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GeneralRe: HI ALL [modified] Pin
73Zeppelin29-Apr-08 20:02
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johnzfrebbies@30-Apr-08 11:28
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73Zeppelin1-May-08 0:11
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johnzfrebbies@1-May-08 8:32
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73Zeppelin1-May-08 9:08
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johnzfrebbies@1-May-08 18:49
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73Zeppelin2-May-08 22:06
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QuestionCan someone who is good at math please check my working!? - Differential equation [modified] Pin
MarkB77722-Apr-08 3:58
MarkB77722-Apr-08 3:58 
Problem:

<br />
Let w = 2000<br />
V = 6V<br />
L = 0.2H<br />
R = 400<br />
<br />
Given the differential equation L*(di/dt) + R*i = V*cos(w*t)<br />
<br />
Find i(t). - The above equation is true for t = 0<br />
<br />


This is what I have come up with. - NOTE: When im integrating ive used the '!' symbol

I solved the above using the steady state and transient method.


For the transient part:

<br />
-> 0.2*(di/dt) + 400*i = 6*cos(w*t)     <br />
-> (di/dt) + 2000*i = 30*cos(w*t) <br />
<br />
-> di/dt + 2000*i = 0      <br />
    <br />
-> di = -2000*i*dt<br />
<br />
-> !(1/i)*di = !-2000*dt               ('!' means integrate)<br />
<br />
-> ln(i) = -2000t + c<br />
<br />
-> i(transient) = e^(-2000t)*e^c<br />
<br />
Let e^c = A (a constant)<br />
<br />
-> i(transient) = A*e^-2000t<br />


For the steady state part:

Ohms law states that the current i = V/Z, where V is voltage, and Z is impedance.

So, V = 6>0 V The '>' is the polar form (or phasor some might know it as) representation of the complex angle associated with the voltage, so 6V, 0 angle.

<br />
Z = R + j*w*L<br />
Z = 400 + j*2000*0.2<br />
Z = 400 + j400<br />
Z = 565.685>0.785             (all angles are in radians)<br />
<br />
-> i = V/Z<br />
<br />
-> i = 6>0 / 565.685>0.785<br />
<br />
-> i = 10.61>-0.785 mA<br />
<br />
-> i(steady state) = 10.61*sin(w*t - 0.785) mA<br />


Now to find the final equation for i(t)

<br />
To derrive i(t):<br />
<br />
-> i(t) = i(steady state) + i(transient)<br />
<br />
-> i(t) = A*e^-2000t + 10.61*sin(w*t - 0.785) mA    <br />
<br />
To find A make t = 0<br />
<br />
-> i(0) = A(1) + (10.61*10^-3)*sin(-0.785)<br />
<br />
-> 0 = A - 7.5*10^-3<br />
-> A = 7.5*10^-3<br />
<br />
So the final equation for i(t) is:<br />
<br />
-7.5*e^-2000t + 10.61*sin(w*t - 0.785) mA<br />
<br />


However this is not working out - I should be able to substitute this equation back into the original:

<br />
<br />
-> L*(di/dt) + R*i = V*cos(w*t)<br />
<br />
L = 0.2<br />
R = 400<br />
i(t) = 7.5*e^-2000t + 10.61*sin(2000*t - 0.785) mA<br />
i'(t) = -15*e^-2000t + 21.22*cos(2000*t - 0.785)<br />
V = 6<br />
<br />
-> 0.2*(-15*e^-2000t + 21.22*cos(2000*t - 0.785)) + 400((7.5*10^-3)*e^-2000t + (10.61*10^-3)*sin(2000*t - 0.785)) = 6*cos(2000*t)<br />
<br />
Let t = 1<br />
<br />
-> 0.2(0 + 8.434) + 400(0 + (9.8677*10^-3)) = 6cos(2000)<br />
<br />
-> 1.6869 + 3.947 = -2.205<br />
<br />
This does not satisfy the equation!!!<br />
<br />



Can anyone see where I went wrong in my working?

[Edit] - Fixed a typo.

Cheers,


modified on Tuesday, April 22, 2008 10:28 AM

AnswerRe: Can someone who is good at math please check my working!? - Differential equation Pin
73Zeppelin22-Apr-08 4:09
73Zeppelin22-Apr-08 4:09 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
MarkB77722-Apr-08 4:18
MarkB77722-Apr-08 4:18 
AnswerRe: Can someone who is good at math please check my working!? - Differential equation Pin
73Zeppelin22-Apr-08 4:15
73Zeppelin22-Apr-08 4:15 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
MarkB77722-Apr-08 4:22
MarkB77722-Apr-08 4:22 
AnswerRe: Can someone who is good at math please check my working!? - Differential equation Pin
CPallini22-Apr-08 4:20
mveCPallini22-Apr-08 4:20 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
MarkB77722-Apr-08 4:23
MarkB77722-Apr-08 4:23 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
CPallini22-Apr-08 4:26
mveCPallini22-Apr-08 4:26 
AnswerRe: Can someone who is good at math please check my working!? - Differential equation Pin
CPallini22-Apr-08 4:55
mveCPallini22-Apr-08 4:55 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
73Zeppelin22-Apr-08 4:57
73Zeppelin22-Apr-08 4:57 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
CPallini22-Apr-08 5:09
mveCPallini22-Apr-08 5:09 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
MarkB77722-Apr-08 12:54
MarkB77722-Apr-08 12:54 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
CPallini22-Apr-08 22:54
mveCPallini22-Apr-08 22:54 
AnswerRe: Can someone who is good at math please check my working!? - Differential equation Pin
73Zeppelin22-Apr-08 7:41
73Zeppelin22-Apr-08 7:41 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
MarkB77722-Apr-08 12:52
MarkB77722-Apr-08 12:52 
GeneralRe: Can someone who is good at math please check my working!? - Differential equation Pin
73Zeppelin22-Apr-08 22:51
73Zeppelin22-Apr-08 22:51 
Answer[OT] Re: Can someone who is good at math please check my working!? - Differential equation Pin
David Crow28-Apr-08 6:45
David Crow28-Apr-08 6:45 
GeneralRe: [OT] Re: Can someone who is good at math please check my working!? - Differential equation Pin
Roger Wright28-Apr-08 6:55
professionalRoger Wright28-Apr-08 6:55 

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