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Home Electric Networks Filters Transmitted Bands from Propagation Constant  


Transmitted Bands from Propagation ConstantAs shown by equation 58, the propagation constant for a T section such as Fig. 24 is γ = 2 sinh^{1} (0.5*sqrt(Z_{1}/Z_{2})). Since the elements of a filter are assumed to be without resistance and to cause no energy losses, Z_{1} and Z_{2} must be pure reactances. Accordingly, the ratio Z_{1}/Z_{2} 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 Z_{1} and Z_{2} are both inductors with negligible resistance, the ratio Z_{1}/Z_{2} will have a zero angle; if Z_{1} is inductive and Z_{2 }capacitive, then Z_{1}/Z_{2} will have an angle of 180°. The effects of these different ratios of Z_{1}/Z_{2} on the propagation constant will be considered now.^{5} Case I. When Z_{1}/Z_{2} is positive. The propagation constant y is composed of a real and an imaginary term; that is, γ = α + jβ. When Z_{1}/Z_{2} 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 Z_{1}/Z_{2}. Case II. When Z_{1}/Z_{2} is negative and less than 4. If the ratio Z_{1}/Z_{2} has an angle of 180° it is negative in sign; and, therefore, 0.5*sqrt(Z_{1}/Z_{2}) of equation 58 has an angle of 90° and both 0.5*sqrt(Z_{1}/Z_{2})_{ }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(Z_{1}/Z_{2}) 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
therefore,
If Z_{1}/Z_{2} is greater than 4, then 0.5*(sqrt(Z_{1}/Z_{2})) 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 Z_{1}/Z_{2} is negative and less than 4 in magnitude. Case III. When Z_{1}/Z_{2} is more negative than 4. For values of Z_{1}/Z_{2} 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 Z_{1}/Z_{2} greater than 4,_{ }the filter will offer attenuation. From the foregoing, it appears that a nondissipative recurrent structure of the type shown in Fig. 4 having series impedances Z_{1} and shunt impedances Z_{2} will pass readily only those signals of frequencies such that the ratio Z_{1}/Z_{2} will lie between zero and 4. The values of Z_{1} and Z_{2} depend on the frequency because in filters Z_{1} and Z_{2} are inductors and capacitors.


Home Electric Networks Filters Transmitted Bands from Propagation Constant 