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, + a,. (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 + a,. (11)

Within the region of $J 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 RDC. 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 + jx ) , RO

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

csch (x + jx ) .

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.

PROCEEDINGS LETTERS 591

p'.

R, =total distributed resistance, C, =total distributed capacitance.

Fig. 1. (a) pi and @) tee models of the distributed RC network

Fig. 2. Phase angle of Y&) and Z,(x)

and in parallel with Z , and Y,, respectively, are real positive resistors, in- ductors, and/or capacitors. To produce the null, the forms of Z , and Y2 must be those of negative resistors, inductors, and/or capacitors. The phase angle of these quantities determines the type of immittance it corresponds to. A plot of the phase angle of Y2(x) and Z,(x) as a function of the nor- malized frequency variable x2 is given in Fig. 2.

Fig. 2 reveals a great deal of information. For example, it can be seen that for O s x 2 15.593 (obtained by computer to three significant decimal places), Z , has a phase angle between -90" and - 180". Y2 has an angle in this range from 0" to -90. Thus. Z , appears as a negative resistor in series with a negative inductor. Y, appears as a positive conductance in parallel with a negative capacitance. To create a null in the first case, a

TABLE I

0<x2<5.593 5.593

5.593<x2<15.418 15.418

15.418<xz<30.226 30.226

30.226 < x2 < 49.965 49.965

R, L R R, C c

-R, C -R -R, L

L

-G, C C G, C G G, L L

-G, L -G

< TC0

co / 19.12

0 . 5 5 6 Ro

1 0.005 Ro - (b)

Fig. 3. Circuits producing nulls using positive RC immittances outside the ranges indicated in Table I.

positive resistance and a positive inductor should be placed in series with Zb. The values of these components are determined from the magnitude and phase of Z,(x) at the desired frequency of the null. Similarly, to create a null using an admittance in parallel with Y,, a negative conductance and positive capacitance would be selected. Their values would be obtained from the magnitude and phase of Y,(x) at the desired frequency of the null. Continuing in this fashion, the values given in Table I would be obtained. This table shows that type of immittance to be placed in series or in parallel with Z , and Y,, respectively. For values of x 2 > 49.969 the pattern repeats.

It should now be clear how the immittances required to produce a null at the desired frequency are determined. The steps may be summarized as follows. Starting with the prescribed angular frequency o, and given the distributed parameters R , and C,, the normalized frequency variable x is obtained. This value of x , when substituted into the equations for Z(x) and Y(x) , will give the real and imaginary immittances of these functions. The element values necessary to produce the null are then obtained by taking the negative of these numbers.

In many applications inductors and negative resistances are to be avoided. This would Seem to preclude a wide range of frequencies for a given ROCo product. However, there is no need to restrict the circuit to the simple types previously considered. Elements may be placed at several terminals, as shown in Fig. 3, so as to achieve nulls using only resistors and capacitors at frequencies other than those indicated in Table I. Fig. 3(a) shows a circuit producing a null at the normalized frequency x' = 2.25, while Fig. 3(b) has a null at x' = 25.0. Both these circuits were obtained from a knowledge of the values of Z, and Z , at these frequencies and a judicious choice of element values for the lumped immittances. Thus, a distributed RC notch filter employing lumped positive RC elements is not

598

limited in its range of null frequencies to the values indicated in Table I, but may have an extended range of nulls with careful selection of the place- ment and value of its lumped elements.

JACK STElN Dept. of Elec. Engrg. Pennsylvania State University King of Prussia, Pa. 19406

REFERENCE5

[l] L. A. Huelsman, “The distributed-lwnped-active network,” I€€€ Spectrum, vol. 6,

[21 W. M. Kaufman, “Theory of a monolithic null device and some novel circuits,” Proc.

[31 W. D. Fuller and P. S. Castro, “A microsystems bandpass amplifier,” R o c . Narl. € k c -

[4] R. W. Wyndrum, Jr., “Distributed RC notch networks,” Proc. IRE (Correspondence),

[5 ] M. J. Gay, “The design of feedback tuned amplifiers using distributed bridge T net-

[6] K. L. Su, “RC filters with staggered notch frequencies,” Proc. I€€€ (Correspondence)

[7] S. C. Dutta Roy and B. A. Shenoi, “Notch networks using distributed RC elements,’’

pp. 51-58, August 1%9.

IRE, vol. 48, pp. 1540-1545, September 1 9 6 0 .

rronics Conx. vol. 16, pp. 13%151, 1960

vol. 51, pp. 376375, February 1%5.

works,” Microelectronics and Rclirrbiliry, vol. 3, pp. 93-107, May 1964.

vol. 54, pp. 1199-1200, September 1966.

R o c . IEEE(Correspondence), vol. 54 , pp. 122s1221, September 1966.

Modulation Noise of a 6328 A He-Ne Gas Laser Abstract-This letter describes a study of the noise of a 6328 di

Ne-He gas laser which results from the fluctuation in the discharge current. The characteristics of the laser intensity modulation caused by the discharge current are determined.

It has been pointed out [l] that the noise of a dc excited gas laser has a strong correlation with that of the discharge current. In this letter, the noise is treated from the standpoint of the laser intensity modulation caused by the fluctuation in the discharge current.

A discharge current modulation experiment, using a sine wave signal at various frequencies, has been performed for the purpose of clarifying the modulation process. In this experiment the relation between the relative phase and amplitude of the laser signal output and the discharge current has been studied. The results are indicated by the curves with the dotted line in Figs. 1 and 2, where d is the laser signal output, i is the signal com- ponent of the discharge current, and the numbers in the figures are the corresponding frequencies. In the case of a dc discharge current of 20 m.4, a sharp resonance at nearly 100 kHz is observed. ‘As will be made clear later, this resonance results from the saturation effect. For the resonances near 500 kHq a moving striation is responsible.

The relation between d and i is given theoretically by solving the rate equation. By taking into consideration the saturation effect which results from a collision of the second kind [2], the following relation is obtained :

w PO i S,,Uw + r)Uw + S,) _ - -

where

Po= excitation intensity S21 =spontaneous transition rate from Ne 2 p to Ne 1s

r = half-linewidth of the optical cavity S , = phenomenological constant which determines the relation be-

tween the sinusoidal component of the free electron density and that of the discharge current

S , , = spontaneous transition rate from Ne 3s to Ne 2 p S,,=spontaneous transition rate from Ne 1s to the ground state S4, =resonant transition rate from He 2‘s* to Ne 3s

a(io) = saturation parameter.

Manuscript received December 9, 1969.

PROCEEDINGS OF THE IEEE, APRIL 1970

t ( G / t ,

Fig. 1. Re (G/O -Im (GlO diagram for a dc discharge current of 10 m A . The theoretical (solid) curve represents the parameter values of S,,/Zn =20 kHz and S,,/2n =200 kHz.

I Fig. 2. Re (G/O -Im (* /I ) diagram for a dc discharge current of 20 m A . The theoretical

curve represents the parameter values of S,,/2n =80 kHz and S4,/2n =300 kHz.

Here a(io) is strongly dependent on the dc discharge current io, and is given by comparing the dc solution of the rate equation and the measured static characteristics of the laser output.

In Figs. 1 and 2, the computed results are shown by the solid curves. The values of parameters S,, and S,, are those obtained from [3]. The theoretical curves coincide with the experimental curves. Therefore, the uncertain parameters in the laser action S4, and S1, can be obtained. The resonance near 100 kHz results from the term a(io) in formula (1). Since a(io) is the saturation term, the resonance is caused by the saturation effect.

The noise level of the laser beam is calculated by the absolute value of G / i and the measured noise in the discharge current. In Fig. 3, the laser beam noise thus calculated is shown by curve C3. The noise level is indi- cated by the equivalent modulation index m, = linlz/(+i$B), where i, is the ac component of the photocurrent, io is the dc photocurrent, and B is the bandwidth of the receiver. In the figure, curves C1 and C2 indicate the laser beam noise in the free running state and in the state of forced mode locking, respectively. Curves C2 and C3 coincide, and this fact means that in the state of m d e locking (in this case no mode interaction noise exists between the longitudinal modes) the residual laser beam noise is mainly composed of the discharge current modulation noise (DCMN).

It has been found, as a new phenomenon, that the locking of the moving striation by the discharge current modulation signal reduces the low- frequency component of the discharge current noise by about a factor of 10 dB. It is therefore expected that the corresponding DCMN will be re- duced by the same amount. This has been confirmed by measuring the laser beam noise in the state of forced mode locking and, at the same time, the locked moving striation (curve C4 in Fig. 3). Curve C4 coincides with curve C5 which shows the calculated DCMN in the same state. The residual noise indicated by C4 will be composed of the mode interaction noise between the transverse modes (the experiment has been carried out in the