3.1. Numerical Simulation Issues: Simulation Results
As a result of formal limitations, an analytical solution to nonlinear state equations is not generally possible, so system operation was analyzed using numerical methods. Conservative elements making up the system were replaced with associated models, which enabled transforming dynamic systems in corresponding DC systems for a single sampling step. Equations formulated using a modified balance nodal method, given in general form
where
is the system conductance matrix,
is the sought nodal potential vector, and
is the excitation (current) vector, were solved by combining the lower–upper decomposition (LU) method with the sparse matrix technique, which avoided the consideration of trivial cases [
23].
Selected results from the computer simulation are shown in
Figure 6,
Figure 7 and
Figure 8.
Figure 6 and
Figure 7 show the amplitude-phase characteristics of the amplifier for the four configurations of the transistor pairs used: IRFP140/IRFP9140 and IRFP240/IRFP9240. These four transistor pairs were used to show that the DC level value at output does not depend on used semiconductor active elements.
Figure 6 illustrates the results obtained from the simulation for the designed circuit with an OP1A preamplifier attached, while
Figure 7 depicts it without the preamplifier. In both cases, the amplifier is designed to have a uniform signal gain in the frequency band in question (up to 10
Hz): approximately 28 dB for the system with the pre-processing stage attached (
Figure 6a) and approximately 5 dB for the system without the OP1A amplifier (
Figure 7a). The circuit inverts the phase of the input signal (
Figure 6b and
Figure 7b) and has favorable phase characteristics (no undesirable shifts). Since the OP1A preamplifier does not cause any significant changes in the circuit’s operation (apart from the gain value), it was excluded from further considerations.
Figure 8 shows the results of the simulation for the DC component voltage
on the system load for the tested pairs of transistors, with input capacitors C7 and C8 shorted to ground. The voltage
measured at the output of the system was very low (on the order of tens of pV) and did not undergo rapid changes over time. The mean values of these voltages and their standard deviations for each transistor pair are shown in
Table 2.
3.2. Real-World Device Tests
The physically constructed LF amplifier was tested under laboratory conditions. The tests were carried out using an AudioPrecision AP x525 multifunctional measuring and testing system, which is designed to assess the operation of signal processing equipment in the LF domain.
Figure 9 and
Figure 10 present the measurements of basic device parameters. The results are presented for individual pairs of power transistors, which allows us to demonstrate the universality of the solution and its resistance to changing operating or production conditions.
Figure 9 shows the amplitude–phase characteristics of the amplifier for the four configurations of transistor pairs used—IRFP140/IRFP9140 and IRFP240/IRFP9240—for an actual circuit with the preamplifier stage omitted. The circuit shows minimal gain differences (approximately 0.15 dB) for individual pairs of power transistors (
Figure 9a). This is due to their non-uniform internal resistances, which translates directly into differences in the resting currents of the MOSFETs (in the case of components from different manufacturers or production batches, the matched pairs may show statistically insignificant gain differences). The circuit inverts the phase of the input signal (
Figure 9b) and exhibits favorable phase characteristics, while in the case in question, an influence of internal parasitic capacitances of BJT and MOSFET transistors was observed. This becomes apparent in the faster phase drops for the extreme upper frequencies of the frequency response.
Figure 10 shows the following: the frequency response of the amplifier for an output power of 1 W at load resistance R0 (
Figure 10a) and the distortion contributed by the amplifier for all harmonic frequencies (
Figure 10b). The lowest values of the DPR coefficient (below 0.1% in the whole passband under study) were recorded for the pairs 240/9140 and 240/9240.
The measurements of voltage
on a load resistance of 8
for the tested pairs of IRFP transistors are presented in
Figure 11,
Figure 12,
Figure 13,
Figure 14 and
Figure 15. To allow these results to be correlated to other publications, we selected a typical value for load resistance R0, which is commonly used in analogue audio applications. It should be noted that for the intended application of the circuit, the value of R0 is different, but for obvious reasons, it will not be disclosed here. The tests were designed to verify (a) the ability of the amplifier to achieve the assumed values of the signal parameters (cf.
Table 1) and (b) the amplifier’s self-balancing capability.
The measurement of
was first performed with input capacitors C7 and C8 shorted to ground, i.e., with no 1/2
signal on the BJT transistors.
Figure 11 presents sample 100 s waveforms of the voltage measured at R0—for each pair, its values are within ±100
V. The mean values of the voltages and their standard deviations for each transistor pair are shown in
Table 3.
Figure 12,
Figure 13 and
Figure 14 show the measurements of
at R0 during active operation of the amplifier for an input signal frequency of 1 kHz—for the output power at an R0 of 1, 10, and 15 W, respectively. The mean values of the voltages and their standard deviations for these cases are shown in
Table 4.
The higher values of in comparison with those from the simulation studies can be explained, among other things, by the IRFP transistors not being paired in terms of electrical parameters but only being matched in physical complementary pairs. IRFP transistors are voltage-controlled transistors. The different gate opening voltages of IRFP transistors translate directly into differences in the pair’s resting currents. Consequently, the complementary pair is unable to develop the ideal symmetrical voltage to power itself. In the case in question, the two resistances controlled by the gate-opening voltage cyclically seek to deposit exactly half of the asymmetrical voltage on each other, which is only possible in an ideal scenario. Furthermore, the processing path is affected by interference and component noise, which further increases the value of at R0. However, it should be noted that in none of the cases tested did the modulus of voltage exceed 2.5 mV.
An important assumed property of the LF amplifier being tested is its self-balancing capability, which is defined as the ability to (a) bring the value of at R0 to a minimum and to (b) maintain the minimum value of during changes in the DC operating conditions of the system caused by changes in supply voltage. Sudden changes in supply voltage result in changes in the resting currents of the BJT and MOSFET transistors within the circuit. In turn, shifts in operating points lead to significant temperature gradients within the structures of nonlinear components.
The self-balancing capability of the low-frequency amplifier was verified according to the following procedure. A sinusoidal signal with a frequency of 1 kHz was applied to the input of the system under test loads with a resistance of 8 . At time t = 0, the system was supplied with a nominal voltage of 70 V. The supply voltage was increased in increments of 1 V at 50 s intervals until 80 V was reached, which occurred around 550 s into the test. Then, also at 50 s intervals, the voltage was stepped down by 1 V until a nominal value of 70 V was reached, which occurred at 1000 s. The supply voltage spikes affected both the circuit with BJT transistors and the power follower with MOSFET transistors.
A diagram of the test is shown in
Figure 15. The system shows a tendency toward a minimum value of
(and therefore, the desirable self-balancing capability) for all IRFP transistor pairs tested. In particular, regardless of the configuration, the system in question maintained
as low as possible despite sudden changes in the supply voltage. No undesirable increases in the value of
associated with temperature changes in the internal structures of the active elements were observed. Abrupt changes of the supply voltage were accompanied by momentary changes in the value of
at load resistances in the range of ±1.4 mV, which is easily noticeable on the waveforms over time presented in
Figure 15. However, it should be emphasized that between the successive step changes for all the IRFP transistor pairs tested, in the range of supply voltage changes from 70 to 80 V (and consequently for resting current changes ranging from 0.33 to 1 A) and with an increase in operating temperature in the range of 38 to 65
C, the circuit spontaneously brought the value of
to within ±200
V. After the cessation of the step changes of supply voltage (at around 1100 s into the test), the value of
reached that recorded at the beginning of the test—for all pairs of IRFP transistors tested and at an operating temperature of approximately 38
C.