SELF |
44 |
S.B. Karavashkin and O.N. Karavashkina |
|
Actually,
for this circuit the impedances 1 and 2 take the form |
(31) |
In
accord with (9) (11) and (31), the parameter el becomes a complex value |
(32) |
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 Rin can be easy determined from (26): |
(33) |
where |
The
typical form of the amplitude (a) and phase (b) of the input impedance with respect to
frequency and load Rload
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 Rload < R0 and Rload >
R0. It is caused by the fact that at small Rload the
filter behaves as the shorted and consisting of (n + 1) calculation nodes, and at
large Rload 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 Rload 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 Rload from
zero to R0, the peak amplitudes fall, and at Rload = 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 Rload
» 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 Rin at different active load values Rload 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; rL = 10 Ohm; R1 = 20 kOhm; R2 = 33 kOhm; Rload = 0, 51, 102, 158, 358, 558, 758, 958 Ohm. |
Contents: / 35
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