Analysis of the characteristics of linear circuits and linear signal conversions. Signal conversion in parametric circuits Signal conversion in linear circuits and systems
Linear-parametric circuits - radio circuits, one or more parameters of which change over time according to a given law, are called parametric (linear circuits with variable parameters). It is assumed that any parameter is changed electronically using a control signal. In a linear parametric circuit, the parameters of the elements do not depend on the signal level, but can vary independently over time. In reality, the parametric element is obtained from a nonlinear element, the input of which is the sum of two independent signals. One of them carries information and has a small amplitude, so in the region of its changes the circuit parameters are practically constant. The second is a large amplitude control signal, which changes the position of the operating point of the nonlinear element, and therefore its parameter.
In radio engineering, parametric resistances R(t), parametric inductances L(t) and parametric capacitances C(t) are widely used.
For parametric resistance R(t), the controlled parameter is the differential slope
An example of a parametric resistance is the channel of an MOS transistor, the gate of which is supplied with a control (heterodyne) alternating voltage u Г(t). In this case, the slope of its drain-gate characteristic changes over time and is related to the control voltage by the dependence S(t) = S. If you also connect the voltage of the modulated signal to the MOS transistor u(t), then its current is determined by the expression
Parametric resistances are most widely used to convert the frequency of signals. Heterodyning is the process of nonlinear or parametric mixing of two signals of different frequencies to produce oscillations of a third frequency, which results in a shift in the spectrum of the original signal.
Rice. 24. Block diagram of the frequency converter
The frequency converter (Fig. 24) consists of a mixer (SM) - a parametric element (for example, an MOS transistor, varicap, etc.), a local oscillator (G) - an auxiliary generator of harmonic oscillations with a frequency ωg, which serves for parametric control of the mixer, and an intermediate frequency filter (IFF) - a bandpass filter
Let's consider the operating principle of a frequency converter using the example of transferring the spectrum of a single-tone AM signal. Let us assume that under the influence of heterodyne voltage
The slope of the MOS transistor characteristic changes approximately according to the law
where S 0 and S 1 are, respectively, the average value and the first harmonic component of the slope of the characteristic. When an AM signal receiver enters the converting MOS transistor of the mixer
the variable component of the output current will be determined by the expression:
Let the frequency chosen as the intermediate frequency of the parametric converter
In nonlinear electrical circuits, the connection between the input signal U In . (T) and output signal U Out . (T) described by a nonlinear functional relationship
This functional dependence can be considered as mathematical model nonlinear circuit.
Usually nonlinear electrical circuit represents a set of linear and nonlinear two-terminal networks. To describe the properties of nonlinear two-terminal networks, their current-voltage characteristics (CV characteristics) are often used. As a rule, the current-voltage characteristics of nonlinear elements are obtained experimentally. As a result of the experiment, the current-voltage characteristics of the nonlinear element are obtained in the form of a table. This method of description is suitable for analysis nonlinear circuits using a computer.
To study processes in circuits containing nonlinear elements, it is necessary to display the current-voltage characteristic in a mathematical form convenient for calculations. To use analytical methods of analysis, it is necessary to select an approximating function that sufficiently accurately reflects the features of the experimentally measured characteristic. The following methods of approximating the current-voltage characteristics of nonlinear two-terminal networks are most often used.
Exponential approximation. From the theory of work p-n junction it follows that the current-voltage characteristic of a semiconductor diode at u>0 is described by the expression
.
(7.3)
The exponential relationship is often used when studying nonlinear circuits containing semiconductor devices. The approximation is quite accurate for current values not exceeding a few milliamps. At high currents, the exponential characteristic smoothly turns into a straight line due to the influence of the volume resistance of the semiconductor material.
Power approximation. This method is based on the expansion of the nonlinear current-voltage characteristic into a Taylor series, converging in the vicinity of the operating point U0 :
Here are the coefficients... – some numbers that can be found from the experimentally obtained current-voltage characteristic. The number of expansion terms depends on the required accuracy of calculations.
It is not advisable to use the power-law approximation for large signal amplitudes due to a significant deterioration in accuracy.
Piecewise linear approximation It is used in cases where large signals operate in the circuit. The method is based on approximate replacement real characteristics segments of straight lines with different slopes. For example, transfer characteristic of a real transistor can be approximated by three straight segments, as shown in Fig. 7.1.
Fig.7.1.Transfer characteristic of a bipolar transistor
The approximation is determined by three parameters: the characteristic start voltage, the slope, which has the dimension of conductivity, and the saturation voltage, at which the current stops increasing. The mathematical notation of the approximated characteristic is as follows:
(7.5)
In all cases, the task is to find the spectral composition of the current due to the effect of harmonic voltages on the nonlinear circuit. In piecewise linear approximation, circuits are analyzed using the cutoff angle method.
Let us consider, as an example, the operation of a nonlinear circuit with large signals. As a nonlinear element we use a bipolar transistor that operates with a collector current cutoff. To do this, using the initial bias voltage E The operating point is set in such a way that the transistor operates with the collector current cut off, and at the same time we supply an input harmonic signal to the base.
Fig.7.2. Illustration of current cutoff at large signals
The cutoff angle θ is half of that part of the period during which the collector current is not equal to zero, or, in other words, the part of the period from the moment the collector current reaches its maximum to the moment when the current becomes equal to zero - “cut off”.
In accordance with the designations in Fig. 7.2, the collector current for I> 0 is described by the expression
Expanding this expression into a Fourier series allows us to find the constant component I0 and amplitudes of all collector current harmonics. Harmonic frequencies are multiples of the input signal frequency, and the relative amplitudes of the harmonics depend on the cutoff angle. The analysis shows that for each harmonic number there is an optimal cutoff angle θ, At which its amplitude is maximum:
. (7.7)
Fig.7.8. Frequency multiplication circuit
Similar circuits (Fig. 7.8) are often used to multiply the frequency of a harmonic signal by an integer factor. By adjusting the oscillatory circuit included in the collector circuit of the transistor, you can select the desired harmonic of the original signal. The cutoff angle is set based on the maximum amplitude value of a given harmonic. The relative amplitude of a harmonic decreases as its number increases. Therefore, the described method is applicable for multiplication coefficients N≤ 4. Using multiple frequency multiplication, it is possible, based on one highly stable harmonic oscillator, to obtain a set of frequencies with the same relative frequency instability as that of the main generator. All these frequencies are multiples of the input signal frequency.
The property of a nonlinear circuit to enrich the spectrum, creating spectral components at the output that were initially absent at the input, is most clearly manifested if the input signal is the sum of several harmonic signals with different frequencies. Let us consider the case of the influence of the sum of two harmonic oscillations on a nonlinear circuit. We represent the current-voltage characteristic of the circuit as a polynomial of the 2nd degree:
. (7.8)
In addition to the constant component, the input voltage contains two harmonic oscillations with frequencies and , the amplitudes of which are equal to and respectively:
. (7.9)
Such a signal is called biharmonic. Substituting this signal into formula (7.8), performing transformations and grouping terms, we obtain a spectral representation of the current in a nonlinear two-terminal network:
It can be seen that the current spectrum contains terms included in the spectrum of the input signal, second harmonics of both input signal sources, as well as harmonic components with frequencies ω 1 — ω 2 and ω 1 + ω 2 . If the power-law expansion of the current-voltage characteristic is represented by a polynomial of the 3rd degree, the current spectrum will also contain frequencies. IN general case when a nonlinear circuit is exposed to several harmonic signals with different frequencies, combination frequencies appear in the current spectrum
Where are any integers, positive and negative, including zero.
The appearance of combinational components in the spectrum of the output signal during nonlinear transformation determines a number of important effects that have to be encountered when constructing radio-electronic devices and systems. So, if one of the two input signals is amplitude modulated, then modulation is transferred from one carrier frequency to another. Sometimes, due to nonlinear interaction, amplification or suppression of one signal by another is observed.
Based on nonlinear circuits, detection (demodulation) of amplitude-modulated (AM) signals in radio receivers is carried out. The circuit of the amplitude detector and the principle of its operation are explained in Fig. 7.9.
Fig.7.9. Amplitude detector circuit and output current shape
A nonlinear element, the current-voltage characteristic of which is approximated by a broken line, passes only one (in this case positive) half-wave of the input current. This half-wave creates high (carrier) frequency voltage pulses on the resistor with an envelope that reproduces the shape of the amplitude-modulated signal envelope. The voltage spectrum across the resistor contains the carrier frequency, its harmonics and a low-frequency component, which is approximately half the amplitude of the voltage pulses. This component has a frequency equal to the frequency of the envelope, i.e. it represents a detected signal. The capacitor together with the resistor forms a filter low frequencies. When the condition is met
(7.12)
Only the envelope frequency remains in the output voltage spectrum. In this case, the output voltage also increases due to the fact that with a positive half-wave of the input voltage, the capacitor quickly charges through the low resistance of an open nonlinear element almost to the amplitude value of the input voltage, and with a negative half-wave, it does not have time to discharge through the high resistance of the resistor. The given description of the operation of the amplitude detector corresponds to the mode of a large input signal, in which the current-voltage characteristic of a semiconductor diode is approximated by a broken straight line.
In the small input signal mode, the initial section of the diode’s current-voltage characteristic can be approximated by a quadratic dependence. When an amplitude-modulated signal is applied to such a nonlinear element, the spectrum of which contains a carrier and side frequencies, frequencies with sum and difference frequencies arise. The difference frequency represents the detected signal, and the carrier and sum frequencies do not pass through the low-pass filter formed by the elements and .
A common technique for detecting frequency modulated (FM) waveforms is to first convert the FM waveform into an AM waveform, which is then detected in the manner described above. The simplest FM to AM converter can be one detuned relative to the carrier frequency oscillatory circuit. The principle of converting FM signals to AM is explained in Fig. 7.10.
Fig.7.10. Converting FM to AM
In the absence of modulation, the operating point is on the slope of the circuit's resonance curve. When the frequency changes, the amplitude of the current in the circuit changes, i.e., the FM is converted to AM.
The circuit of the FM to AM converter is shown in Fig. 7.11.
Fig.7.11. FM to AM converter
The disadvantage of such a detector is the distortion of the detected signal, which arises due to the nonlinearity of the resonant curve of the oscillatory circuit. Therefore, in practice, symmetrical circuits are used that have best characteristics. An example of such a circuit is shown in Fig. 7.12.
Fig.7.12. FM signal detector
Two circuits are tuned to extreme frequency values, i.e., to frequencies AND. Each of the circuits converts FM to AM, as described above. AM oscillations are detected by appropriate amplitude detectors. Low-frequency voltages are opposite in sign, and their difference is removed from the output of the circuit. The detector response, i.e. the output voltage versus frequency, is obtained by subtracting the two resonance curves and is more linear. Such detectors are called discriminators.
Let at the input of a linear four-port network (Fig. 7.1) with transfer function and the impulse response is a random process with given statistical characteristics; it is required to find the statistical characteristics of the process at the output of the quadrupole.
In ch. 4 the main characteristics were considered random process: probability distribution; correlation function; power spectral density.
Determining the last two characteristics is the simplest task. The situation is different with determining the law of distribution of a random process at the output of a linear circuit. In the general case, with an arbitrary distribution of the process at the input, finding the distribution at the output of the inertial circuit is a very difficult task.
Rice. 7.1. Linear quadripole with constant parameters
Only with a normal distribution of the input process does the problem become simpler, since for any linear operations with a Gaussian process (amplification, filtering, differentiation, integration, etc.) the distribution remains normal, only the functions change.
Therefore, if the probability density of the input process (with zero mean) is given
then the probability density at the output of the linear circuit
Dispersion is easily determined from the spectrum or from the correlation function. Thus, the analysis of the transmission of Gaussian processes through linear circuits essentially comes down to spectral (or correlation) analysis.
The next four paragraphs are devoted to transforming only the spectrum and correlation function of a random process. This consideration is valid for any probability distribution law. The question of transforming the distribution law for non-Gaussian input processes is considered in § 7.6-7.7.
Passage of signals through resistive parametric circuits. Frequency conversion
12.1 (O). An ideal EMF source creates a voltage (V) And= 1.5 cos 2π · l0 7 t. A resistive element with time-varying conductivity (Cm) is connected to the source terminals G(t) = 10 -3 + 2 10 -4 sin 2π l0 6 t. Find the amplitude of the current IT, having a frequency of 9.9 MHz.
12.2(O). The long-wave broadcast receiver is designed to receive signals in the frequency range from f c min = 150 kHz to f c max = 375 kHz. Receiver intermediate frequency f pr = 465 kHz. Determine within what limits the local oscillator frequency should be adjusted f g of this receiver.
12.3(TS). In a superheterodyne receiver, the local oscillator creates harmonic oscillations with a frequency f r = 7.5 MHz. Receiver intermediate frequency f pr = 465 kHz; Of the two possible frequencies of the received signal, the main receiving channel corresponds to the larger one, and the mirror channel - the lower frequency. To suppress the mirror channel, a single oscillating circuit tuned to the frequency of the main channel is switched on at the input of the frequency converter. Find the value of the quality factor Q this circuit, at which the attenuation of the mirror channel will be - 25 dB relative to the main receiving channel.
12.4(O). The differential slope of the resistive parametric element included in the frequency converter changes according to the law S diff ( t) =S 0 +S 1 cos ω G t, Where S 0 ,S 1 - constant numbers, ω r is the angular frequency of the local oscillator. Assuming that the intermediate frequency ω known, find the signal frequencies ω s, at which the effect occurs at the output of the converter.
12.5(R). The flow characteristic of a field-effect transistor, i.e. drain current dependence i c (mA) from gate-source control voltage And zi (B) at And zi ≥ -2 V, approximated by a quadratic parabola: i c = 7.5( u zi + 2) 2 . Local oscillator voltage is applied to the transistor input And zi = Um g cos ω G t. Find the law of change in time of differential slope S diff ( t) characteristics i c = f(And zi).
12.6(UO). In relation to the conditions of Problem 12.5, select the amplitude of the local oscillator voltage Um g in such a way as to ensure the slope of the transformation S pr = 6 mA/V.
12.7(O). The frequency converter uses a semiconductor diode, the current-voltage characteristic of which is described by the dependence (mA)
Local oscillator voltage (V) is applied to the diode u g = 1.2 cos ω G t. Calculate Transform Slope S pr of this device.
12.8(UO). In a diode frequency converter, which is described in problem 12.7, a voltage (V) is applied to the diode u(t) =U 0 + 1.2cos ω G t. Determine
at what bias voltage U 0 < 0 крутизна преобразования составит величину 1.5 мА/В.
12.9(OS). The circuit of a frequency converter based on a field-effect transistor is shown in Fig. I.12.1. Oscillatory circuit tuned to intermediate frequency ω pr = | ω With - ω g |. Resonant circuit impedance R res = 18 kOhm. The sum of the useful signal voltage (μV) is applied to the converter input u With ( t) = 50 cos ω c t and local oscillator voltage (V) u G ( t) = 0.8cos ω G t. The characteristics of the transistor are described in the conditions of problem 12.5. Find the amplitude Um pr output signal at intermediate frequency.
Passage of signals through parametric reactive circuits. Parametric amplifiers
12.10(R). Differential capacitance of a parametric diode (varactor) in the vicinity of the operating point U 0 depends on applied voltage And in the following way: WITH diff ( u) =b 0 +b 1 (u-U 0), where b 0 (pF) and b 1 (pF/V) - known numerical coefficients. A voltage is applied to the varactor u=U 0 +Um cos ω 0 t. Get the formula describing the current i(t) through a varactor.
12.11 (UO). The differential capacitance of the varactor is described by the expression C diff ( u) =b 0 +b 1 (u-U 0) +b 2 (u-U 0) 2 . Voltage is applied to the varactor terminals u=U 0 +Um cos ω 0 t. Calculate amplitude I 3 third harmonics of the current through the varactor, if f 0 = 10 GHz, Um=1.5 V, b 2 = 0.16 pF/V 2.
12.12(O). Varactor has the following parameters: b 0 = 4 pF, b 2 = 0.25 pF/V 2. A high-frequency voltage with an amplitude is applied to the varactor Um = 0.4 V. Determine how many times the amplitude of the first harmonic of the current will increase I 1 if value Um will become equal to 3 V.
12.13(UO). The capacitance of a parametric capacitor changes over time according to the law WITH(t) =WITH 0 exp (- t/τ) σ ( t), Where WITH 0, τ are constant values. A source of linearly increasing voltage is connected to the capacitor u(t) =atσ( t). Calculate the law of change over time of current i(t) in the capacitor.
12.14 (UO). In relation to the conditions of problem 12.13, find the moment of time t 1, at which the instantaneous power consumed by the capacitor from the signal source is maximum, as well as the moment of time t 2, in which the maximum power is given by the capacitor to the external circuits.
12.15(R). A single-circuit parametric amplifier is connected from the input side to an EMF source (generator) with an internal
resistance R g = 560 Ohm. The amplifier operates on a resistive load with resistance R n = 400 Ohm. Find the value of the introduced conductivity G Vn, which provides power gain TOR= 25 dB.
12.16(O). For the parametric amplifier described in Problem 12.15, find the critical value of the introduced conductance G vn cr, at which the system is on the threshold of self-excitation.
12.17 (UO). A signal voltage is applied to the terminals of the controlled parametric capacitor u(t) =Um cos( ω c t+π/3). The capacitance of a capacitor changes over time according to the law C(t) =C 0" where φ n is the initial phase angle of pump oscillation. Select the smallest absolute value φ n, which provides zero value of introduced conductivity.
12.18(O). In relation to the conditions of problem 12.17 for the parameter values WITH 0 = 0.3 pF, β = 0.25 and ω s = 2π · 10 9 s -1 calculate the largest absolute value of negative conductivity G vn max, as well as the smallest absolute phase angle sra, providing such a regime.
12.19(R). A two-circuit parametric amplifier is designed to operate at a frequency f c = 2 GHz. Amplifier idle frequency f cold = 0.5 GHz. The varactor used in the amplifier changes its capacitance (pF) with pump frequency ω n by law WITH(t) = 2(1 + 0.15 cos ω n t). The signal source and the load device have the same active conductances G g = G n = 2 10 -3 See. Calculate the value of the resonant resistance of the idle circuit R rez.col, at which self-excitation occurs in the amplifier.
Parametric (linear circuits with variable parameters), are called radio circuits, one or more parameters of which change over time according to a given law. It is assumed that the change (more precisely, modulation) of any parameter is carried out electronically using a control signal. In radio engineering, parametric resistance R(t), inductance L(t) and capacitance C(t) are widely used.
An example of one of the modern parametric resistances can serve as a channel of a VLG transistor, the gate of which is supplied with a control (heterodyne) alternating voltage u g (t). In this case, the slope of its drain-gate characteristic changes over time and is related to the control voltage by the functional dependence S(t)=S. If you also connect the voltage of the modulated signal u(t) to the VLG transistor, then its current will be determined by the expression:
i c (t)=i(t)=S(t)u(t)=Su(t). (5.1)
As with the class of linear circuits, the principle of superposition is applicable to parametric circuits. Indeed, if the voltage applied to the circuit is the sum of two variables
u(t)=u 1 (t)+u 2 (t), (5.2)
then, substituting (5.2) into (5.1), we obtain the output current also in the form of the sum of two components
i(t)=S(t)u 1 (t)+S(t)u 2 (t)= i 1 (t)+ i 2 (t) (5.3)
Relation (5.3) shows that the response of a parametric circuit to the sum of two signals is equal to the sum of its responses to each signal separately.
Conversion of signals in circuits with parametric resistance. Parametric resistances are most widely used to convert the frequency of signals. Note that the term “frequency conversion” is not entirely correct, since the frequency itself is unchanged. Obviously, this concept arose due to an inaccurate translation of the English word “heterodyning”. Heterodyning – it is the process of nonlinear or parametric mixing of two signals of different frequencies to produce a third frequency.
So, frequency conversion is a linear transfer (mixing, transformation, heterodyning, or transposition) of the spectrum of a modulated signal (as well as any radio signal) from the carrier frequency region to the intermediate frequency region (or from one carrier frequency to another, including a higher one) without changing type or nature of modulation.
Frequency converter(Fig. 5.1) consists of a mixer (SM) - a parametric element (for example, an MOS transistor, a varicap or a conventional diode with a quadratic characteristic), a local oscillator (G) - an auxiliary self-oscillator of harmonic oscillations with a frequency ω g, which serves for parametric control of the mixer, and an intermediate frequency filter (usually an oscillating circuit of an amplifier or UHF).
Fig.5.1. Block diagram of the frequency converter
Let's consider the operating principle of a frequency converter using the example of transferring the spectrum of a single-tone AM signal. Let us assume that under the influence of heterodyne voltage
u g (t)=U g cos ω g t (5.4)
The slope of the characteristic of the MOS transistor of the frequency converter changes over time approximately according to the law
S(t)=S o +S 1 cos ω g t (5.5)
where S o and S 1 are, respectively, the average value and the first harmonic component of the slope of the characteristic.
When an AM signal u AM (t)= U n (1+McosΩt)cosω o t arrives at the MOS transistor of the mixer, the alternating component of the output current in accordance with (5.1) and (5.5) will be determined by the expression:
i c (t)=S(t)u AM (t)=(S o +S 1 cos ω g t) U n (1+McosΩt)cosω o t=
U n (1+McosΩt) (5.6)
Let us choose as the intermediate frequency of the parametric converter
ω pch =|ω g -ω o |. (5.7)
Then, having isolated it using the IF loop from the current spectrum (5.6), we obtain a converted AM signal with the same modulation law, but a significantly lower carrier frequency
i pch (t)=0.5S 1 U n (1+McosΩt)cosω pch t (5.8)
Note that the presence of only two side components of the current spectrum (5.6) is determined by the choice of an extremely simple piecewise linear approximation of the transistor characteristic slope. In real mixer circuits, the current spectrum also contains components of combination frequencies
ω pc =|mω g ±nω o |, (5.9)
where m and n are any positive integers.
The corresponding time and spectral diagrams of signals with amplitude modulation at the input and output of the frequency converter are shown in Fig. 5.2.
Fig.5.2. Diagrams at the input and output of the frequency converter:
a – temporary; b – spectral
Frequency converter in analog multipliers. Modern frequency converters with parametric resistive circuits are built on a fundamentally new basis. They use analog multipliers as mixers. If two harmonic oscillations of a certain modulated signal are applied to the inputs of the analog multiplier:
u c (t)=U c (t)cosω o t (5.10)
and the local oscillator reference voltage u g (t) = U g cos ω g t, then its output voltage will contain two components
u out (t)=k a u c (t)u g (t)=0.5k a U c (t)U g (5.11)
Spectral component with difference frequency ω fc =|ω g ±ω o | is isolated by a narrow-band IF filter and is used as the intermediate frequency of the converted signal.
Frequency conversion in varicap circuits. If only heterodyne voltage (5.4) is applied to the varicap, then its capacitance will approximately change over time according to the law (see Fig. 3.2 in part I):
C(t)=C o +C 1 cosω g t, (5.12)
where C o and C 1 are the average value and the first harmonic component of the varicap capacitance.
Let us assume that the varicap is affected by two signals: heterodyne and (to simplify calculations) unmodulated harmonic voltage (5.10) with amplitude U c . In this case, the charge on the varicap capacity will be determined:
q(t)=C(t)u c (t)=(C o +C 1 cosω g t)U c cosω o t=
С o U c (t)cosω o t+0.5С 1 U c cos(ω g - ω o)t+0.5С 1 U c cos(ω g + ω o)t, (5.13)
and the current flowing through it is
i(t)=dq/dt=- ω o С o U c sinω o t-0.5(ω g -ω o)С 1 U c sin(ω g -ω o)t-
0.5(ω g +ω o)С 1 U c sin(ω g +ω o)t (5.14)
By connecting an oscillatory circuit in series with a varicap, tuned to the intermediate frequency ω fc =|ω g -ω o |, the desired voltage can be selected.
With a varicap-type reactive element (for ultra-high frequencies this is varactor) you can also create a parametric oscillator, power amplifier, frequency multiplier. This possibility is based on the conversion of energy in a parametric capacitance. It is known from a physics course that the energy accumulated in a capacitor is related to its capacitance C and the charge q on it by the formula:
E = q 2 /(2C). (5.15)
Let the charge remain constant and the capacitance of the capacitor decrease. Since energy is inversely proportional to the size of the capacitance, when the latter is reduced, the energy increases. We obtain a quantitative relationship for such a connection by differentiating (5.15) with respect to the parameter C:
dE/dC= q 2 /2C 2 = -E/C (5.16)
This expression is also valid for small increments of capacity ∆C and energy ∆E, so we can write
∆E=-E (5.17)
The minus sign here shows that a decrease in capacitance of the capacitor (∆C<0) вызывает увеличение запасаемой в нем энергии (∆Э>0). The increase in energy occurs due to the external costs of performing work against the forces of the electric field when the capacitance is reduced (for example, by changing the bias voltage on the varicap).
When the parametric capacitance (or inductance) of several signal sources with different frequencies is simultaneously exposed, there will be a redistribution (exchange) of vibration energies. In practice, the oscillation energy of an external source called pump generator, is transmitted through a parametric element to the useful signal circuit.
To analyze the energy relationships in multi-circuit circuits with a varicap, let us turn to the generalized diagram (Fig. 5.3). It contains three circuits parallel to the parametric capacitance C, two of which contain sources e 1 (t) and e 2 (t), creating harmonic oscillations with frequencies ω 1 and ω 2. The sources are connected through narrow-band filters Ф 1 and Ф 2, which respectively transmit oscillations with frequencies ω 1 and ω 2. The third circuit contains a load resistance R n and a narrow-band filter Ф 3, the so-called idle circuit, tuned to a given combination frequency
ω 3 = mω 1 +nω 2, (5.18)
where m and n are integers.
For simplicity, we will assume that the circuit uses filters without ohmic losses. If in the circuit the sources e 1 (t) and e 2 (t) supply power P 1 and P 2, then the load resistance R n consumes power P n. For a closed system, in accordance with the law of conservation of energy, we obtain the power balance condition:
P 1 + P 2 + P n = 0 (5.19)
