PROCEEDINGS LETTERS 595

I, n i?

6 G

( b )

I L/I 1 hl I

G ( C )

y’ IS, ” I Is2

Fig. 5 . Static large-signal FET models. (a) Depletion-enhancement mode IGFET (b) Enhancement-mode IGFET, (c) Junction-gate FET.

purity profiles [9F[12]. The problem of a forward-biased gate junction may be handled by simply placing an exponential semiconductor diode in parallel with the ideal fieldeffect diode. This effectively clamps the for- ward voltage that may be applied to the fieldeffect diode, and provides a gate-current path for the forward-bias condition. This arrangement is shown in Fig. 4.

Substitution of the models of Figs. 2 through 4 into the general form of Fig. 1 results in the final static large-signal models shown in Fig. 5. In Fig. 5(b), a source transformation was used to combine the two V, sources and locate them in the gate lead by noting that a voltage source which is in series with a current source may be deleted. The development of an equivalent set of models for p-channel devices is straightforward.

CONCLUSIONS A series of static large-signal models for the field effect have been de-

veloped. These models consist of familiar circuit building blocks plus a single new nonlinear element, the ideal fieldeffect diode, which has a mathematically simple voltampere characteristic. These models unify the large-signal description of the various types of FETs, and permit genera- tion of the complete set of static characteristics, including the usually neglected behavior of inverted operation.

BRUCE D. WEDLOCK Dept. of Elec. Engrg. and Ctr. for Materials S c i . and Engrg. Mass. Inst. Tech. Cambridge, Mass. 02139

REFERENCES [ I ] B. D. Roberts, Jr., and C. 0. Harbourt, “Computer models of the field-effect tran-

[2] H. Shichman and D. A. Hodges, “Modeling and simulation of insulated-gate field- sistor,” Proc. IE€E, vol. 5 5 , pp. 1921-1929, November 1967.

effect transistor switching circuits,” IEEE 1. Solid-State Circuits, vol. SC-3, pp. 285- 289, September 1968.

[3] F. A. Lindholm and D. I. Hamilton,“‘A systematic modeling theory for solid-state devices,” Soli&Srate Electronics, vol. 7, pp. 771-783, November 1964.

[4] C. L. Searle et a/., Elementary Circuit Properties of Transistors. New York: Wiley, 1964, pp. 37-51.

[51 R. R. Bockemuehl, “Analysis of field-effect transistors with arbitrary charge dis- tribution,” IE€€ Trans. Electron Deoices, vol. ED-IO, pp. 31-34, January 1963.

[6] S. R. Hofstein and F. P. Heiman, “The silicon insulated-gate field-effect transistor,” Proc. I€E€, vol. 51, pp. 119C-1202, September 1963.

[71 C. T. Sah, “Characteristiaofthemetaldxide semiconductor transistors,” I€€€ Trans.

[E] 6. D. Wedlock, “On the field-effect transistor characteristics,” I€€E Trans. Electron Electron Decices, vol. ED-1 1, pp. 324-345, July 1964.

[9] I. Richer and R. D. Middlebrook, “Power-law nature of field-effect transistor experi- Deoices (Correspondence), vol. ED-15, pp. 181-182, March 1968.

mental characteristics,” Proc. IEEE (Correspondence), vol. 51, pp. 1145-1146, August 1963.

[IO] L. J. Sevin, Field-Efect Transistors. New York: McGraw-Hill, 1965, pp. 21-22. [ I l l R. D. Middlebrook, “A simple derivation of field-effect transistor characteristics,”

Proc. IEEE(Correspondence). vol. 51, pp. 114&1147, August 1963. [I21 B. D. Wedlock, “Direct determination of the pinch-off voltage of a depletion-mode

field-effect transistor,” Proc. IEEE (Letten), vol. 57, pp. 75-77, January 1%9.

Sensitivity Minimum in Active Transfer Function Synthesis Abstract-A formula for Schoeffler‘s sensitivity criterion is de-

veloped and used to prove the existence of a sensitivity minimum in an active RC transfer function synthesis procedure.

In a letter by Cooper and Harbourt [l] a method of reducing the sensi- tivity of an RC active network in transfer function synthesis has been discussed. It is shown here that even when that method is not applicable, Schoeffler’s performance criterion [2] can always be minimized for a voltage transfer ratio of the form [3]

where Y,, Y,, Y,, Y4 are RC admittances, and A , , A , are the conversion factors of the active elements.

The synthesis procedure considered requires that a real rational func- tion T ( s ) = N ( s ) / D ( s ) of order m be expanded by selecting a polynomial Q(s) constrained to have (m - 1) distinct negative real zeros :

and then decomposing N(s)/Q(s), D(s)/Q(s) into partial fractions such that

where Kli 20, KZi 2 0. The network elements are determined by identifying the terms in (3) with the corresponding RC admittances in (1):

‘ C l i G l i s m-l c s - c - z

A,C,iGliS

T(s) = le + G 1 o -t zl Clis + G l i i = k + l C,;s + G , ;

czns + G z o + z, C Z i s + G z i i = n + l C Z i s + G Z i ” C,,G, ,s n-l

‘ (4) A2C2iG~is

Schoeffler’s performance criterion [2] is defined as the sum of the mag- nitudes squared of the element sensitivities :

Manuscript received November 12, 1969

596 PROCEEDINGS OF THE IEEE, APRIL 1970

2.5

2.0

1.5 a n Q 1.0

,$

5

I -1

.5 1.0 1.5 2.0 2.5 3. .5 4.0 4.5 ALPHA I

Fig. 1. Sensitivity as a function of the zcrap of MI).

where

Applying this definition to the elements in (4) and substituting into (S),

where

j = 1 j = 1

By virtue of the choice of Q(s), the denominator of K,, is of degree (m - 1) inai,andifD(-ai)isofdegeem,then

K,, + to for ai = 0, ai = a j , a, + co. (9)

The same applies to Kli except for the possibility of

Kli = 0 for ai + a, (10)

when the degree of N ( - ai) is less than m. By applying these conditions of K l i and K,, to (7), it follows that

4 - w for ai = 0, ai = aj, ai + co. (11)

Within the region of 4 bounded by the values of a, in (1 l), Q is h i t e and continuous and must therefore have at least one minimum. Hence there exists a Q(s) for which Q is a minimum.

EXAMPLE Consider the third-order Buttenvorth transfer function:

N ( s ) 1 D(s) (s + l)(s2 + s + 1)

T(s) = - =

and Q(s) of the form

Q M = Ns + a d s + ad. (13)

To reduce the sensitivity, Cooper and Harbourt [ l ] suggest choosing al = 1 to cancel the real zero of D(s) and performing the optimum Horowitz decomposition [4] of the remaining quadratic factor, which yields a, = 1. This choice, however, is not permissible since the zeros of Q(s) must be

distinct. On the other hand, Schoeffler’s sensitivity criterion, when evalu- ated for Tu1) as shown in Fig. 1, has a minimum at a1 =2.20 and a2 =0.85.

In conclusion, (7) can easily be modified and used to find the minimum sensitivity of other networks [l], [ 5 ] , that are synthesized by the above procedure.

HERBERT M. AUMANN Dept. of Elec. Engrg. University of Wisconsin Madison, Wis. 53706

REFxRENm [I] R. E. Coop and C. 0. Harbour& “Sensitiviiy reduction in ampli6er-RC synthe&“

[2] J. D. scborf&r, +“Tke synthesis of minimum sensitivity netwo%” IEEE Tram. Cimrir Theory, vol. (TT-11, pp. 271-276, J u m 1964.

[3] W. F. Lovering, “Analog computer simulation of transfer h. IEEE

[41 I. M. Horoaitz, “Optimization of negative-impedance conversion methods of active (Comspondena), vol. 53, pp. 304-307, March 1%5.

RC synthesis,‘ IRE Trans. Circuir Theory, vol. (TT-6, pp. 29&303, September 1959. [51 L. s. Bobrow, “00 active RC synthesis using an operational ampiik,” Roe. IEEE

(CorreSp0adena). vol. 53, pp. 164.8-1649, October 1965.

ROC. m , w m t c D ) , V O ~ . 54, pp. 1 ~ 7 a 1 5 7 9 , ~ o v c m b e r 1%.

A New Look at Distributed RC Notch Filters Abstract-A new approach to the study of distributed RC notch

filters is presented. The method is intended to impart a clearer under- standing into the mechanism which causa the null to appear. It also leads to the introduction of different forms of notch filters than those previously considered.

In a recent paper, Huelsman [ l ] discussed the necessity of solving a transcendental equation of the form

tanh Jm = - tan ,,/= in order to determine the notch frequency of a distributed RC network em- ploying a lumped resistor. It is the purpose of this letter to indicate an alternative method for obtaining the null. The method introduced is also intended to impart a clearer insight into the mechanism which causes the null to appear, to be accomplished by looking at the notch filter through the parameters of the distributed circuit rather than the mathematics in- volved in obtaining the transfer function of the network.

As a first step, pi and tee equivalent networks for a uniform distributed line are introduced, as shown in Fig. 1. It is well known in circuit theory that a null occurs when either a short circuit in a shunt branch or an open circuit in a series branch is encountered. Thus, if an impedance Z , is placed in series with Z,, and 2, has the value -Zb, or if an admittance Y, is placed in parallel with Y2, and Y, = - Y,, then a null is obtained.

To determine the type of immittances, and their magnitudes, to be placed in series or in parallel with 2, and Y,, it is first necessary to examine the variations of Z , and Y, as a function of frequency. Setting p = j w and introducing the variable

the immittances Y,(x) and Z,(x) are found to be

Y,(x) = (x + j x ) csch (x + j x ) , RO

Z,(X) = - RO (x - j x )

csch (x + j x ) .

The above terms are generally complex. However, the real and imaginary parts, and thus the magnitude and phase, may readily be obtained for dif- ferent values of the variable x. The phase, however, is more informative, for herein lies the information on the type of immittances needed to produce the null. To see why this is so, recognize that what must be placed in series

Manuscript received September 25, 1969.