2021-08-10
Calculation of a parametric generator of the second kind.
Parametric inductance
Parametric inductance
P_{R } =
P_{1 } =
P_{2 } =
K_{η2 } =
Voltage G_{1}
Change inductance L
Load current R_{n}
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The purpose of this calculator is to provide a visual representation of transient and stationary processes resulting from parametric generation of electrical energy, and also, - obtaining some initial parameters of a real device: the shape and frequency of oscillations, phase shift between generators, inductance and Q-factor of the coil, load resistance. It is assumed that the influence of the first generator (G_{1}) on the second (G_{2}), and the second on the first, is minimal and is not taken into account here (see figure) ... This property is typical for parametric devices of the second kind. In this case, such a device consists of two generators. The first one works in a series circuit consisting of the generator itself, an inductor L and an active resistance R_{n} (or load). The inductance L is changed by the second generator, which creates conditions for a parametric of the second kind, in which the inductor can become a negative resistance, and therefore a generator of additional electrical power. A full description of the mathematical model and the principle of operation of such a scheme are presented in this work.
To generate energy in the circuit, generator G_{1 } spends active power P_{1 }, which is shown in the results of calculations, above the graph. The generator G_{2 } also spends a certain power P_{2 } to change the inductance, the value of which is still unknown to us and we can assume or measure its value in a real device. It must be entered into a special field of the second generator: Generator power, W . Together with the received power in the load P_{R }, these values are substituted into the calculation of the balance of capacities and growth efficiency of the second kind: K_{η2} = P_{R}/(P_{1} + P_{2}). If its value exceeds one, then for the researcher it means obtaining the initial parameters for the development of his real device. This parameter is also shown in the calculation results, above the graph.
Input parameters
Below is a description of the parameters entered in the fields of this calculator, by section.
Frequency, Hz
Generator frequency G_{1 } in Hertz.
Generator frequency G_{1 } in Hertz.
Amplitude, V
The amplitude value of the voltage at the generator output G_{1 }. The minimum reading is 10 volts.
The amplitude value of the voltage at the generator output G_{1 }. The minimum reading is 10 volts.
Waveform
Waveform at the output of generator G_{1 }. The voltage and waveform of this generator is shown in the graph in green.
Waveform at the output of generator G_{1 }. The voltage and waveform of this generator is shown in the graph in green.
Pulse duty cycle,%
This field works only with rectangular pulses and is responsible for the relative width that the pulse fills in comparison with the entire period of the oscillation. In the reference literature [1], this is the "D" parameter.
This field works only with rectangular pulses and is responsible for the relative width that the pulse fills in comparison with the entire period of the oscillation. In the reference literature [1], this is the "D" parameter.
Initial inductance, mH
Inductance of coil L without generator G_{2 }. The parameter is entered in millihenry.
Inductance of coil L without generator G_{2 }. The parameter is entered in millihenry.
Active resistance, Ohm
The diagram does not show the active resistance of the coil wire L, which can be either large or small, depending on the parameters of the winding conductor. Also, at relatively high frequencies, the skin effect [2] can affect, which increases this resistance. This parameter affects the efficiency of the entire device and, ideally, should be as low as possible. We must not forget that the resistance entered in the calculator will need to be obtained in a real coil.
The diagram does not show the active resistance of the coil wire L, which can be either large or small, depending on the parameters of the winding conductor. Also, at relatively high frequencies, the skin effect [2] can affect, which increases this resistance. This parameter affects the efficiency of the entire device and, ideally, should be as low as possible. We must not forget that the resistance entered in the calculator will need to be obtained in a real coil.
The minimum value is 0.001 ohm.
Active resistance, Ohm
The load is the resistance Rn (see figure), on which the power P_{R } is dissipated. For parametric circuits, it is necessary to select its optimal value, but usually it should be made as small as possible, not forgetting that the active resistance of the coil L, in this case, must be even less, otherwise most of the output power will be dissipated on it.
The load is the resistance Rn (see figure), on which the power P_{R } is dissipated. For parametric circuits, it is necessary to select its optimal value, but usually it should be made as small as possible, not forgetting that the active resistance of the coil L, in this case, must be even less, otherwise most of the output power will be dissipated on it.
The minimum value is 0.01 ohm.
Frequency ratio
This means the ratio of the frequency of the second generator to the frequency of the first. For parametric schemes, based on classical concepts, at least a two-fold ratio is required, but in practice, in some cases, the ratio 1/1 is quite working.
This means the ratio of the frequency of the second generator to the frequency of the first. For parametric schemes, based on classical concepts, at least a two-fold ratio is required, but in practice, in some cases, the ratio 1/1 is quite working.
Phase difference, degrees
This parameter is responsible for the phase difference between the oscillations of the first and second generator. In many cases, the phase shift is critical to obtain an efficiency gain of more than unity. The same shift will need to be ensured in a real device.
This parameter is responsible for the phase difference between the oscillations of the first and second generator. In many cases, the phase shift is critical to obtain an efficiency gain of more than unity. The same shift will need to be ensured in a real device.
Modulation form
In this parametric generator, the inductance L is changed using the generator G_{2 }. The shape of the inductance curve is referred to herein as the "modulation shape". It could be measured if there was a device that measures the inductance of a coil and displays the results in real time. This curve is shown in blue on the graph.
In this parametric generator, the inductance L is changed using the generator G_{2 }. The shape of the inductance curve is referred to herein as the "modulation shape". It could be measured if there was a device that measures the inductance of a coil and displays the results in real time. This curve is shown in blue on the graph.
Pulse duty cycle,%
This field works only with rectangular pulses and is responsible for the relative width that the pulse fills in comparison with the entire period of the oscillation. In the reference literature [1], this is the "D" parameter.
This field works only with rectangular pulses and is responsible for the relative width that the pulse fills in comparison with the entire period of the oscillation. In the reference literature [1], this is the "D" parameter.
Generator power, W
This field is filled with active power, which will be spent by the second generator to change the inductance L. Above the graph, this power is displayed as P_{2 }. At this stage of the design, it is usually unknown, but it will need to be incorporated into the final calculations. When searching for over-unity values of K_{& eta; 2 }, it is recommended to put zero in this field, and when these values are found, select its acceptable value. When calculated in a calculator, this power affects only the increase in efficiency. It is quite clear that in a real device it will not be possible to go beyond this power.
This field is filled with active power, which will be spent by the second generator to change the inductance L. Above the graph, this power is displayed as P_{2 }. At this stage of the design, it is usually unknown, but it will need to be incorporated into the final calculations. When searching for over-unity values of K_{& eta; 2 }, it is recommended to put zero in this field, and when these values are found, select its acceptable value. When calculated in a calculator, this power affects only the increase in efficiency. It is quite clear that in a real device it will not be possible to go beyond this power.
Perhaps an exception to the rule may be devices in which the change in inductance occurs due to the properties of the ferromagnetic core material, for example, in amorphous metals. In this case, such materials can be heated, which can be considered the active power of the second generator.
Only one parameter is indicated here: modulation coefficient L in percent, let's call it "M" below. In a real device, the second generator can change the inductance L using a second coil wound so that the magnetic lines are perpendicular to the lines of the first coil (example). This is how the principle of maximum mutual non-influence of generators on each other is realized. The second coil changes the inductance L according to the following law: L = L_{0}*(1 + F*M/100), where L_{0} is the initial inductance of the L coil, which is entered in the "Initial inductance, mH" field; M - modulation coefficient L in percent; F - oscillations of the second generator, the shape, frequency and phase of which we set in the fields described above. It should also be recalled that the larger the parameter M, the more power you need to spend to obtain it (to change the inductance).
This calculator can also be used to study nonparametric transients in RL circuits. To do this, set the M parameter to zero. In this case, K_{η2} turns into a classic efficiency.
Graph
The graph displays three functions of time: voltage at the output of the first generator (green curve), change in inductance L by the second generator (blue curve) and the resulting graph of the current in the circuit (red curve). Each curve can be excluded from the graph by clicking on the corresponding button, which are located below the graph. Based on the current data, the consumed power from the first generator P_{1 } and the power in the load P_{R } are calculated. The current values are displayed on the graph on the right, in amperes. Approximately the same current values, the form of its oscillations and their displacement relative to zero, we will have to obtain in a real device.
After entering new parameters, it is necessary to press the "Counting" button to recalculate and update the data. Recalculation can take several seconds of processor time, the duration of which depends on the number of calculated oscillation periods. This parameter is entered in the "Display periods" field and is limited to a maximum of 35 oscillation periods. But the more such periods are calculated, the more accurate the result is. Usually, a sufficient condition is when the transient process ends, and the subsequent stationary (or steady-state) process takes about half of the schedule.
For the correct display of the graph, a monitor resolution of 1000 pixels or more horizontally is required. If you are viewing the chart on a mobile phone, it should be expanded horizontally.
Disregarded parameters
This calculator does not take into account some parameters of a real device. For example, it is assumed here that the second generator can change the inductance L, but the first one cannot. In reality, this can only be partially achieved. Also, the calculator assumes that the device is operating on a relatively smooth section Stoletov's curve, which characterizes all ferromagnetic materials. In a real coil, this can also be realized only partially.
Examples
Two offset sines (classic)
This is a classic version of a parametric oscillator, when the frequency of the inductance change is twice the frequency of the master oscillator. But to get the desired effect, you need to shift these frequencies by 135 (-45) degrees. Note that the resulting current shape is similar to the fins of fish swimming from right to left.
This is a classic version of a parametric oscillator, when the frequency of the inductance change is twice the frequency of the master oscillator. But to get the desired effect, you need to shift these frequencies by 135 (-45) degrees. Note that the resulting current shape is similar to the fins of fish swimming from right to left.
Two rectangular pulses without offset
In principle, this is the same classic version as in the previous example, with the only difference that the pulses are rectangular. And if so, then there is no phase difference between them. By the way, try to play around with this parameter. Also, unlike the previous example, the change in inductance here is 2.5 times less, and the current shape is triangular.
In principle, this is the same classic version as in the previous example, with the only difference that the pulses are rectangular. And if so, then there is no phase difference between them. By the way, try to play around with this parameter. Also, unlike the previous example, the change in inductance here is 2.5 times less, and the current shape is triangular.
Two offset sines (non-classical)
This option is similar to the first, but unlike it, here the frequency ratio of the two generators is 1/1, which is not quite a classical value. The phase shift here is 60 degrees, and the current waveform is a quasi-sinusoid.
This option is similar to the first, but unlike it, here the frequency ratio of the two generators is 1/1, which is not quite a classical value. The phase shift here is 60 degrees, and the current waveform is a quasi-sinusoid.
Pulse scheme
In this case, the pulse shape of the first and second generator is rectangular, with a duty cycle of 10%, and the frequency ratio is 1/1. A small phase shift between the two generators improves the output efficiency. The shape of the current is a saw. This option is suitable for pulsed excitation circuits and in cases where the inductance of the coil is changed by the internal parameters of the ferromagnet.
In this case, the pulse shape of the first and second generator is rectangular, with a duty cycle of 10%, and the frequency ratio is 1/1. A small phase shift between the two generators improves the output efficiency. The shape of the current is a saw. This option is suitable for pulsed excitation circuits and in cases where the inductance of the coil is changed by the internal parameters of the ferromagnet.
Here you need to pay attention to the negative offset of the pulses from the first generator (green graph). If no offset is done, then the efficiency of the device returns to the classic values.
Magnetic wheel
These parameters are suitable for an electro-mechanical device in which the coil (s) are located on the stator, and the magnet (s), which change their inductance, are located on the wheel rotor. In this case, the first generator supplies rectangular pulses to the coil (coils), and a rotating magnet (magnets) acts here as the second generator. Here you need to pay special attention to some phase shift of the generator pulse and the transit time of the magnet near the coil. This diagram allows us to draw analogies with the Adams generator.
These parameters are suitable for an electro-mechanical device in which the coil (s) are located on the stator, and the magnet (s), which change their inductance, are located on the wheel rotor. In this case, the first generator supplies rectangular pulses to the coil (coils), and a rotating magnet (magnets) acts here as the second generator. Here you need to pay special attention to some phase shift of the generator pulse and the transit time of the magnet near the coil. This diagram allows us to draw analogies with the Adams generator.
Current Amplifier
If we change the inductance of the coil, through which a sufficiently large current flows, rather quickly, and for a relatively short interval from the full period of the oscillation, then the effect of current amplification can be obtained. It is interesting here that the shape of the current fluctuations almost does not change in comparison with the one at which inductance remains constant. This means that in a real circuit, such an effect is rather difficult to track using instruments, for example, an oscilloscope. Also, you need to pay attention to the phase difference between the generators: the necessary effect appears only at a certain value.
If we change the inductance of the coil, through which a sufficiently large current flows, rather quickly, and for a relatively short interval from the full period of the oscillation, then the effect of current amplification can be obtained. It is interesting here that the shape of the current fluctuations almost does not change in comparison with the one at which inductance remains constant. This means that in a real circuit, such an effect is rather difficult to track using instruments, for example, an oscilloscope. Also, you need to pay attention to the phase difference between the generators: the necessary effect appears only at a certain value.
Short pulse, sinusoid and quadruple frequency
This is a rather exotic version of the parametric oscillator, in which the modulation frequency is 4 times higher than the main one. At the same time, the duty cycle of the master oscillator and the phase between frequencies are very important here. With a slight change in these parameters, the necessary effect of energy amplification disappears and the system goes over to the classical efficiency.
This is a rather exotic version of the parametric oscillator, in which the modulation frequency is 4 times higher than the main one. At the same time, the duty cycle of the master oscillator and the phase between frequencies are very important here. With a slight change in these parameters, the necessary effect of energy amplification disappears and the system goes over to the classical efficiency.
Saving data
This calculator can save the obtained calculations to your account. To do this, you must be registered on this site. You can save the result of the calculation, which is called the word "project" here, by clicking on the "Save to account" button, and then completely restore the data from the section "My projects".
Materials used
- Wikipedia. Duty cycle.
- Wikipedia. Skin Effect.