Gyc1_12 y1 0 x12 0 0.941822507697304
Gyc1_13 y1 0 x13 0 -0.935052890161621
Gyc1_14 y1 0 x14 0 0.988857841712279
Gyc1_15 y1 0 x15 0 0.953630211881379
Gyc1_16 y1 0 x16 0 -1
Gyd1_1 y1 0 u1 0 0.603239996699853
Gyd1_2 y1 0 u2 0 -0.352279713304301
Gyc2_1 y2 0 x1 0 1
Gyc2_2 y2 0 x2 0 -1
Gyc2_3 y2 0 x3 0 0.900686135364326
Gyc2_4 y2 0 x4 0 0.996969962778983
Gyc2_5 y2 0 x5 0 -0.991563989741649
Gyc2_6 y2 0 x6 0 0.997628084371699
Gyc2_7 y2 0 x7 0 0.961370589404134
Gyc2_8 y2 0 x8 0 -1
Gyc2_9 y2 0 x9 0 -0.265661405863358
Gyc2_10 y2 0 x10 0 0.0684840088505534
Gyc2_11 y2 0 x11 0 -1
Gyc2_12 y2 0 x12 0 -1
Gyc2_13 y2 0 x13 0 1
Gyc2_14 y2 0 x14 0 -1
Gyc2_15 y2 0 x15 0 -1
Gyc2_16 y2 0 x16 0 0.999975410715647
Gyd2_1 y2 0 u1 0 -0.33716515482015
Gyd2_2 y2 0 u2 0 0.700178512220146
.ENDS