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3 3.1 Theory The lowpass model
(12)
Since b = coC, the bandwidth / c is given by
/c
.,2
T
1/2
=
g
2TTC
a
-(a2 + 1 -302)
(13)
In the following theory it is assumed that the X/4 line is a quarter wavelength at all frequencies, that is, it is an ideal X/4 line. Referring again to Fig. 1 and assuming that each load is a parallel CR lowpass, such as network shown in Fig. 2a, then eqn. 3 gives
|p o l)
(1-IPol)
(14)
where b = coc. From eqn. 5 we get
Y2
l — IPOI
P
n
2
-
go+g
n (,,
+
Y2
)
+
where So is the v.s.w.r. when b is zero, so
IV
+
V +ft2[1
I'
g2 + b'\
Y2
Yi
a = (l+(32)S0
Eqn. 13 expresses the bandwidth in terms of the coefficients, a and (3, but a and j3 are themselves related, in terms of |pol, by eqn. 9. For a given |pol there are two solutions for a and j3 depending on whether a > (1 + /32) or +j3 2 )then a = Now 1 + lPol
Consider the circuit shown in Fig. 1. If a passive load admittance YL is connected to the end of a A/4 transmission line of characteristic admittance Yo, it is easily shown that the input admittance of the line is
Similarly, ii
(15)
g2+b2
(7)
(16) It can be shown that a Chebyshev-type frequency response is obtained when a < ( l 4-132) and a Butterworth response is obtained when a > (1 + ft2). As the derivation of eqn. 13 assumed a Chebyshev response, a can be eliminated by substituting eqn. 16 into it, which gives
ห้องสมุดไป่ตู้Yl
" YL
(1)
Paper T32 2 M, first received 16th October 1978 and in revised form 16th January 1979 Mr. Aitchison is, and Mr. Chapman was formerly, with the Electronics Department, Chelsea College, Pulton Place, London SW6 5PR, England Mr. Chapman is now with the Royal Signals and Radar Establishment, St Andrew's Road, Malvern, Worcs., England
|2 _
+ ft2^2 U 2 + 2 ( 1 - | 3 2 ) -
go - Yin go + Yit
2
= 0 (11)
The third term of eqn. 11 is zero, hence (5) b2 =
8
so, clearly, by varying the value of Yo the degree of match of the loads to the source can be changed. The values of g0 and Yo for minimum mismatch and maximum bandwidth for different types of load and circuit configuration are analysed in the next Section.
If yL = g + jb then
(2)
One of the most common problems encountered in electronic circuit design is that of providing a broadband impedance match to a resistive source of a complex load impedance. Conventional impedance-matching techniques at low frequencies involve ladder networks of lumped inductances and capacitances, while at microwave frequencies, resort is frequently made to distributed elements in the form of lengths of open-circuit or short-circuit transmission lines which approximate to lumped inductances and capacitances. Usually matching networks are designed to have a specific frequency response in the passband1 such as the Chebyshev or Butterworth responses. Occasionally techniques are used which simply reduce the mismatch without intending to achieve a particular frequency response ? A technique which has been used with considerable success in increasing the gain-bandwidth product of parametric amplifiers3'4 and Impatt amplifiers5 and increasing the tuning range of Gunn oscillators6'7 is active reactance compensation. Active reactance compensation takes advantage of the impedance-inverting property of a quarter-wavelength transmission line. By connecting two identical devices together with such a line the inverted impedance of one device compensates the impedance of the other by reducing the total circuit reactances. So far the technique has only been applied to the problem of broadbanding active loads, that is devices with a negative-resistance characteristic. The object of this paper is to consider its application to either increasing the bandwidth for a given mismatch or reducing the mismatch over a given band for passive loads, that is loads with a positiveresistance characteristic.
Fig. 1 Basic shunt compensation circuit 43 0308-6976/79/020043 + OS $01-50/0
MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2
at the expense of not cancelling the susceptances. The reflection coefficient of the circuit shown in Fig. 1 is given by