# A Mathematical Description of MOSFET Behavior

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1 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical relaionships beween i D, v, or v DS. How can we deermine he correc numeric values for MOSFET volages and currens? A: A mahemaical descripion of enhancemen MOSFET behavior is relaively sraighforward! We acually need o concern ourselves wih jus 3 equaions. Specifically, we express he drain curren i D in erms of v and v DS for each of he hree MOSFET modes (i.e., Cuoff, Triode, Sauraion). Addiionally, we need o mahemaically define he boundaries beween each of hese hree modes!

2 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc /8 Bu firs, we need o examine some fundamenal physical parameers ha describe a MOSFET device. These parameers include: k Process Transconducance Parameer A/V W L = Channel Aspec Raio The Process Transconducance Parameer k is a consan ha depends on he process echnology used o fabricae an inegraed circui. Therefore, all he ransisors on a given subsrae will ypically have he same value of his parameer. The Channel Aspec Raio W L is simply he raio of channel widh W o channel lengh L. This is he MOSFET device parameer ha can be alered and modified by he circui designer o saisfy he requiremens of he given circui or ransisor. We can likewise combine hese parameer o form a single MOSFET device parameer K : K 1 W k A L V = Now we can mahemaically describe he behavior of an enhancemen MOSFET! Well do his one mode a a ime.

3 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 3/8 CUTOFF This relaionship is very simple if he MOSFET is in cuoff, he drain curren is simply zero! i D = 0 (CUTOFF mode) TRIODE When in riode mode, he drain curren is dependen on boh v and v DS : W 1 id = k ( v V ) vds vds L ( ) = K v V v v DS DS (TRIODE mode) This equaion is valid for boh NMOS and PMOS ransisors (if in TRIODE mode). Recall ha for PMOS devices, he values of v and v DS are negaive, bu noe ha his will resul (correcly so) in a posiive value of i D. SATURATION When in sauraion mode, he drain curren is (approximaely) dependen on v only:

4 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 4/8 1 W id = k v V L ( V ) = K v ( ) (SATURATION mode) Thus, we see ha he drain curren in sauraion is proporional o excess gae volage squared! This equaion is likewise valid for boh NMOS and PMOS ransisors (if in SATURATION mode). Q: OK, so know we know he expression for drain curren i D in each of he hree MOSFET modes, bu how will we know wha mode he MOSFET is in? A: We mus deermine he mahemaical boundaries of each mode. Jus as before, we will do his one mode a a ime! CUTOFF A MOSFET is in cuoff when no channel has been induced. Thus, for an enhancemen NMOS device: if v V < 0 hen NMOS in CUTOFF

5 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 5/8 Like wise, for an enhancemen PMOS device: if v V > 0 hen PMOS in CUTOFF TRIODE For riode mode, we know ha a channel is induced (i.e., an inversion layer is presen). Addiionally, we know ha when in riode mode, he volage v DS is no sufficienly large for NMOS, or sufficienly small (i.e., sufficienly negaive) for PMOS, o pinch off his induced channel. Q: Bu how large does v DS need o be o pinch off an NMOS channel? How can we deermine if pinch off has occurred? A: The answer o ha quesion is surprisingly simple. The induced channel of an NMOS device is pinched off if he volage v DS is greaer han he excess gae volage! I.E.: if v > v V DS hen NMOS channel is "pinched off"

6 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 6/8 Conversely, for PMOS devices, we find ha: if v < v V DS hen PMOS channel is "pinched off" These saemens of course mean ha an NMOS channel is no pinched off if vds < v V, and a PMOS channel is no pinched off if v > v V. Thus, we can say ha an NMOS DS device is in he TRIODE mode: if v V > 0 and v < v V hen NMOS in TRIODE DS Similarly, for PMOS: if v V < 0 and v > v V hen PMOS in TRIODE DS SATURATION Recall for SATURATION mode ha a channel is induced, and ha channel is pinched off.

7 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 7/8 Thus, we can sae ha for NMOS: if v V > 0 and v > v V hen NMOS in SAT. DS And for PMOS: if v V < 0 and v < v V hen PMOS in SAT. DS We now can consruc a complee (coninuous) expression relaing drain curren i D o volages v DS and v. For an NMOS device, his expression is: 0 if v V < 0 id = K ( v V ) vds vds if v V > 0 and vds < v V ( ) K v V if v V > 0 and vds > v V

8 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 8/8 Likewise, for a PMOS device we find: 0 if v V > 0 id = K ( v V ) vds vds if v V < 0 and vds > v V ( ) K v V if v V < 0 and vds < v V Le s ake a look a wha hese expressions look like when we plo hem. Specifically, for an NMOS device le s plo i D versus v DS for differen values of v :

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