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Transmitted Bands from Propagation Constant

As shown by equation 58, the propagation constant for a T section such as Fig. 24 is γ = 2 sinh-1 (0.5*sqrt(Z1/Z2)). Since the elements of a filter are assumed to be without resistance and to cause no energy losses, Z1 and Z2 must be pure reactances. Accordingly, the ratio Z1/Z2 must have an angle that is either 0 or 180°; that is, the ratio must be a real positive quantity, or a real negative quantity, but cannot be complex. For instance, if Z1 and Z2 are both inductors with negligible resistance, the ratio Z1/Z2 will have a zero angle; if Z1 is inductive and Z2 capacitive, then Z1/Z2 will have an angle of 180°. The effects of these different ratios of Z1/Z2 on the propagation constant will be considered now.5

Case I. When Z1/Z2 is positive. The propagation constant y is composed of a real and an imaginary term; that is, γ = α + jβ. When Z1/Z2 of equation 58 gives a zero angle, y is a real number since the jβ part becomes zero. There will accordingly be undesired attenuation in the filter for all positive values of Z1/Z2.

Case II. When Z1/Z2 is negative and less than -4. If the ratio Z1/Z2 has an angle of 180° it is negative in sign; and, therefore, 0.5*sqrt(Z1/Z2) of equation 58 has an angle of 90° and both 0.5*sqrt(Z1/Z2) and γ are pure imaginaries. Equation 58 can be written

From hyperbolic trigonometry, sinh (x + jy) = sinh x cos y + j cosh x sin y. Thus,

Since it was previously shown that 0.5*sqrt(Z1/Z2) and γ are pure imaginaries for the case being considered, then the real part of equation 62, that is, sinh α/2 cosβ/2, must be zero. For this to be true, either sinh α/2 or cosβ/2 must equal zero. If sinh α/2 = 0, then cosh α/2 = 1, because of the numerical relations between these two hyperbolic functions. When these relations exist, equation 62 becomes


If Z1/Z2 is greater than -4, then 0.5*(sqrt(-Z1/Z2)) would exceed 1, and equation 63 cannot hold because the sine of an angle cannot exceed unity. Thus, from equation 64 a filter will transmit without attenuation when Z1/Z2 is negative and less than -4 in magnitude.

Case III. When Z1/Z2 is more negative than -4. For values of Z1/Z2 more negative than -4, the real part sinh α/2 cosβ/2 of equation 62 can be made zero by cos β/2 = 0. At the values cos β/2 = 0, sin β/2 = ±1. Thus the angle β/2 must be some odd multiple of 90° to give these zero and unity values. Expressed in terms of radians (90° = π/2 radians),

where K is any whole number. This relation must hold to give cosβ/2 = 0 values.

When cos β/2 = 0 and sinβ/2 = ±1 are placed in equation 62, this becomes

From this relation

Then, from equations 65 and 67, the propagation constant becomes

This relation shows that, for negative values of Z1/Z2 greater than -4, the filter will offer attenuation.

From the foregoing, it appears that a non-dissipative recurrent structure of the type shown in Fig. 4 having series impedances Z1 and shunt impedances Z2 will pass readily only those signals of frequencies such that the ratio Z1/Z2 will lie between zero and -4. The values of Z1 and Z2 depend on the frequency because in filters Z1 and Z2 are inductors and capacitors.

Last Update: 2011-05-18