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how can i analysis this symbol transcendental function? "SOS"

Posted on 2008-08-11 03:59 By andy.zhou.nuaa (Maple Rating 1 17)
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The function is shown in following ;

-(1/2)*Rs*(-2*Rs*Cs+t+2*exp(-t/(Rs*Cs))*Rs*Cs+exp(-t/(Rs*Cs))*t)/(Lp*(-1+exp(-Pi*sqrt(Cp*Lp)/(Rs*Cs))*exp(-t/(Rs*Cs))))=n;

we assume Cs,Lp, n is constant, then we know "t" and "Cs" is the function relationship. but this is the transcendental function,

when Cs is increase , how does "t" change? increase or decrease? i don't know how to analysis this by higher mathmatics.

Best regards!

 

andy

12/08/08.

 

 

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alec's picture

Constants

Comment on 2008-08-11 04:56 from alec (Maple Rating 4 1476)

That may depend on the constants. You may try to substitute some values for the constants, then find symbolic solution using solve (solving for t), then plot the result (as a function of Cs) and see from the plot, either t is decreasing or increasing.

Alec

  i know it can be solved

Comment on 2008-08-11 07:33 from andy.zhou.nuaa (Maple Rating 1 17)

  i know it can be solved by substitue constants with values. i want to know does the value has generality, i worry some problems come up:

 Does the following problem will be exsit:

 when Lp=1mH, the function is  the monotone function.

 when Lp=10mH, the function is the monotonous fuction.

that is what i worry!

use adimensional variables

Comment on 2008-08-11 15:10 from jakubi (Maple Rating 4 911)

to simplify the analysis, eg like:

-(1/2)*Rs*(-2*Rs*Cs+t+2*exp(-t/(Rs*Cs))*Rs*Cs
+exp(-t/(Rs*Cs))*t)/(Lp*(-1+exp(-Pi*sqrt(Cp*Lp)/(Rs*Cs))
*exp(-t/(Rs*Cs))))=n;
subs(Rs=ts/Cs,t=tau*ts,(Cp*Lp)^(1/2)=alpha*ts,%);
collect(%,ts);
subs(ts^2=Cs*Lp/beta^2,%);

                -2 + tau + 2 exp(-tau) + exp(-tau) tau
           -1/2 -------------------------------------- = n
                                                    2
                (-1 + exp(-Pi alpha) exp(-tau)) beta

So, the adimensionalized time tau depends on two parameters: alpha and n*beta^2=gamma, say. You can make a 3D plot then.

don't overlook implicitdiff

Comment on 2008-08-11 15:54 from Alex Smith (Maple Rating 3 289)

Perhaps you can use implicitdiff:

implicitdiff(eq,t,Cs);

It looks a little messy, but if you can get a handle on its sign, you question is answered.

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