the Technology Interface JournalFall  Cartwright Volume  No
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the Technology Interface JournalFall Cartwright Volume No

1 ISSN 15239926 httptechnologyinterfacenmsueduFall09 Derivation of the Exact Transconductance of a FET without Calculus by Kenneth V Cartwright PhD School of Mathem atics Physics and Technology College of The Bahamas httpkvcartwrightgooglepagescom A

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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ Derivation of the Exact Transconductance of a FET without Calculus by Kenneth V. Cartwright, Ph.D. School of Mathem atics, Physics and Technology College of The Bahamas http://kvcartwright.googlepages.com/ Abstract It is shown that the exact transconductance of a field effect transistor (JFET, MESFET or MOSFET) can be derived without calculus. Th e method simply requires the solution of two simultaneous equations, one invol ving a quadratic equation

a nd the other a linear equation. I. Introduction The transconductance of a field effect transistor (FET) is used in the sma ll signal analysis of FET circuits. It is well known th at the transconductance at a particular bias point depends upon the slope of the tangent line of the transfer characteristic curve of the FET, at that point. Since slopes of tangent lines are conventionally found with calculus, most tec hnology textbooks simply assume that calculus is necessary to derive the slope: he nce, the transconductance is simp ly presented without derivation. However, it is known in the

mathematical literature that slopes of tangent lines of some simple functions can be found without calculus. (See, for example, [1] and [2]: note the content of [1] is reproduced online in [3]). Unfortunately, this fact is not well known to engineering te chnologists. Hence, the purpose of this paper is to apply th is mathematical result to the deriva tion of the slope of the transfer characteristic curve, i.e. the transconductance of a FET. This will be done for the JFET, the depletion and enhancement mode MESFET and MOSFET. II. Transconductance of the JFET or Depletio n Mode MESFET or MOSFET

without Calculus The transfer equation (Shockley s equation [4]) for the JFET or depletion mode MESFET or MOSFET is given by 1, GS DDSS II (1)
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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ where is the drain current, is the pinch-off voltage and GS is the gate-to-source voltage. Eq. (1) is plotted in Fig. 1. -2.5 -2 -1.5 -1 -0.5 -1 x 10 -3 GS (V) (A) Fig. 1. Plot of the transfer equation of the JF ET or depletion mode MESFET or MOSFET with DSS mA and 2.5 . VV Also shown is the tangent line that

touches the transfer curve at the point 1.5 , 0.96 . GS D VVImA The slope of this tangent line is by definition the transconductance of the JFET or depletion mode MESFET or MOSFET, at this point. Note that the equation of the straight lin e also shown in Fig. 1 is given by , DGS mV B (2) where is the slope of the line in /, i.e. Siemens, and is the intercept of that line. Furthermore, the straight line in (2) is a tangent line to the curve in Fig. 1 if it intercepts the curve at exactly one point. This particular definition of a tangent line is not a definition that works for any function.

Fortunately, however, this definition is valid for the quadrat ic curve of Fig. 1, and other functions such as reciprocals, squa re roots and ellipses [1]. Hence, in order for (2) to be a tangent line to the quadratic curve, we must have
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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ 1. GS DSS GS mV B (3) Equation (3) can be rewritten as 10 12 0 0, GS DSS GS GS GS DSS GS PP DSS DSS GS GS DSS PP GS GS ImVB VV ImVB VV II VmVIB VV aV bV c (4) where DSS DSS PP II ab m VV and . DSS cI B The quadratic

formula can easily be applied to (4) to get or ac ac GS (5) Eq. (5) is stating that the straight line intersects the curve at two values of GS . However, in order for (2) to be a tangent, we require that there be only one point of intersecti on. Hence, we require that or 0, ac ac and GS This means that
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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ or or or GS DSS GS DSS DSS GS DSS DSS GS DSS DSS DSS GS (6) As is negative for n-channel FET, VV and so (6) is quite often written as 1. DSS GS PP IV

VV (7) Note that (7) is also valid for the p-channel FET. Also, note that (7) is giving the slope of the straight line which is touching the curve of Fig.1 at exactly one point: that is, (7) is giving the slope of the tangent line of (1). This is in fact the definition of the transconductance of the JFET or deple tion mode MESFET or MOSFET, i.e., . gm Hence, the transconductance of the JFET depletion mode MESF ET or MOSFET has been derived without using calculus, which is thankfully the same expr ession that is deri ved with calculus. III. Transconductance of an Enha ncement Mode MESFET or

MOSFET without Calculus The transfer equation for the enhancemen t mode MESFET or MOSFET is given by DGST IkVV (8) where is the drain current, GS is the gate-to-source voltage, is the minimum gate-to-source voltage that is needed to turn the device on, DON GSON T VV and GSON DON VI is a point on the curve when the device is on. Eq. (8) is plotted in Fig. 2 for representative values of and .
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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ 10 0.002 0.004 0.006 0.008 0.01 0.012 GS (V) (A) Fig. 2.

Plot of the transfer equation of th e enhancement mode MESFET or MOSFET with 0.00024 / kAV and 3. VV Also shown is the tangent line that touches the transfer curve at the point 7 , 3.84 . GS D VVI mA The slope of this tangent line is by definition the transconductance of the enhancement mode MESFET or MOSFET at this point. Note that the equation of the straight lin e also shown in Fig. 2 is given by 11 DGS mV B (9) where is the slope of the line in /, AV i.e. Siemens, and is the intercept of that line. Furthermore, the straight line in (9) is a tangent line to the curve in Fig. 2 if it

intercepts the curve at exactly one point, as mentioned earlier. Hence, in or der for (9) to be a tangen t line to the quadratic curve, we must have 11 GS T GS kV V mV B (10)
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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ Eq. (10) can be rewritten as 11 22 11 22 11 111 20 20 0, GS T GS GS T GS T GS GS T GS T GS GS kV V mV B kV VV V mV B kV kV m V kV B aV bV c (11) where 11 1 ,2 akb kVm and 11 ckV B The quadratic formula can easily be applied to (11) to get 22 111 111 11 11 11 44 or . 22 22 GS

bac bac bb aa aa (12) Eq. (12) is stating that the straight lin e intersects the curve at two values of GS . However, in order for (9) to be a tangent, we require th at there be only one point of intersection. Hence, we require that 111 111 0, or 4 , bac bac and GS This means that ,or 22 ,or 22,or 2. GS GS T GS T GS T kV m kV kV m mkV kV mkVV (13) Notice that (13) is giving the slope of the straight line which is touching the curv e of Fig.2 at exactly one point: that is, (13) is giving the sl ope of the tangent line of (8). This is in fact the definition of the transconductance of the

enhancemen t mode MESFET or MOSFET, i.e., gm Hence, the transconductance of the enhancem ent mode MESFET or MOSFET has been derived without using calculus, which is thankfully the same expression that is derived with calculus. VI. Conclusion It has been shown that the tr ansconductance of a JFET, MESFET or MOSFET can be derived without using calculus. This was possible because the transconduc tance is simply the slop e of the tangent of the transfer characteristic curve, which happens to be a quadratic function. Happily, for such a function, the slope of the tangent line can be f ound quite

easily by purely algebraic m eans, as has been demonstrated.
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the Technology Interface Journal/Fall 2009 Cartwright Volume 10 No. 1 ISSN# 1523-9926 http://technologyinterface.nmsu.edu/Fall09/ References [1] Wahl, M., “Derivatives without Limits, Math Horizons , pg 12 and pg. 30, Nov. 2008. [2] Rabin, J. M., “Tangent Lines without Calculus, Mathematics Teacher , Vol. 101, No. 7, pp. 499- 503, March 2008. [3] CTK Wiki Math, “Deriv atives without Limits, http://www.cut-the-knot.org/wiki-math/inde x.php?n=Calculus.Der ivativesWithoutLimits , accessed 7 th March 2009. [4] Boylestad,

R. L. and Nashelsky, L., Electronic Devices and Circuit Theory , 10 th edition, Prentice Hall, 2009. Biography KENNETH V. CARTWRIGHT ( kvcartwright@yahoo.com ) was born in Nassau, N. P., Bahamas. He received the B.E.Sc, M.S., and PhD degrees in elec trical engineering from th e University of Western Ontario in 1978, and Tulane University in 1987 and 1990, respectively. Presently, he is a professor of electrical engineering in the School of Mathematics, Physics and Technology and the Coordinator for the Electrical Engineering Technolog y Program, at The College of The Bahamas. His research

interests include digital signal processing, analog and dig ital communication systems, and analog electronics. Some of this research is published in IEEE Signal Processing Letters, IEEE Communication Letters , IEEE Transactions on Communications, IEEE conferences' proceedings, and the Technology Interface Journal . Dr. Cartwright has also served as a reviewer for many journals. He is a member of Eta Kappa Nu, Tau Beta Pi, and numerous IEEE societies. (A more complete bi ography is available at http://kvcartwright.googlepages.com/ ).