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'phase control and RMS voltage'
1997\09\06@212221
by
Harold Hallikainen
From page 240 of the GE SCR Manual (1979) is the following:
E 1
Eo =  sqrt(pia+  sin(2a))
sqrt(2*pi) 2
where
Eo is the RMS output voltage
E is peak AC voltage
a is turnon phase delay in radians (trigger triac a radians past
zerocross)
This is for fullwave control (triac or inverse parallel SCRs)
into a resistive load.
Glad I didn't have to derive it!
Harold

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1997\09\08@203101
by
sdattalo

Harold Hallikainen wrote:
> From page 240 of the GE SCR Manual (1979) is the following:
>
> E 1
> Eo =  sqrt(pia+  sin(2a))
> sqrt(2*pi) 2
>
> where
>
> Eo is the RMS output voltage
> E is peak AC voltage
> a is turnon phase delay in radians (trigger triac a radians past
> zerocross)
>
> This is for fullwave control (triac or inverse parallel SCRs)
> into a resistive load.
> Glad I didn't have to derive it!
But you could have if you wanted, because the equation from which
this expression was derived is:
____________________
/ 1 / T
E_eff = /  *  E_per(t)^2 dt
v T / 0
E_per(t) is a periodic signal whose "effective DC component" we wish
to ascertain. In the GE manual (which was destined to an early death
before I philanthropically rescued it from a circular file) they provide
the solution to this integral when the periodic signal is that obtained
from a SCR chopped sine wave. While drawing the equation using ASCII
art is almost doable, drawing a SCR chopped sine wave is nearly
impossible. But the equation is simple:
E_per(t) = 0 0 < t <= a
= E * sin(t) a < t <= pi
= 0 pi < t <= pi+a
= E * sin(t) pi+a < t < 2*pi (negative over this interval)
E_per(t) = E_per(t + n*2*pi) < Just emphasizing the periodicity.
The squaring operation in the integral makes the negative portion
positive. Consequently, we can simplify the calculation with the
newly found symmetry:
____________________________
/ 1 / pi
E_eff = /  *  E^2 * sin(t)^2 dt
v pi / a
________________________________
/ E^2 / pi
E_eff = /  *  (1  cos(2*t))/2 dt
v pi / a
______________________________
/ 1  pi
E_eff = E * /  * (t  sin(2*t)/2) 
v 2*pi  a
______________________________
/ 1
E_eff = E * /  * (pi  a + sin(2*a)/2)
v 2*pi
However, Stephen's question was
> Has anyone got a formula to calculate the change in power compared to the
> change in time in firing within the phase so as to allow me to linearise
> the
> response  or do you know of any articles which could point the way.
In short, the answer is no. The reason is transcendentalism.
If you have a purely resitive load, then the power delivered to it
would be:
P = E_eff^2/R
Where R is the load resistance. If I understand you correctly, you
want something like:
P(i) = Pmax * i / N
Where the variable "i" is a linear parameter that varies from 0 to N
or perhaps (N1). And so my guess is that you wish to derive the
functional relationship between the linear paramter "i" and the
phase angle "a". In which case your stuck with the burden of solving
N transcendental equations of this form:
i 2 1
 = E *  * (pi  a + sin(2*a)/2)
N R*4*pi
In other words, substitute i=0 and solve for a (that one's easy). Then
substitue i=1 and solve for a......
At any rate, the table approach suggested by Andy B. sounds like a
very reasonable solution.
BTW, The GE SCR manual I have is the 6th edition and was published in
1979 (which I presume is the same as yours Harold  the page number
matches). There is no ISDN number or anything of that sort.
Scott

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