Finding an expression for charge (Q) given an I-V equation (1 Viewer)

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An ideal semiconductor diode is a nonlinear element that obeys the following I-V equation:

[tex]I\,=\,I_s\,\left(\,e^{\frac{V}{V_{th}}}\,-\,1\right)[/tex]

where [itex]I_s[/itex] is a constant (saturation current) and [itex]V_{th}[/itex] is a constant (thermal voltage, [itex]V_{th}\,=\,\frac{k_B\,T}{q}[/itex]).

Assuming the applied voltage is given by

[tex]\begin{displaymath}
V\,=\,\left\{ \begin{array}{ll}
0 & for\,t\,<\,0 \\
Bt & for\,t\,\geq\,0 \\
\end{array} \right.
\end{displaymath}[/tex]

where B is a known constant.

Find an analytic expression for the charge Q(t) that has passed through the diode over the period from 0 to t. Also find the analytic expressions for the power dissipated by the diode p(t) and for the total energy dissipated by the diode w(t) over the period from 0 to t.

Now assuming [itex]I_s\,=\,1\,\times\,10^{-14},\,V_{th}\,=\,25.85\,mV[/itex] and [itex]B\,=\,90\,\frac{mV}{s}[/itex] use MATLAB to plot I(t), Q(t), p(t), and w(t). Do your plots for t = 0 to 10s.

Find the time [itex]\tau[/itex] at which a total of 1 C of charge has passed through the diode ([itex]Q(\tau)\,=\,1\,C[/itex]) and find the values of [itex]p(\tau)[/itex] and [itex]w(\tau)[/itex].


MY WORK SO FAR:

[tex]Q\,=\,\int_0^t\,i\,dt\,=\,\int_0^t\,\left(I_s\,e^{\frac{V}{V_{th}}}\,-\,I_s\right)\,dt[/tex]

[tex]Q(t)\,=\,\left[I_s\,e^{\frac{V}{V_{th}}}\,t\,-\,I_s\,t\right]_0^t\,=\,\left(I_s\,e^{\frac{V}{V_{th}}}\,-\,I_s\right)\,t[/tex]

Do I have the first part (equation for Q) of the question right?
 
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Mabye the integral is wrong? Is this better?:

[tex]Q\,(t)\,=\,\int_0^t\,\left(I_s\,e^{\frac{B\,t}{V_{th}}}\,-\,I_s\right)\,dt[/tex]

[tex]Q\,(t)\,=\,\left[I_s\,e^{\frac{B\,t}{V_{th}}}\,t\,-\,I_s\,t\right]_0^t\,=\,\left(I_s\,e^{\frac{B\,t}{V_{th}}}\,-\,I_s\right)\,t[/tex]
 
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