Triac voltage and delay time relation

Question

I am trying to find a formula for voltage and delay time, but I don't know if there is a relation between delay time and voltage. For example if I need 100V, I should use X ms delay time.

I found this formula in a physics forum. User Jony130 gives the formula.

I will give the Vload to MCU and expect it to calculate the delay time with a formula. I tried to create a formula between degrees as radian and voltage according to Jony130's formula. I couldn't do it.

1. Is there linear relation between voltage and delay time?
2. If yes, how can I create the formula?

Update:

clc;
clear all;

piValue = 1/(2*pi);

Vm = 220; %Vmax = 220V

for i=1:1:180
    a = (i*pi)/180;
    Vload(i) = real(Vm*sqrt(piValue*((pi-a)+(sin(2*a)/2)))*1.41);
end

I made calculation using MATLAB. These are the results that I got. Shouldn't I get 220V when the degree is 0 and also 110V when the degree is 90 ? It's 110V when it is 90 degree. However, it's not 220V when degree is 0.

155.563404112011    155.562789995711    155.561123774353    155.557880953464    155.552538876660    155.544577300795    155.533478951120    155.518730056150    155.499820862054    155.476246126473    155.447505591725    155.413104437476    155.372553712985    155.325370749122    155.271079550396    155.209211167293    155.139304049274    155.060904378791    154.973566386766    154.876852649953    154.770334370667    154.653591639364    154.526213680574    154.387799082713    154.237956012293    154.076302413056    153.902466190581    153.716085382880    153.516808317522    153.304293755813    153.078211024532    152.838240135765    152.584071895297    152.315408000090    152.031961125294    151.733455001265    151.419624481049    151.090215598751    150.744985619221    150.383703079457    150.006147822127    149.612111021565    149.201395202625    148.773814252722    148.329193427403    147.867369349747    147.388190003921    146.891514723153    146.377214172419    145.845170326089    145.295276440790    144.727437023720    144.141567796631    143.537595655705    142.915458627518    142.275105821279    141.616497377539    140.939604413529    140.244408965302    139.530903926819    138.799092986146    138.048990558880    137.280621718955    136.494022126944    135.689237955972    134.866325815370    134.025352672167    133.166395770519    132.289542549193    131.394890557179    130.482547367539    129.552630489576    128.605267279393    127.640594848957    126.658759973713    125.659918998847    124.644237744270    123.611891408394    122.563064470775    121.497950593698    120.416752522774    119.319681986622    118.206959595706    117.078814740398    115.935485488350    114.777218481229    113.604268830912    112.416900015201    111.215383773135    110 108.771036642087    107.528789591311    106.273562579758    105.005667074269    103.725422171137    102.433154491031    101.129198074241    99.8138942763547    98.4875916644782    97.1506459141236    95.8034197068806    94.4462826290131    93.0796110711142    91.7037881289713    90.3192035057960    88.9262534159873    87.5253404906026    86.1168736847286    84.7012681869512    83.2789453311404    81.8503325107803    80.4158630960921    78.9759763542133    77.5311173727192    76.0817369867938    74.6282917103783    73.1712436716575    71.7110605532673    70.2482155376443    68.7831872579690    67.3164597551979    65.8485224417214    64.3798700722323    62.9110027224475    61.4424257763812    59.9746499229401    58.5081911626850    57.0435708256872    55.5813156015068    54.1219575824270    52.6660343211985    51.2140889046909    49.7666700450011    48.3243321897511    46.8876356535082    45.4571467724991    44.0334380850534    42.6170885405246    41.2086837397909    39.8088162108545    38.4180857235362    37.0370996478256    35.6664733611017    34.3068307102131    32.9588045353168    31.6230372634575    30.3001815811531    28.9909011967950    27.6958717055220    26.4157815714677    25.1513332450013    23.9032444359183    22.6722495676338    21.4591014425216    20.2645731548901    19.0894602960822    17.9345835063348    16.8007914410299    15.6889642357922    14.6000175768676    13.5349075123076    12.4946361784364    11.4802586689942    10.4928913473174    9.53372200427160    8.60402241092457    7.70516402845776    6.83863795693910    6.00608069463758    5.20930805580732    4.45036086952568    3.73156826581312    3.05563829033722    2.42579313150406    1.84598185936478    1.32123929241027    0.858353109415476   0.467299422404012   0.165230394247177   0

Answer 1

I don't know if there is a relation between delay time and voltage

Yes there is.

I found this formula in a physics forum.

The triac load voltage RMS formula is also detailed on this website: -

And, that website appears to give a decent explanation of the formula and waveforms: -

Is there linear relation between voltage and delay time

No; use the formula if you want accuracy or plot it in excel to convince yourself and make up some approximate piecewise linear equation.

Answer 2

The formula calls for the use of peak value of the waveform and you used RMS. That accounts for the square-root-of-2 error in your results.

If you are implementing a controller you would typically want to use the desired output power rather than RMS voltage, to avoid the majority of the nonlinearity.

Anyway, either way you can implement the nonlinear curve in any number of ways, for example a polynomial or cubic spline interpolation or just use a lookup table with linear interpolation.

To get the numbers for the curve fit, you can use simple numerical solver techniques such as a binary search if there is no obvious closed form solution. Those calculations only have to be done once and you have the immense resources of a modern computer to work with.