SELF

46

S.B. Karavashkin and O.N. Karavashkina

Fig. 9. The schematic diagram of the experimental setup to measure the amplitude-frequency and phase-frequency characteristics of the LC-type ladder filter. The setup 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.

To check up the obtained calculation regularities, we have carried out the experiment whose circuit is shown in Fig. 9. Its principal aim was to investigate the amplitude-frequency and phase-frequency characteristics of the input impedance at the ladder filter pass band with the constant input current amplitude I (t). The introduced condition of the input current (not voltage) constancy was caused, on one hand, by the dependence of solutions (26) on the input current amplitude. And on the other hand, when experimenting, it is convenient to compare voltages, not currents. Moreover, the phase characteristics measurement between the voltages excludes a number of essential systematic errors.

Basing on the above arguments, we worked out the experimental scheme. One can see from Fig. 9 that the phase of the filter input impedance was measured as the difference between the voltage phases at the source output and at the ladder filter input. For it between the source output and filter input we inserted quite large resistor R₁ stabilising the phase of the source output voltage. The second task of this resistor was to bring the experimental conditions at the filter input nearer to the calculated conditions, according to which the filter input was loaded (see Fig. 5).

The pattern of input impedance variation with frequency was investigated by the voltage variation at the filter input being equivalent to the constant input current amplitude. The experimental data were picked off the screen of oscillograph having a large input resistance (> 1 MOhm), small input capacitance (~ 40 F) and the resolving ability in frequency more than 10 MHz. To reduce the measurement error, we used the one-ray oscillograph with the measurement channels switching at its input (the switch SA1). To reduce the standard error of the oscillograph (about 5%) when the phase characteristics measuring, we used the maximal sweep with the external synchronisation, and when investigating the amplitude characteristics – the maximal amplification of the signal and measuring the oscillation swing.

Fig. 10. The experimental 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.

The results – the amplitude-frequency (a) and phase-frequency (b) characteristics – are presented in Fig. 10. This first has the form of regularity of the acting voltage amplitude at the filter input with respect to frequency and input impedance. And the second has the form of displacement phase in radians, also with respect to frequency and input impedance. Comparing them with the calculation diagrams in Fig. 6, we see a full accord of the patterns accurate to the experimental error. The only difference is in the amplitude values of the resonance peaks when comparing the experimental and calculation data. The calculated amplitude is some higher. This is caused by the finiteness of resistance R₁. Its value is well more than the filter impedance R₀ and is quite large to ensure a good accord of the phase characteristics, but is not quite sufficient, the amplitude characteristics to be in full accord with the calculation circuit. Because, as one can determine by the data in Fig. 6, it is comparable with the maximal value of the input impedance amplitude in the resonance peaks domain. Basically, the presented method allows getting over this disparity too. In this order we should complicate a little the initial mechanical prototype model and consider an elastic line having heterogeneities not only at its output but also at the input. But since our main aim here is to investigate just the load impedance influence on the vibration pattern, such refinement is out of frames of this paper. While within these frames we can surely state that the presented coincidence of the calculated and experimental results demonstrates quite conclusively that ladder filters cannot be modelled by a simple assemblage of elementary two-ports in general case. Their amplitude and phase characteristics have the complex resonance form well described by the combined method based on the exact analytical solutions for mechanical elastic lines as the analogue and the dynamical electromechanical analogy DEMA. With it the ladder filter structure can be much complicated, if necessary, even in frames of the problem solved here, since one can think under the impedances ₁ and ₂ any complex impedance, in that number the input impedance of the filter branches. In all these cases the DEMA relationship retain true, because they take into account the similarity of just the dynamical processes in mechanical lines and ladder filters.