User:Msiddalingaiah/BJT transition frequency

From Millman and Grabel second edition, page 465

$I_{b}=V_{\pi }\left[{\frac {1}{r_{\pi }}}+s(C_{\pi }+C_{\mu })\right]$

$I_{c}=V_{\pi }(g_{m}-sC_{\mu })$

$g_{m}r_{\pi }=\beta _{o}$

$\beta (s)={\frac {I_{c}}{I_{b}}}={\frac {\beta _{o}(1-{\frac {sC_{\mu }}{g_{m}}})}{1+s(C_{\pi }+C_{\mu })r_{\pi }}}$

$\beta (s)={\frac {\beta _{o}(1-{\frac {s}{\omega _{z}}})}{1+{\frac {s}{\omega _{\beta }}}}}$

$\omega _{z}={\frac {g_{m}}{C_{\mu }}}$

$\omega _{\beta }={\frac {1}{r_{\pi }(C_{\pi }+C_{\mu })}}$

Assuming single pole response (ignore the zero):

$\beta (s)={\frac {\beta _{o}}{1+{\frac {s}{\omega _{\beta }}}}}$

$\omega _{T}=\beta _{o}\omega _{\beta }$ $f_{T}=\beta _{o}f_{\beta }$

$f_{T}={\frac {g_{m}}{2\pi (C_{\pi }+C_{\mu })}}$

$g_{m}={\frac {I_{cq}}{V_{T}}}={\frac {I_{cq}}{26mV}}$

$f_{T}={\frac {I_{cq}}{2\pi (C_{\pi }+C_{\mu })V_{T}}}$