Signal conversion in linear parametric circuits. Signal conversion in parametric circuits. Objectives of the course work

Parametric ( linear circuits with variable parameters), are called radio circuits, one or more parameters of which change in time according to a given law. It is assumed that a change (more precisely, modulation) of any parameter is carried out by electronic method with a control signal. In radio engineering, parametric resistances R(t), inductances L(t) and capacitances 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) AC voltage u g(t). In this case, the slope of its drain-gate characteristic changes with time and is related to the control voltage by the functional dependence S(t)=S. If the voltage of the modulated signal u(t) is also connected to the VLG transistor, then its current is determined by the expression:

i c (t)=i(t)=S(t)u(t)=Su(t). (5.1)

As a class of linear, 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.

Converting signals in circuits with parametric resistance. The most widely parametric resistances are 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". Heterodyne - it is the process of non-linear or parametric mixing of two signals of different frequencies to produce a third frequency.

So, frequency conversion- this is a linear transfer (mixing, transformation, heterodyning, or transposition) of the spectrum of the modulated signal (as well as any radio signal) from the carrier frequency region to the intermediate frequency region (or from one carrier carrier frequency to another, including a higher one) without changing the type or nature of the modulation.

Frequency converter(Fig. 5.1) consists of a mixer (SM) - a parametric element (for example, an MIS 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 IF or UHF resonant circuit).

Fig.5.1. Structural scheme frequency converter

Let us consider the principle of operation of the frequency converter using the example of transferring the spectrum of a single-tone AM signal. Let us assume that under the influence of a heterodyne voltage

u g (t)=U g cos ω g t (5.4)

the steepness of the characteristics of the MIS transistor of the frequency converter changes with 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 arrives at the MIS transistor of the mixer u AM (t) = U n (1+McosΩt)cosω o t, the variable 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 as the intermediate frequency of the parametric converter is chosen

ω pc \u003d | ω g - ω about |. (5.7)

Then, having selected it with the help of the IF circuit from the current spectrum (5.6), we obtain the converted AM signal with the same modulation law, but with a significantly lower carrier frequency

i pc (t)=0.5S 1 U n (1+McosΩt)cosω pc 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 steepness of the transistor characteristic. In real mixer circuits, the current spectrum also contains combination frequency components

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 a certain modulated signal is applied to the inputs of the analog multiplier:

u c (t)=U c (t) cosω o t (5.10)

and the reference voltage of the local oscillator u g (t) \u003d 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 pc =|ω g ±ω o | is separated by a narrow-band IF filter and used as the intermediate frequency of the converted signal.

Frequency conversion in a varicap circuit. If only a heterodyne voltage (5.4) is applied to the varicap, then its capacitance will approximately change in 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 two signals act on the varicap: heterodyne and (to simplify calculations) unmodulated harmonic voltage (5.10) with amplitude U c . In this case, the charge on the capacitance of the varicap will be determined by:

q(t)=C(t)u c (t)=(C o +C 1 cosω g t)U c cosω o t=

C o U c (t) cosω o t + 0.5С 1 U c cos (ω g - ω o) t + 0.5 С 1 U c cos (ω g + ω o) t, (5.13)

and the current flowing through it

i (t) \u003d dq / dt \u003d - 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 turning on in series with the varicap an oscillatory circuit tuned to an intermediate frequency ω pch \u003d | ω g - ω about |, you can select the desired voltage.

With a reactive element of the varicap type (for microwave frequencies, this varactor) you can also create a parametric generator, power amplifier, frequency multiplier. This possibility is based on the conversion of energy into a parametric capacitance. It is known from the course of physics that the energy accumulated in a capacitor is related to its capacitance C and the charge on it q by the formula:

E \u003d q 2 / (2C). (5.15)

Let the charge remain constant, and the capacitance of the capacitor decreases. Since the energy is inversely proportional to the value of the capacitance, as the latter decreases, the energy increases. We obtain the quantitative ratio of such a connection by differentiating (5.15) with respect to the parameter С:

dE / dC \u003d q 2 / 2C 2 \u003d -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 the decrease in the capacitance of the capacitor (∆C<0) вызывает увеличение запасаемой в нем энергии (∆Э>0). The increase in energy occurs due to external costs for performing work against the forces of the electric field with a decrease in capacitance (for example, by changing the bias voltage on the varicap).

With simultaneous action on the parametric capacitance (or inductance) of several signal sources with different frequencies, between them will occur redistribution (exchange) of oscillation energies. In practice, the vibrational energy of an external source called pump generator, is transmitted through the parametric element to the useful signal circuit.

To analyze the energy relationships in multi-circuit circuits with a varicap, let's turn to the generalized scheme (Fig. 5.3). In it, parallel to the parametric capacitance C, three circuits are included, 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 F 1 and F 2 , passing respectively vibrations 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 filters without ohmic losses are used in the circuit. If in the circuit sources e 1 (t) and e 2 (t) give power R 1 and R 2, then the load resistance R n consumes power R 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 \u003d 0 (5.19)

MOSCOW STATE TECHNICAL UNIVERSITY OF CIVIL AVIATION

Department of Fundamentals of Radio Engineering and Information Security

COURSE WORK

Analysis of the characteristics of linear circuits

And linear transformations signals

Completed:

Supervisor:

Ilyukhin Alexander Alekseevich

Moscow 2015

1. Goals term paper. 3

2. Individual task.3

3. Calculations 4

4. Program for calculating and constructing the amplitude-frequency, phase-frequency, transient and impulse characteristics of the circuit for given parameters10

5. The program for calculating and constructing the response of a given circuit to a given signal11

6. Graphs 13

1. Objectives of the course work.

1. To study the nature of transient processes in linear circuits.

2. Fix analytical methods for calculating the frequency and time characteristics of linear circuits.

3. Master the superposition analysis of signals.

4. Master the superposition method for calculating the reactions of linear circuits.

5. Understand the influence of the circuit parameters on the type of its reaction.

2. Individual task.

Option 27 (circuit number 7, signal number 3).

Fig. 1. Electrical circuit

Fig.2.Signal

E=2V

t and \u003d 10 μs

R \u003d 4 kOhm

C=1000pF

Operator transfer characteristic of the circuit;

Complex frequency response of the circuit;

Amplitude-frequency characteristic of the circuit;

Phase-frequency response of the circuit;

Transient response of the circuit;

The impulse response of the circuit.

2. Perform superposition analysis of the signal.

4. Compile a program for calculating and constructing the amplitude-frequency, phase-frequency, transient and impulse characteristics of the circuit with its given parameters.

5. Compile a program for calculating and constructing the response of a given circuit to a given signal.

6. Calculate the characteristics and response of the circuit indicated in p.p. 4 and 5, plot their graphs.

3. Calculations

3.1. Circuit characteristics calculation

1. Operator transfer characteristic

Fig.3. Generalized circuit diagram

For a given schema:

According to the formula:

For a given circuit shown in Fig. 1,

Where θ=RC is the time constant.

2. Complex frequency response

The complex frequency response is determined from the relationship:

3. Frequency response (AFC)

4. Phase response (PFC)

For this chain:

5. Step response

For this chain:

Because , where x 1 and x 2 are the roots of the equation x 2 + bx + c = 0,

Linear-parametric circuits - radio circuits, one or more parameters of which change in time according to a given law, are called parametric (linear circuits with variable parameters). It is assumed that the change of any parameter is carried out electronically using a control signal. In a linear-parametric circuit, the parameters of the elements do not depend on the signal level, but can independently change over time. In reality, a parametric element is obtained from a non-linear element, the input of which is the sum of two independent signals. One of them carries information and has a small amplitude, so that in the region of its changes the circuit parameters are practically constant. The second is a high-amplitude control signal that changes the position of the operating point of the non-linear element and, consequently, its parameter.

In radio engineering, parametric resistances R(t), parametric inductances L(t) and parametric capacitances C(t) are widely used.

For the parametric resistance R(t), the controlled parameter is the differential slope

An example of a parametric resistance is the channel of an MIS transistor, the gate of which is supplied with a control (heterodyne) alternating voltage u Г (t). In this case, the steepness of its drain-gate characteristic changes with time and is related to the control voltage dependence S(t) = S. If the voltage of the modulated signal is also connected to the MIS transistor u(t), then its current is determined by the expression

The most widely parametric resistances are used to convert the frequency of signals. Heterodyning is the process of non-linear or parametric mixing of two signals of different frequencies to obtain oscillations of the third frequency, as a result of which the spectrum of the original signal is shifted.

Rice. 24. Structural diagram of the frequency converter

The frequency converter (Fig. 24) consists of a mixer (SM) - a parametric element (for example, an MIS transistor, a 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 (PLF) - a bandpass filter

Let us consider the principle of operation of the frequency converter using the example of transferring the spectrum of a single-tone AM signal. Assume that under the influence of a heterodyne voltage

the steepness of the characteristics of the MIS transistor varies approximately according to the law

where S 0 and S 1 - respectively, the average value and the first harmonic component of the slope characteristics. When an AM-signal receiver receives an MOS-converting transistor of the mixer

the variable component of the output current will be determined by the expression:

Let the frequency be chosen as the intermediate frequency of the parametric converter

In non-linear electrical circuits, the relationship between the input signal U Vx . (T) and output signal U Exit . (T) is described by a nonlinear functional dependence

Such a functional dependence can be considered as a mathematical model of a nonlinear circuit.

Usually non-linear electrical circuit represents a set of linear and non-linear two-terminal networks. To describe the properties of nonlinear two-terminal networks, their current-voltage characteristics (CVC) are often used. As a rule, the CVC of nonlinear elements is obtained experimentally. As a result of the experiment, the CVC of a nonlinear element is obtained in the form of a table. This description method is suitable for the analysis of nonlinear circuits using a computer.

To study processes in circuits containing non-linear elements, it is necessary to display the CVC in mathematical form convenient for calculations. To use analytical methods of analysis, it is required to choose an approximating function that accurately reflects the experimental features removed characteristics. Most often, the following methods of approximating the I–V characteristics of nonlinear two-terminal networks are 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)

An exponential relationship is often used in the study of 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 a nonlinear current-voltage characteristic in 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 calculation accuracy.

It is impractical to use a 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 the approximate replacement of the real characteristic by segments of straight lines with different slopes. For example, the transfer characteristic of a real transistor can be approximated by three line segments, as shown in Figure 7.1.

Fig.7.1.Transfer characteristic of bipolar transistor

The approximation is determined by three parameters: the voltage of the beginning of the characteristic , slope , which has the dimension of conductivity and saturation voltage , at which the current increase stops. 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 impact on the nonlinear circuit of harmonic voltages. In the piecewise linear approximation, the circuits are analyzed by the cutoff angle method.

Consider, for example, the operation of a nonlinear circuit with large signals. As a non-linear element, we use a bipolar transistor operating with a cutoff of the collector current. To do this, using the initial bias voltage E See the operating point is set in such a way that the transistor operates with a collector current cutoff, and at the same time we apply 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 notation in Fig. 7.2, the collector current for I> 0 is described by the expression

The expansion of this expression into a Fourier series allows us to find the constant component I0 and amplitudes of all harmonics of the collector current. The 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 schemes (Fig. 7.8) are often used to multiply the frequency of a harmonic signal by an integer number of times. 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 value of the amplitude of the given harmonic. The relative amplitude of the harmonic decreases as its number increases. Therefore, the described method is applicable for multiplication factors N≤ 4. Using multiple frequency multiplication, it is possible, on the basis of one highly stable harmonic oscillation generator, 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 non-linear circuit to enrich the spectrum, creating at the output spectral components that were originally absent at the input, is most pronounced if the input signal is the sum of several harmonic signals with different frequencies. Consider the case of the impact on a nonlinear circuit of the sum of two harmonic oscillations. We represent the current-voltage characteristic of the circuit by a polynomial of the 2nd degree:

. (7.8)

The input voltage, in addition to the constant component, contains two harmonic oscillations with frequencies and , the amplitudes of which are equal and, respectively:

. (7.9)

Such a signal is called biharmonic. Substituting this signal into formula (7.8), performing transformations and grouping the terms, we obtain the 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, the 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 the 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 or negative, including zero.

The occurrence of combination components in the spectrum of the output signal during a nonlinear transformation causes a number of important effects that one has to deal with when constructing radio electronic devices and systems. So, if one of the two input signals is modulated in amplitude, then the modulation is transferred from one carrier frequency to another. Sometimes, due to nonlinear interaction, one signal is strengthened or suppressed by another.

Non-linear circuits are used to detect (demodulate) amplitude modulated (AM) signals in radio receivers. The scheme 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 waveform

A non-linear element, whose current-voltage characteristic is approximated by a broken line, passes only one (in this case, positive) half-wave of the input current. This half-wave creates voltage pulses of high (carrier) frequency on the resistor with an envelope that reproduces the shape of the envelope of the amplitude-modulated signal. The voltage spectrum across the resistor contains the carrier frequency, its harmonics, and a low-frequency component, which is about half the amplitude of the voltage pulses. This component has a frequency equal to the frequency of the envelope, i.e., it is a detected signal. The capacitor together with the resistor forms a filter low frequencies. When the condition

(7.12)

Only the envelope frequency remains in the output voltage spectrum. At the same time, the output voltage also increases due to the fact that with a positive half-wave of the input voltage, the capacitor is quickly charged 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 above description of the operation of the amplitude detector corresponds to the regime of a large input signal, in which the I–V characteristic of a semiconductor diode is approximated by a broken line.

In the small input signal mode, the initial section of the I–V characteristic of the diode can be approximated by a quadratic dependence. When an amplitude-modulated signal is applied to such a nonlinear element, the spectrum of which contains the carrier and side frequencies, frequencies arise with sum and difference frequencies. The difference frequency is the detected signal, and the carrier and sum frequencies do not pass through the low-pass filter formed by the and elements.

A common technique for detecting frequency modulated (FM) oscillations is that the FM oscillation is first converted to an AM oscillation, which is then detected in the manner described above. An oscillatory circuit detuned with respect to the carrier frequency can serve as the simplest FM to AM converter. The principle of converting FM signals to AM is explained in Fig. 7.10.

Fig.7.10. FM to AM conversion

In the absence of modulation, the operating point is on the slope of the resonant curve of the circuit. When the frequency changes, the amplitude of the current in the circuit changes, i.e., the FM is converted to AM.

The FM to AM converter circuit 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 due to the nonlinearity of the resonant curve of the oscillatory circuit. Therefore, in practice, symmetrical circuits are used that have the best performance. An example of such a scheme is shown in Fig. 7.12.

Fig.7.12. FM signal detector

Two circuits are tuned to the extreme values ​​of the frequency, i.e., to frequencies AND. Each circuit converts FM to AM as described above. AM oscillations are detected by appropriate amplitude detectors. Low-frequency voltages and are opposite in sign, and their difference is taken from the output of the circuit. The characteristic of the detector, i.e. the dependence of the output voltage on the frequency, is obtained by subtracting the two resonance curves and is more linear. Such detectors are called discriminators (distinguishers).

Passage of signals through resistive parametric circuits. Frequency conversion

12.1 (O). An ideal EMF source produces a voltage (V) and= 1.5 cos 2π l0 7 t. A resistive element with a time-varying conductivity (Sm) is connected to the source terminals G(t) \u003d 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 broadcast receiver of the long-wave range is designed to receive signals in the frequency range from f c min = 150 kHz up to f c max = 375 kHz. Receiver Intermediate Frequency f pr = 465 kHz. Determine within what limits the local oscillator frequency should be tuned f g of this receiver.

12.3(TO). 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 a larger one, and the mirror channel to a smaller frequency. To suppress the mirror channel at the input of the frequency converter, a single oscillatory circuit is switched on, tuned to the frequency of the main channel. Find the value of the quality factor Q this circuit, at which the attenuation of the image channel will be - 25 dB in relation 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 pr is known, find the frequency of the signal ω s, at which the effect occurs at the output of the converter.

12.5(P). The pass characteristic of the 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 . A local oscillator voltage is applied to the input of the transistor and zi = Um g cos ω G t. Find the law of time variation of the differential slope S diff ( t) characteristics i c = f(and zi).

12.6(TO). With regard 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 steepness of the transformation S pr \u003d 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)

LO applied to diode (V) u r = 1.2 cos ω G t. Compute Slope of Conversion S pr this device.

12.8(TO). In the diode frequency converter, which is described in problem 12.7, a voltage (V) is applied to the diode u(t) =U 0 + 1.2 cos ω G t. Define,

at what bias voltage U 0 < 0 крутизна преобразования составит величину 1.5 мА/В.

12.9(MO). The circuit of the frequency converter on a field-effect transistor is shown in fig. I.12.1. The oscillatory circuit is tuned to an intermediate frequency ω pr = | ω With - ω g |. Loop resonant impedance R cut = 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) = 0.8 cos ω G t. The characteristic of the transistor is 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: FROM diff ( u) =b 0 +b 1 (u-U 0), where b 0 (pF) and b 1 (pF/V) - known numerical coefficients. Voltage 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 applied to the varactor terminals u=U 0 +Um cos ω 0 t. Calculate amplitude I 3rd harmonic current through the varactor if f 0 = 10 GHz, Um=1.5 V, b 2 \u003d 0.16 pF / V 2.

12.12(O). Varactor has parameters: b 0 = 4 pF, b 2 \u003d 0.25 pF / V 2. A high-frequency voltage with 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 becomes 3 V.

12.13(UO). The capacitance of a parametric capacitor changes with time according to the law FROM(t) =FROM 0 exp (- t/τ) σ ( t), where FROM 0 , τ are constant values. A linearly increasing voltage source is connected to the capacitor u(t) =atσ( t). Compute law of change with time of current i(t) in the condenser.

12.14(MO). In relation to the conditions of problem 12.13, find the moment in time t 1 , at which the instantaneous power consumed by the capacitor from the signal source is maximum, as well as the time t 2 , in which the maximum is the power given by the capacitor to 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 r = 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 a power gain ToR= 25 dB.

12.16(O). For the parametric amplifier described in problem 12.15, find the critical value of the introduced conductivity G ext cr, at which the system is on the threshold of self-excitation.

12.17(MO). 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 with time according to the law C(t) =C 0" where φ n is the initial phase angle of the pump oscillation. Choose the smallest modulo value φ n, which provides a zero value of the introduced conductivity.

12.18(O). As applied to the conditions of problem 12.17 for parameter values FROM 0 = 0.3 pF, β = 0.25 and ω c \u003d 2π 10 9 s -1 calculate the largest modulo value of negative conductivity G ext max , as well as the smallest modulo 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 the pump frequency ω n by law FROM(t) = 2(1 + 0.15 cos ω n t). Signal source and load device have the same conductance G r = G n \u003d 2 10 -3 See Calculate the value of the resonant resistance of the idle circuit R rez.hol, at which self-excitation occurs in the amplifier.