11-26-2016, 10:21 AM
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:
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