Single-Phase AC Regulator Study

[Ángel Martin]

Single-phase AC Regulator. [SPACREG ]
From the author’s Engineering Collection, included in the ETSII5 module.


This program is useful to calculate the RMS voltage on the output load of a single-phase thyristor AC regulator; based on the thyristor gate delay angle (alpha). Also the current extinction angle "beta" is determined, showing the message “NO REGULA” in non-regulating conditions of the delay angle. The program allows for different combination of load types, resistor and inductance, arranged in series or in parallel.

Let w = 2pi f the input frequency, and Z = R + jL the load impedance. The expressions used for the four different cases considered are shown below:

1. Resistive Load

Extinction angle: beta = pi

VRMS = Vmax sqrt{ (1/pi [ ( pi – alpha ) + (½) sin 2 alpha ] }

IRMS = 1/R (VRMS)


2. Inductive load

Extinction angle: beta = 2pi - alpha

VRMS = Vmax sqrt{ (1/2pi [ 2( beta – alpha ) + sin 2alpha - sin 2beta }

IRMS^2 = (1/pi).(Vmax/wL)^2 f(a, b)

Where f(a, b) = { (1/2 + cos^2(alpha))(beta-alpha) + ¼(sin 2beta + 3 sin 2alpha) – 2 cos(alpha) sin(beta) }


3. R-L load in series.

Let the load natural power angle phi = atan (wL/R). The extinction angle is obtained solving for beta in the equation below:

sin (beta-phi) exp[ beta.R/wL] = sin (alpha-phi) exp (R.alpha/wL)

in addition, for the regulation to occur the delay angle must also be greater than the load natural power angle, that is: phi <= alpha <= pi

The expression for the load voltage is the same as the case 2) above, but not so for the current RMS, which is this case is a much more elaborate one (no kidding!):

IRMS^2 = Umax^2 / pi.|z|^2 { [ ¼( 2(beta-alpha) + sin 2(alpha-phi) – sin 2(beta-phi)] –
- (wL/2R) sin^2(alpha-phi). exp (2R.alpha/wL) .[ exp (-2Rbeta/wL) – exp(-2R.alpha/wL) ] –
- 2 sin (alpha-phi) (wL/R)^2 exp [ (R/wL) (alpha-phi). f(a, b, f) }

with:
f(a, b, f) = exp (-R(alpha-phi)/wL) [cos (alpha-phi) + R/wL sin (alpha-phi) ] –
- exp (-R(beta-phi)/wL) [cos (beta-phi) + R/wL sin (beta-phi) ]


4. R-L load in Parallel.

The extinction angle is obtained solving for beta in the equation below:

sin(beta) { 1 + exp [ R(beta-alpha-phi)/wL] } = (R/wL) [cos beta - cos alpha]

with the same condition as the case before for regulation: phi <= alpha <= pi

Finally, the expression for the voltage RMS is below:

VRMS = [Vmax/2sqr(pi)] . sqrt{ [ 2(beta – alpha) + sin(2.alpha) -
- sin(2.beta)] + 2wL sin^2 (beta-pi) [ 1- exp [ 2R( beta – pi - alpha ) /wL ] }



Program Details.

The U/I guides the user during the data entry stage, with several prompts to determine the configuration used. You should use zero values to determine simple resistive or inductive cases, and answer “S/P” for the series or parallel case.

Parameter values can be changed at any time by executing the routine “ND” – new data – which will prompt for a parameter choice until you answer “N” in the “CHANGES? prompt.

To calculate the extinction angle b the program uses a built-in root-finding routine, “SLV”, also included in the module.


Example .

Obtain the extinction angles and the RMS values of the load voltage for a configuration with R= 200 Ohm and L = 1.1026 H in series - using trigger angles between 0 and 180, in increments of 10 deg. Compare the results with a simple resistive load case. The network data are: input voltage RMS Vinpt = 220 V, and frequency f = 50 Hz

The results are shown in the table below:

Code:

Delay angle   Extinction angle  VRMS Series R-L    VRMS Resistive
----------------------------------------------------------------------
10            NO REG            220                219.88
20            NO REG            220                219.038
30            NO REG            220                216.81
40            NO REG            220                212.67
50            NO REG            220                206.26
60            240               220                197.33
70            238.17            208.01             185.82
80            235.92            193.42             171.82
90            233.22            176.41             155.56
100           230.01            157.29             137.39
110           226.27            136.44             117.77
120           221.94            114.37              97.27
130           216.97             91.64              76.54
140           211.31             68.91              56.30
150           204.89             46.97              37.36
160           197.61             26.82              20.65
170           189.37              9.95               7.37
180           180.01           9.6 E-5               0.00