SELF

44

S.B. Karavashkin and O.N. Karavashkina

Actually, for this circuit the impedances ₁ and ₂ take the form

In accord with (9) – (11) and (31), the parameter _el becomes a complex value

where  

With it the parameter _el becomes a complex function of the frequency and circuit parameters, and the relationship between _el and the unity losses its sense. As we said before, in this case (26) is true for all the range, while (27) and (28) loss their sense too. We see that to reveal the complex pattern of _el, we need not consider complex ladder filters. It is sufficient for it even in LC filters simply to note the real parameters of the elements, such as inductive parasitic impedance, capacitive parasitic impedance, and sometimes the connecting wire resistance. In all these cases we can apply (26) to all the range. And if the parasitic parameters influence is small, this expression will simply describe the regularity, practically identical to (26) – (28) for an ideal filter. At the same time, a clear differentiation of the pass and stop bands and the calculation simplicity are the merits of the system (26) – (28). For the case of complex _el the calculation essentially complicates when real and imaginary parts of (26) separating, and it is difficult to determine the boundary frequency from the solution itself. In the studied specific problem, the input impedance R_in can be easy determined from (26):

where  

The typical form of the amplitude (a) and phase (b) of the input impedance with respect to frequency and load R_load is shown in Fig. 6. First of all we see that for the finite ladder filter the amplitude and phase of the input impedance has the resonance form. And the number of peaks is different at R_load < R₀ and R_load > R₀. It is caused by the fact that at small R_load the filter behaves as the shorted and consisting of (n + 1) calculation nodes, and at large R_load the input resistance corresponds to an unloaded filter consisting of n nodes, which is in full correspondence with the above analysis of the solutions (26) – (28) transformation. With the growing load impedance the phase characteristic also transforms, up to changing the sign at definite bands. This transformation of the amplitude and phase characteristics takes place at a quite narrow range, where the values R_load are lower or higher than the impedance R₀ that is equal in this calculation to 159,15 Ohm (the plots at this value of the load resistance are shown in red). One can see that with growing R_load from zero to R₀, the peak amplitudes fall, and at R_load = R₀ reach some minimum. With the following increase of the impedance, the resonance peaks at the low and medium bands displace, the first resonance peak vanishes, and already at R_load » 600 Ohm (for the present circuit parameters) the resonance frequencies location stabilises at the new number of the resonance frequencies. Further only the resonance peaks themselves grow. And this last concerns to a large extent to the amplitude-frequency characteristic. The resonance peaks on the phase-frequency characteristic reach fast the saturation amplitude not exceeding /2 for the first peak that determines the limits of the input impedance phase variation.

Fig. 6. The calculated amplitude-frequency (a) and phase-frequency (b) characteristics of the input resistance R_in at different active load values R_load  and constant input current value I (t)  with respect to frequency. The investigated filter parameters: L = 12,6 mH; C = 0,5 mF; R₀ = 159,15 Ohm; r_L = 10 Ohm; R₁ = 20 kOhm; R₂ = 33 kOhm; R_load = 0, 51, 102, 158, 358, 558, 758, 958 Ohm.