1. Derivation of fT And fMAX of a MOSFET

2. Derivation of fT (MOSFETs)
The unity current gain frequency* (aka cutoff frequency)
Defined under the condition that the output is loaded with an AC short.
fT does not depend on Rg and ro

3. Derivation of fT (MOSFETs) (Continued)
Assume the zero (sCgd) is smaller compared to gm.

4. fT with Parasitic RS and RD

5. Derivation of fT (MOSFETs) (Continued) (RS and RD are included)
Miller’s Approximation

6. Derivation of fMAX (MOSFETs)
fMAX * is the frequency at which the maximum power gain =1 (*aka maximum oscillation frequency)
fMAX is defined with its input and output ports conjugate-matched for maximum power transfer
So, we need to know the input and output impedance to define the input and output power as well as achieve the max power transfer matching condition.

7. Derivation of fMAX (MOSFETs)
At high frequency (close to fmax), we can assume that
So, Rg is independent of RL

8. Derivation of fMAX (MOSFETs) (Continued)
Conjugate match at the input:
Conjugate match at the output:
For the matching conditions,

9. Power Gain (Under Conjugate Match)
Using the definition of fT

10. Derivation of fMAX (MOSFETs)(Continued) (RS and RD are included)
For high frequency condition, Cgs → short
Hence, replace Rg by Rg+Rs
w/ (RS+RD) term
w/o (RS+RD) term

11. Derivation of fT And fMAX of a BJT

12. Derivation of fT (Bipolar)
For Bipolar Transistors,
CDE is due to minority carriers caused by FB

13. Derivation of fT (Bipolar) (Continued)
QE = minority holes stored in emitter
QB = minority electrons stored in base
QBE = electrons induced by the current
through the depletion region of BE-junction
QBC = electrons induced by the current
through the depletion region of BC-junction

14. Derivation of fT (Bipolar) (Continued)
Width of Neutral Region
Width of Depletion Region
if drift current is considered.
is greater than because of reverse-biasing.

15. Derivation of fT (Bipolar) (RS and RD are included)
For bipolar, the result is similar.
The only difference is that the term must be included.

16. Derivation of fMAX (Bipolar)
For bipolar transistors, there is no term.