\$Hybrid~ Model\$
more complex
\$h_{ie}=\dfrac{\delta V_{BE}}{\delta I_B} @Q_{point}\$ This is the small signal input impedance. This is a product of the dynamic current gain and many variables. \$h_{ie}\$ is normally in kΩ range for low linear currents. The transistor has both a DC current gain of \$\beta =H_{fe}\$ and a smaller AC current gain of \$h_{fe}\$ .
Simpler \$r_e~Model\$
-simpler model for DC to mid-frequency use with only 2 variables \$h_{fe}\$ and temperature.
Vt = γ = kT/q as the thermal voltage with temp. T [°K] and k=Boltzman's constant and charge, q
Vt = 26mV @ 30°C , 25mV @ 20°C
\$r_e = V_t/I_e~~~~\$ [Ω= mV / mA ] while \$r_e\$ is typically in the 10~70 ohm range.
since \$I_e=I_c+I_b ~,~~ ~ I_c=βI_b~,~~ I_e/I_b=(β+1)\$
\$h_{ie}=(β+1)r_e\$
ref's
Berkeley
Xi'an University of Electronic Science and Technology
Other stuff
Now if we were to add an emitter resistor Re=100Ω;
\$r_e=50,~~ Z_{in(f)} = β * [r_e+Ze(f)]=100 * (50+100)=15kΩ\$
But if Vce is saturated as a switch, β drops to 10-20 and Zin(f) drops to 10% of the above value.
When Vbe saturates this become more linear and depends on the PN junction size inverse to Pd of the diode junction. But we know that Ic doubles for every 17mV rise in Vbe if at constant temp.
But consider when Vbe is saturated and consider just diodes with no current gain.
I generalize this /$r_e~/$ as ESR~1/Pd so a 20mW junction is 50 Ohms, a 65mW LED as 16 Ohms, a 1W LED as 0.5 to 1 Ohm at 85'C but due to tolerances this is generally +/-25% typ and +/-50% worst case over temp.
Thermal design affects this as does temp rise, so generally SMD parts for ESR=k/Pd(max) k is drops from 1 to 0.5 and the very best thermal bulk power diodes are k=0.25 .
You won't find this in datasheets or textbooks, but it is my observation of doing above over thousands of devices.
The same can be done for the Rce value when a transistor is used as a switch.
Vce /Ice is saturated Vce(sat) @ I spec, then you can expect an Rce value similar to a power diode Rce = x/Pd but depends on design of chip and thermal capacity. so k has a wider tolerance and an offset that depends on doping differences of BE-CB, but x = 0.25 to 1 is a good fit from SMD power transistor to small signal plastic transistors.