A simple method of analysing distributed RC notch-filters

A simple method of analysing distributed RC notch-filters by S. Ahmadt and R. Singhtt A new, simple method of analysing distributed parameter notch filters, is presented. The unique feature of this method is that the parameters of the circuit leading to the null conditions are clearly seen.

1. Introduction The modern trend in electronics towards the achievement of microminiaturisation and higher reliability in electronic systems together with the development of thin-film technology have resulted in the development of distributed RC (RC) circuits. The use of an RC transmission line as a low-pass or high-pass filter and as an attenuator has been discussed by many authors, L2. Its use in conjunction with a lumped capacitor as a phase-shift network has been studied by Ahmad et al. Application of the RC transmission line in conjunction with a lumped resistor as a notch filter has been studied by Kaufman 4, Kaufman and Garrett s, Ahmad and Singh 6,7and Stein 8. The analysis of the null ne_.ttworks of Fig lb and lc using the uniform or tapered R C transmission line of Fig la together with their transfer functions is tedious and lengthy because in this method the ratio of total distributed capacitance to the required lumped capacitance or the ratio of the required lumped resistance to the total distributed resistance to obtain optimum null output conditions is obtained by a trial and error method which is very time consuming and inconvenient.

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Notch filter using distributed RC transmission line in conjunction with lumped capacitor C

I Aligarh Muslim University, Aligarh (UPI, India t t Department of Electronics and Communication Engineering, University of Roorkee, Roorkee (UP), India 18

2. Null mechanism and theoretical analysis The impedances ofpi and tee lumped-equivalents of the E R C transmission line of Fig la shown in Fig 2a and 2b respectively are given below:

RT

RT

I0

A simple and straightforward method (giving a clearer insight into the mechanism which causes the null) for obtaining the optimum notch parameters of an exponentially tapered distributed RC (ERC) transmission line is discussed in this paper. The optimum notch parameters are achieved by analysing only one component of the pi or tee lumped-equivalents of the R C transmission line of Fig 1a as shown in Figs 2a and 2b respectively.

Fig lc

Notch filter using distributed RC transmission line in conjuction with lumped resistor R

where A, B, G, H and y are the matrix parameter functions of the ERC transmission line and are defined as"

MICROELECTRONICS JOURNALVol. 12 No. 1 © 1981 Mackintosh Publications Ltd., Luton.

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Fig 2a Pi lumped-equivalent of ERC transmission line of Fig la

Fig 2b Tee lumped-equivalent of ERC transmissionline of Fig la

A = O Cosech 0 . eD=B

3.

G = O Coth O + D

3.1

H = ( O Coth O - D ) em

Results and discussions

Notch filter with lumped capacitor

Figures 3a, 3b and 3c show the variations of the

y = RT/Ct O=[D2+j U0] u2

normalised resistive part

R

(=

RT

1

U0=c0Rocot~

CT

U=toRxCx=normalised frequency

of the normalised capacitive part - C

C~= .

phase ~bof

D

), the reciprocal

aR

- e -° for 0 < k < 0

Z2p

RT

(=ac) and the

of the circuit of Fig 2a against

C~= 1.0 for k=0, uniformly distributed D=

ke 2

=Taper parameter

k=Tapering constant ~e= 1.375 cm=length of the transmission line R0 and Co are the distributed resistance and distributed capacitance per unit length atthe input end of the ERC transmission line. RT and CT are total distributed resistance and total distributed capacitance respectively. The null will occur when either an open circuit between terminals 1 and 2 of Fig 2a or a short circuit between the terminals 0 and 3 of Fig 2b is achieved. Therefore to obtain optimum null conditions only the impedance Z2p or 7-,zrneed be analysed. Any impedance consisting of resistive and capacitive parts may be written as Z=R-

J

-k-T-g z2p

Zn-

Normalised impedances ~ and have been RT RT analysed. For k=0, 1 and 2 the variations of the normalised resistive part

the normalised capacitive part be -

CT

. If now a

C. lumped capacitor of value C, is connected in parallel with Z2p i.e. as shown in Fig lb, then the network will give an open circuit condition between terminals 1 and 2 at normalised frequency U,. This will result in null output.

3.2

Notch filter with lumped resistor

The variations of the normalised resistive part R/Rr and the reciprocal of the nomalised capacitive part CT/C and

toC

and when normalised with respect to total distributed resistance RT it cart be written as

R,

normalised frequency U for k=0, I and 2 respectively. It can be seen that the phase of the impedance varies between 0 and 90° in the range 0~
R

, the reciprocal of the

RT CT Z2p normalised capacitive part - and the phase o f - C RT against normalised frequency U are plotted in Fig 3,

Z2T

while the same results for ~ are shown in Fig4. RT

Z2T

the phase th of the impedance ~ of Fig 2b against RT normalised frequency U for k=0, 1 and 2 are shown in Figs 4a, 4b and 4c respectively. It can be seen that the impedance has a phase variation between - 9 0 ° and - 180° in the range 0~U~
zzr behaves as a purely RT

normalised frequency U,, ~

negative resistor with the capacitive part equal to zero and phase ~bequal to -180 °. Let this value of the normalised resistive part be -

Rn

. Now if a lumped

RT resistance equal to Rn is connected in series with Z,z.T,i.e. as shown in Fig lc, this will provide a short circuit across the shunt branch at normalised frequency U. and this will give a null output. 19

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P/OP,MALIZEO IN[QUINCY U ( : ~ R T C T ) '

Fig4

R Cr Variation ofnormalised resistive part - ~ r ' reciprocal ofnormalisedcapacitive part • C and phase d) of

normalised impedance

Zzr of Fig 2b with normalised frequency U for (a) k=0, uniformly distributed, (b) k= 1.0, (c) k=2.0 Rr 21

A simple method of analysing distributed RC notch-filters continued from page 21

Table 1 Notch parameters ac., tapering constant k

k

CT Cn =aCn

Rn RT

0 ----1 --+2

17.79853 21.00620 33.60461

0.0561840 0.04760477 0.0297578

--

1 aRn

CT Table 1 gives the ratios C. '

O~Rnand

U. with

OtR n

Un

17.7987 11.186635 21.0063 13.363755 33.6046 22.071355

output and a simple method of analysing notch filters to obtain the optimum null condition with any degree and type of taper. For a given RC transmission line, optimum notch parameters acn or aR, and U, can easily be calcu.__lated.The use of pi and tee lumped equivalents of an RC transmission line makes the analysis and study of other complicated networks fabricated with RC transmission line, in conjuction with other distributed, lumped and active elements easier and more straightforward.

R, R----~and the

normalised frequency U. at which null output is achieved when a lumped capacitor value C, or a lumped resistor of value Rn are connected as shown in Fig lb and lc respectively for k=0, --1, -+2. It is interesting to note that for a given value of taper coefficient k, the notch frequency U, and also the ratio of the total distributed to

1

[1] Hagar, C. K., "Network Design of Microcircuits", Electronics, 32, pp. 44-49, Sept. 4, 1959.

[2] Castro, P. S. and Happ, W. W., "Distributed Parameter [3]

thelumpedelement(C--~, r and R,RT)arethesame' whether the notch filter is fabricated with lumped capacitor Cn or lumped resistor R~. The behaviour of

z2p -

-

RT

and

Zrr RT

with frequency for k = - 1 and - 2 is the

I'4] [5]

same as for k= 1 and 2 except that there is a phase difference of 180° .

[6]

4.

[7]

Conclusion

The pi and tee lumped-equivalents of an RC transmission line give a clearer insight and a more straightforward approach to the mechanism causing null

22

References

[8]

Circuits and Microsystem Electronics", Proc. NEC (USA), 16, pp. 448-480, Oct. 1960. Ahmad, S., Pal, K., Singh, R., "Distributed-Lumped Phase Shift Networks", Electro-Technology, pp. 26-29, June 1979. Kaufman, W. M., "Theory of Some Monolithic Null Devices and Some Novel Circuits", Proc. IRE, 48, pp.!540-1545, 1960. Kaufman, W. M. and Garrett, S. J., "Tapered Distributed Filters", IRE Trans. on Circuit Theory, CT-9, pp. 329-336, Dec. 1962. Ahmad, S. and Singh, R., "A Composite Dielectric Thin-Film Notch Filter, Thin Solid Films, 67, pp. L63-L65, 1980. Ahmad, S., Singh,.R., "High-Quality Dielectric Film for Distributed RC Filters and Amorphous Semiconductors", Thin Solid Films, 74, pp. 165-171, 1980. Stein Jack, "A New Look at Distributed RC Notch Filters", Proc. IEEE, pp. 596-598, April 1970.