eem2am
Banned
Hello,
I am trying to find out how to find the maximum current that will occur for the LRES inductor in this CFL schematic.
Please could I be assisted to do this.?
CFL Schematic
*************
I must know this so I can ensure that this inductor does not saturate.
Mains = 265VAC
LRES = 2.3mH
CRES = 6.8nF
Flourescent Tube ON’ resistance, R = 228 Ohms.
Now:-
iL(t) = inductor current at time t
iR(t) = Resistor current at time t
iC(t) = Capcitor current at time t
vL(t) = inductor voltage at time t
vR(t) = Resistor voltage at time t
vC(t) = Capcitor voltage at time t
V = Step input voltage of 188V at t=0
Z = impedance of L in series with parallel RC.
To cut a long story short………
…..this problem boils down to an inductor (LRES) in series with a parallel RC (where R = 227.8R and C= 6.8nF)………
That is, this circuit here
********************
LRC circuit
A voltage of 188V appears across this LRC network at t = 0 (C initially uncharged and I(L) = 0 at t = 0.
…so here is the situation on LTSpice, with the step input of 188V, and the rising inductor current…….
*************
It is then needed to solve (by calculation, not simulation) for what current appears in the inductor (LRES) at t=16us.
(This is due to 16us being the bridge half period time at the ‘RUN’ frequency)
Here is the circuit to be solved as above…..
L in series with parallel RC Circuit
**********
I believe the quickest way to do this is to use Laplace Transforms.
The differential equation to start with is………
(Using Kirchoffs Laws…….iL(t) = iR(t) + iC(t)
…….iL(t) = vC(t) / R + C * dvC(t) / dt
differentiating both sides with respect to t and rearranging……
[V - vc(t)] / L = C * [d^2]vc(t)/[dt]^2 + d.vc(t)/dt (1)
rearranging…
V/L = C * [d^2]vc(t)/[dt]^2 + d.vc(t)/dt + vc(t)/L (2)
This is then converted to the S domain…………
V/sL = C( [s^2]Vc(S) ) + s Vc(S) + Vc(S) / L (3)
Rearranging………
Vc(S) = V / { s * (CL[s^2] + sL + 1) } (4)
I have somehow got to convert the above equation (4) to a standard Laplace Transform so that I can convert back to one of the time-domain solutions as given in standard Laplace tables.
Unfortunately I am struggling with this.
I believe I must use Partial Fractions to help, but the ……………
“s * (CL[s^2] + sL + 1) “
…..term in the denominator of (4) appears to defy all the partial fraction forms given in my maths book.
Any help to get equation (4) into a standard Laplace Transform much appreciated.
When I have solved equation (4) I will have an expression for vC(t). (Cap voltage)
-I will then be able to do Q = C * vC(t) to find the Q (i.e. charge) in the cap at time t .
the current in the capacitor will be iC(t) = dQ/dt
current in the resistor will be iR(t) = vC(t)/R
then the required inductor current , iL(t) will be iR(t) + iC(t)
Anyway, I have not been able to convert equation (4) to the time domain so ,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-HERE is ANOTHER method to find the inductor current at time t = 16us.:-
(again I am not sure if it is right so please could you check?
First of all
iL(t) = iC(t) + iR(t) (5) …..Kirchoff’s Law
And
iL(t) = V / Z (6)
where V = 188V ,
and Z = sL + { [R * 1/sC] / [R + 1/sC ] } (7)
rearranging (7)
Z = sL + { R / (sCR + 1) } (8)
Putting (8) and (6) into (5)…………..
V / sL + [ R / (sCR + 1) ] = iC(t) + iR(t) (9)
But: = iC(t) = C*dvC(t)/dt (10)
And : iR(t) = vC(t)/R (11)
So: V / {sL + [ R / (sCR + 1) ] } = C*dvC(t)/dt + vC(t)/R (12)
Converting LHS of (12) to S domain……
: V / {sL + [ R / (sCR + 1) ] } = sC Vc(S) + Vc(S)/R (13)
Rearranging (13)
Vc(s) = (V/R) / ( s^2 .LCR + sL + R) (14)
And again, (14) is not in any standard Laplace form.
Please does any reader know how to get (14) into a standard Laplace form so I can convert it to time domain and find vC(t) at t = 16us as part of the “journey” of getting the inductor current, iL(t) at t = 16us ?
Tables of partial fractions don’t appear to help in factorisong the denominator of (14).
I am trying to find out how to find the maximum current that will occur for the LRES inductor in this CFL schematic.
Please could I be assisted to do this.?
CFL Schematic
*************
I must know this so I can ensure that this inductor does not saturate.
Mains = 265VAC
LRES = 2.3mH
CRES = 6.8nF
Flourescent Tube ON’ resistance, R = 228 Ohms.
Now:-
iL(t) = inductor current at time t
iR(t) = Resistor current at time t
iC(t) = Capcitor current at time t
vL(t) = inductor voltage at time t
vR(t) = Resistor voltage at time t
vC(t) = Capcitor voltage at time t
V = Step input voltage of 188V at t=0
Z = impedance of L in series with parallel RC.
To cut a long story short………
…..this problem boils down to an inductor (LRES) in series with a parallel RC (where R = 227.8R and C= 6.8nF)………
That is, this circuit here
********************
LRC circuit
A voltage of 188V appears across this LRC network at t = 0 (C initially uncharged and I(L) = 0 at t = 0.
…so here is the situation on LTSpice, with the step input of 188V, and the rising inductor current…….
*************
It is then needed to solve (by calculation, not simulation) for what current appears in the inductor (LRES) at t=16us.
(This is due to 16us being the bridge half period time at the ‘RUN’ frequency)
Here is the circuit to be solved as above…..
L in series with parallel RC Circuit
**********
I believe the quickest way to do this is to use Laplace Transforms.
The differential equation to start with is………
(Using Kirchoffs Laws…….iL(t) = iR(t) + iC(t)
…….iL(t) = vC(t) / R + C * dvC(t) / dt
differentiating both sides with respect to t and rearranging……
[V - vc(t)] / L = C * [d^2]vc(t)/[dt]^2 + d.vc(t)/dt (1)
rearranging…
V/L = C * [d^2]vc(t)/[dt]^2 + d.vc(t)/dt + vc(t)/L (2)
This is then converted to the S domain…………
V/sL = C( [s^2]Vc(S) ) + s Vc(S) + Vc(S) / L (3)
Rearranging………
Vc(S) = V / { s * (CL[s^2] + sL + 1) } (4)
I have somehow got to convert the above equation (4) to a standard Laplace Transform so that I can convert back to one of the time-domain solutions as given in standard Laplace tables.
Unfortunately I am struggling with this.
I believe I must use Partial Fractions to help, but the ……………
“s * (CL[s^2] + sL + 1) “
…..term in the denominator of (4) appears to defy all the partial fraction forms given in my maths book.
Any help to get equation (4) into a standard Laplace Transform much appreciated.
When I have solved equation (4) I will have an expression for vC(t). (Cap voltage)
-I will then be able to do Q = C * vC(t) to find the Q (i.e. charge) in the cap at time t .
the current in the capacitor will be iC(t) = dQ/dt
current in the resistor will be iR(t) = vC(t)/R
then the required inductor current , iL(t) will be iR(t) + iC(t)
Anyway, I have not been able to convert equation (4) to the time domain so ,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-HERE is ANOTHER method to find the inductor current at time t = 16us.:-
(again I am not sure if it is right so please could you check?
First of all
iL(t) = iC(t) + iR(t) (5) …..Kirchoff’s Law
And
iL(t) = V / Z (6)
where V = 188V ,
and Z = sL + { [R * 1/sC] / [R + 1/sC ] } (7)
rearranging (7)
Z = sL + { R / (sCR + 1) } (8)
Putting (8) and (6) into (5)…………..
V / sL + [ R / (sCR + 1) ] = iC(t) + iR(t) (9)
But: = iC(t) = C*dvC(t)/dt (10)
And : iR(t) = vC(t)/R (11)
So: V / {sL + [ R / (sCR + 1) ] } = C*dvC(t)/dt + vC(t)/R (12)
Converting LHS of (12) to S domain……
: V / {sL + [ R / (sCR + 1) ] } = sC Vc(S) + Vc(S)/R (13)
Rearranging (13)
Vc(s) = (V/R) / ( s^2 .LCR + sL + R) (14)
And again, (14) is not in any standard Laplace form.
Please does any reader know how to get (14) into a standard Laplace form so I can convert it to time domain and find vC(t) at t = 16us as part of the “journey” of getting the inductor current, iL(t) at t = 16us ?
Tables of partial fractions don’t appear to help in factorisong the denominator of (14).
