V.2 No 1

45

Mismatched ladder filters

Fig. 7. The combined plot of the amlitude-frequency and phase-frequency characteristics of the low-frequency RLC filter consisting of 5 sections and loaded onto the imedance R₀=159,15 Ohm

Especially note that at the band near the cutoff frequency the transformations are practically absent. It evidences that it is basically impossible to choose the load impedance so that it provided the plate of the amplitude-frequency characteristic at the filter pass band.

Furthermore, we see from the plots that at the pass band the input impedance phase not only is not zero, which would correspond to the real value R_in , but it multiply alternates the sign, taking by turns the inductive or capacitive pattern. With it the clear correlation between the amplitude and phase resonance peaks is absent too. To demonstrate it, we show in Fig. 7 the combined plot of the input impedance amplitude and phase for the case R_load = R₀. This evidences that in ladder filters one can obtain plate neither an amplitude-frequency nor a phase-frequency characteristic of the filter input impedance, at no active load value. Even if at some load impedance a definite smoothing of the characteristics were possible in the low-frequency domain of the pass band, near the cutoff frequency the regularities have resonance pattern and practically do not transform with the load impedance variation.

The same we see in case of complex loading. As an example, in Fig. 8 we show the amplitude-frequency and phase-frequency characteristics over the r_L LC filter investigated before, when it is loaded by the sequentially connected active R_load = R₀ resistance and capacitive C_load impedance. In the plots we see alike transformations of the regularities with the C_load variation, only the resonance peaks amplitude monotonously grows with the capacitance falling. Now some value of capacitance approximately equal to 1 F at the chosen circuit parameters serves the transition border. In the region of this capacitive load, in the low-frequency domain of the pass band, the curves displace, and the new resonance peak merges with the first peak of the previous characteristic. Thus, with the falling load capacitance, the quantity of resonance peaks does not change, and even their location along the frequency axis retains out of the transition domain. And again, all described transformations occur in the low- and middle-frequency domains of the pass band, with the practically flat pattern at the cutoff frequency.

Fig. 8. The calculated amplitude-frequency (a) and phase-frequency (b) characteristics of the input resistance R_in at different load capacitance values C_load and constant value of the input current amplitude 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 = 158 Ohm.