Transients in RLC circuits

Linear circuits of the 2nd order contain two different types of reactive elements L and C. Examples of such circuits are series and parallel resonant circuits (Fig. 1).

Rice. one. Linear circuits of the second order: a - series resonant circuit; b - parallel resonant circuit

Transient processes in oscillatory circuits are described by differential equations of the 2nd order. Consider the case of capacitance discharge on the RL circuit (Fig. 2). We compose the circuit equation according to the first Kirchhoff law:

After differentiation (1) we get

Rice. 2.

The solution U c (t) of equation (2) is found as the sum of free U st (t) and forced U pr components

U s \u003d U St + U Ave (3)

U pr depends on E, and U st (t) is determined by the solution of a homogeneous differential equation of the form

The characteristic equation for (4) has the form

LCpІ + RCp + 1 = 0, (5)

Roots of the characteristic equation

The value of R / 2L \u003d b is called the attenuation coefficient, - the resonant frequency of the circuit. Wherein

The nature of transient processes in the circuit depends on the type of roots p 1 and p 2 . They can be:

1) real, different for R > 2с, Q< 0,5;

2) real and equal at R = 2s, Q = 0.5;

3) complex conjugate at R< 2с, Q > 0,5.

Here - characteristic impedance, Q \u003d c / R - quality factor of the circuit.

In the scheme of Fig. 2 before switching at t<0 емкость заряжена до напряжения U c (0 -) = E. После коммутации емкость начинает разряжаться и в контуре возникает переходный процесс. В случае 1 при Q < 0,5 решение уравнения (2) имеет вид

To find the integration constants A 1 and A 2, we write the expression for the current in the circuit

Using the initial conditions U c (0 -) = E and i(0 -) = 0, we obtain the system of equations

From the solution of the system we have

As a result, for the current and voltages in the circuit, we obtain

Transient processes in second-order circuits

Definition of the independent variable.

I L - independent variable

We compose a differential equation for the transient process in the circuit and write down the general solution.

I L (t) \u003d i sv (t) + i pr

Let us define the initial conditions.

IL(0)=E/R=19.799А

Let us write down the solution of the diff. equations for the free component.

i sv (t) \u003d A * e bt * sin (wt + i)

Z in =2R+jwL+1/jwC

p=-883.833-7.016i*10 3

f=1/|b|=1.131*10 -3

T \u003d 2p / w \u003d 8.956 * 10 -4

Let's define forced components at t=?

We define the constant of integration A and

U L (t)=LAbwe bt *sin(wt+u)

i L (t)=Ae bt *sin(wt+i)

LAb*sin and + LAw*cos and =0

p Acos u=2.494

tg u=19.799/Acos u=7.938

Spectral representation of periodic processes in electrical circuits

In many cases, in steady state, the curves of periodic emf, voltages and currents in electrical circuits may differ from sinusoidal ones. In this case, the direct application of the symbolic method for the calculation of AC circuits becomes impossible. For linear electrical circuits, the calculation problem can be solved based on the superposition method using the spectral expansion of non-sinusoidal voltages and currents in a Fourier series. In the general case, the Fourier series contains a constant component, the first harmonic, the frequency of which coincides with the frequency w 1 =2p/T of a current or voltage that is periodic with a period T, and a set of higher harmonics with frequencies w n =nw 1 that are multiples of the fundamental frequency w 1 . For most periodic functions, the Fourier series contains an infinite number of terms. In practice, they are limited to a finite number of members of the series. In this case, the original periodic function will be represented using the Fourier series with some error.

Let there be a periodic emf with period T e(t)=e(t±nT), which satisfies the Dirichlet conditions (the function on the interval T has a finite number of discontinuities and extrema). Such a function can be represented by the sum of harmonic components with different amplitudes E n , frequencies w n =nsh 1 and initial phases q n in the form of a Fourier series

The Fourier series can be represented in another form:

The constant component E 0 and the coefficients of the Fourier series B n and C n are calculated by the formulas

For odd functions e(t) the coefficients C n =0, and for even B n =0, The relationship between the coefficients B n , C n and the amplitudes E n and the phases q n of the harmonics is determined by the relations

The diagram, which depicts the dependence of the amplitude of the harmonics E n on the frequency w n =nsh 1, is called the spectrum.

Using the superposition method and the spectral representation of the periodic emf. in the form of a Fourier series, an electric circuit can be calculated using the following method:

1. Non-sinusoidal periodic emf. e(t) is expanded into a Fourier series and the amplitudes E n and phases q n of all harmonics of the emf are determined.

2. In the branch of interest, the currents i 0 , i 1 ,...i n , created by each harmonic of the emf are calculated.

3. The desired current in the branch is found as the sum of the currents

Since the components of the current i(t) are either a constant value i 0 or sinusoidal currents i n , the known methods for calculating circuits of direct and alternating sinusoidal currents are used to determine them.

Reactive circuit L and WITH stores energy in both magnetic and electric fields, so there are no current and voltage surges in it. Let's find transitional i and associated with energy reserves in RLC-circuit (Fig. 7.13), when it is turned on for an arbitrary voltage u, counting the capacitor WITH pre-discharged.

The circuit state equation satisfies the second Kirchhoff law:

.

Expressing the current in terms of capacitive voltage:

,

we get the equation

,

the order of which is determined by the number of elements in the circuit capable of storing energy. Dividing both sides of the equation by the coefficient LC with a higher order derivative, we find the equation of the transient process:

, (7.17)

whose general solution consists of the sum of two terms:

The forced component is determined by the type of applied voltage. When the circuit is switched on, the steady state current and all the voltage will be applied to the capacitance. When the circuit is turned on steady current and voltage on the elements R, L, C will be sinusoidal. The forced component is calculated by the symbolic method, and then it is passed from the complex to the instantaneous value .

The free component is determined from the solution of the homogeneous equation

(7.18)

as the sum of two exponentials (two energy storage elements L, C):

where are the roots of the characteristic equation

.

The nature of the free component depends on the type of roots

, (7.20)

which can be real or complex, and is determined by the ratio of parameters RLC-chains.

There are three options for the transition process:

- aperiodic when the transient current and voltage approach the final steady state without changing sign. Occurrence condition:

(7.21)

where - critical resistance. In this case, the roots of the characteristic equation are real, negative and
different: ; time constants are also different: ;

- limiting mode of aperiodic.Occurrence condition:

. (7.22)

The roots of the characteristic equation are real, negative and equal: ; the time constants are also equal: . The limiting regime corresponds to the general solution of the homogeneous equation (7.18) in the form

; (7.23)

- periodical or oscillatory , when the transient current and voltage approach the final steady state, periodically changing sign and damping in time along a sinusoid. Occurrence condition:

. (7.24)

The roots of the characteristic equation are complex conjugate with the negative real part:

where α - damping factor:

ω St. - angular frequency of free (natural) oscillations:

. (7.26)

The transient process in this case is the result of an oscillatory exchange of energy with a frequency of free oscillations between reactive elements L and C chains. Each oscillation is accompanied by losses in the active resistance R, providing damping with time constant .

The general solution of equation (7.18) with an oscillatory transient process has the form

where A and γ are constants of integration determined from the initial conditions.

Let's write down the voltage u C and current i associated with the energy reserves in the circuit, for the case of real and different roots of the characteristic equation:

From initial conditions

(7.30)

define constants of integration A 1 and A 2 .

Consider the inclusion RLC- voltage circuits. The forced components of the capacitive voltage and current are determined from the final steady state at and equal to:

. (7.31)

Then the system of equations (7.30) for determining the constants of integration takes the form

(7.32)

The solution of system (7.32) gives:

; (7.33)

. (7.34)

As a result of the substitution of forced components and constants A 1 and A 2 in expressions for transient voltage u C(t) (7.28) and current i(t) (7.29) we get:

; (7.35)

since according to the Vieta theorem .

Knowing the transient current, we write the transient voltages:

;

. (7.37)

Depending on the type of roots, three variants of the transient process are possible.

1. When the transient process is aperiodic, then

On fig. 7.14, a, b curves and their components are given; in fig. 7.14, v curves , , are presented on the same plot.

As follows from the curves (Fig. 7.14, v), the current in the circuit increases smoothly from zero to a maximum, and then smoothly decreases to zero. Time t 1 reaching the maximum current is determined from the condition . The current maximum corresponds to the inflection point of the capacitive voltage curve ( ) and zero inductive voltage ( ).

The voltage at the moment of switching increases abruptly up to U 0 , then decreases, passes through zero, changes sign, increases in absolute value to a maximum and decreases again, tending to zero. Time
me t 2 reaching the maximum voltage across the inductance is determined from the condition . The maximum corresponds to the inflection point of the current curve, since .

In the area of ​​current growth (), the self-induction EMF, which prevents growth, is negative. The voltage expended by the source to overcome the EMF, . In the area of ​​​​decreasing current () EMF, and the voltage that balances the EMF,.

2. When occurs in the circuit ultimate (borderline)mode aperiodic transition process; curves , and are similar to the curves in Figs. 7.14, the nature of the process does not change.

3. When the circuit occurs periodic(oscillatory) a transient process when

where - resonant frequency, on which RLC-circuit will resonance.

Substituting conjugate complexes into the equation for capacitive voltage (7.35), we obtain:

Substituting the conjugate complexes into the equation for the current (7.36), we obtain:

Substituting the complexes into (7.37), we obtain for the voltage across the inductance

To plot dependencies , , it is necessary to know the period of natural oscillations and time constant .

On fig. 7.15 shows the curves , and for a sufficiently large constant . The construction order is as follows: first, envelope curves are built (dashed curves in Fig. 7.15) on both sides of the final steady state. Given the initial phase on the same scale as t, postpone quarters of the period in which the sinusoid reaches a maximum or vanishes. The sinusoid is inscribed in the envelopes in such a way that it touches the envelopes at the maximum points.

As follows from the curves u C(t), i(t) and u L(t), the capacitive voltage lags the current in phase by a quarter of a period, and the inductive voltage leads the current by a quarter of a period, being in antiphase with the capacitive voltage. Zero inductive voltage ( ) and the inflection point of the capacitive voltage curve ( ) correspond to the maximum current./Maximum inductive voltage corresponds to the inflection point of the current curve ( ).

Current i(t) and voltage u L(t) perform damped oscillations around zero, the voltage u C(t) - about the steady state U 0 . The capacitive voltage in the first half of the period reaches its maximum value, not exceeding 2 U 0 .

When ideal oscillatory circuit w

called logarithmic damping decrement .

The ideal oscillatory circuit corresponds to .

Consider transient processes in RLC circuits using the example of a series oscillatory circuit circuit in Fig. 4.3, a, the losses in which will be taken into account by including a resistor R in the circuit.

Fig.4.3. RLC circuit (a) and transients in it (b) and (c).

Transient processes in a series oscillatory circuit under zero initial conditions. Set the key K to position 1, and connect the input action to the circuit. Under the action of the connected source u, current i will flow in the circuit, which will create voltages uR, uL, uC.

Based on Kirchhoff's second law, the following equation can be written for this circuit

.

Considering that we will have

. (4.34)

We will seek the general solution of equation (4.34) in the form of the sum of free uС s and forced uС pr components:

. (4.35)

The free component is determined by the solution of the homogeneous differential equation, which is obtained from (4.34) for u = 0

. (4.36)

Solution (4.36) depends on the roots of the characteristic equation, which is obtained from (4.36) and has the form

. (4.37)

The roots of this equation are determined only by the circuit parameters R, L, C and are equal to

, (4.38)

where α = R/2L is the attenuation coefficient of the circuit;

The resonant frequency of the circuit.

From (4.38) it can be seen that the roots p1 and p2 depend on the characteristic resistance of the circuit and can be:

for R > 2ρ real and distinct;

at R< 2ρ комплексно-сопряженными;

for R = 2ρ real and equal.

For R > 2ρ, the free component will be equal to:

. (4.39)

Let the input action u = U = const, then the forced component upr = U. Taking into account the expression (4.39) and that upr = U, expression (4.35) will take the form:

Knowing uC we find the current in the circuit

. (4.41)

To determine the integration constants A1 and A2, we write the initial conditions for uC and i at t = 0:

(4.42)

Solving the system of equations (4.42) we get:

;

Substituting A1 and A2 into equations (4.40) and (4.41) and taking into account that in accordance with (4.38) p1 p2=1/LC we will have:

; (4.43)

. (4.44)

Since then

. (4.45)

Graphs of changes in uС, i, uL in a series oscillatory circuit under the condition R > 2ρ are shown in fig. 4.3b).

Time points t1 and t2 are determined, respectively, from the conditions

; .

An analysis of the graphs described by expressions (4.43 - 4.45) shows that at R > 2ρ (with large losses) aperiodic processes occur in the circuit.

Consider the processes in the loop at R< 2ρ. В этом случае из (4.38) имеем:

where - frequency of free damped oscillations. The solution of equation (4.36) has the form

where A and θ are integration constants

Taking into account (4.47) and that upr = U we find the law of change in voltage across the capacitance

Under the action of uC, a current flows in the circuit

Assuming in (4.48) and (4.49) t = 0 and taking into account the commutation laws, we obtain

(4.50)

Solving the system of equations (4.50) we find

Substituting A in (4.48) and (4.49) and taking into account that we find equations describing changes in uС, i, uL in the circuit for the case R< 2ρ:

. (4.51)

. (4.52)

. (4.53)

The graph of voltage change uС, determined by expression (4.51) is shown in fig. 4.3b with a dotted line. It can be seen from the figure and expression (4.51) that if the series circuit has low losses (R< 2ρ), то при подключении к нему источника постоянного напряжения в контуре возникает затухающий колебательный процесс.

Transient processes in a series oscillatory circuit under non-zero initial conditions. Install the key K in the circuit fig. 4.3, and to position 2. In this case, the input action will be disconnected from the circuit and the circuit will close. Since the capacitor was charged to a voltage of uC = U before switching the circuit, then at the moment the circuit is closed, it will begin to discharge and a free transient will occur in the circuit.

If the condition R> 2ρ is satisfied in the contour, then the roots p1 and p2 in (4.38) will be real and different, and the solution of equation (4.36) will have the form

The voltage uC creates a current in the circuit

. (4.55)

To determine the integration constants A1 and A2, we set t = 0 and take into account that at the time of switching uC = U, i = 0, then from (4.54) and (4.55) we obtain

(4.56)

Solving the system of equations (4.56) we find

Substituting A1 and A2 in (4.54) and (4.55) we obtain the equations for voltage uC and current i in the circuit circuit

. (4.57)

. (4.58)

From expressions (4.57) and (4.58) it can be seen that when the input action is turned off from the circuit of the circuit, which has a large attenuation (R> 2ρ), an aperiodic discharge of the capacitance C occurs. The energy stored before the input action is turned off in the capacitance covering the heat losses in the resistor R and the creation of a magnetic field in the inductance L. Then the energy of the electric field of the capacitance WC and the magnetic energy of the inductance WL is consumed in the resistor R.

Let us find the law of change of voltage uC and current i in the circuit when the circuit has low losses, i.e. under the condition R< 2ρ. В этом случае корни р1 и р2 носят комплексно-сопряженный характер (4.46) и решение уравнения (4.36) имеет вид:

Under the action of uC, a current flows in the circuit

To determine the integration constants A and θ, we take into account that at the time of switching t = 0, uC = U, i = 0 and substituting these values ​​into (4.59) and (4.60) we obtain

(4.61)

Solving the system of equations (4.61) we find

Substituting A and θ into (4.59) and (4.60) and taking into account that we obtain equations that determine the law of voltage and current change in a low-loss circuit

(4.62)

Analysis of equations (4.62) shows that when the input action is disconnected from the circuit with low losses (R< 2ρ) в нем возникают затухающие колебания с частотой ωС, которая определяется параметрами R, L, C цепи. Графики изменения uC и i изображены на рис. 4.3,в.

The damping rate of a periodic process is characterized by the damping decrement, which is defined as the ratio of two neighboring current or voltage amplitudes of the same sign

. (4.63)

In logarithmic form, the damping decrement has the form

. (4.64)

It can be seen from (4.64) that the greater the losses in the circuit, which are determined by the value R, the greater the attenuation. For R ≥ 2ρ, the transient process in the circuit becomes aperiodic. At R = 0, a continuous harmonic oscillation takes place in the circuit with a frequency . In real circuits, R ≠ 0; therefore, damped oscillations take place in them.

Consider two cases of transients in series RLC circuits:

consistent RLC circuit connects to a source of constant E.D.S. E;

The pre-charged capacitor is discharged to RLC circuit.

1) When connecting a serial RLC circuits to the source of constant E.D.S. E(Fig. 6.3.a) the equation of electrical equilibrium of the circuit according to the second Kirchhoff law has the form:

U L +U R +U C =E (6.10)

taking into account the ratios

U R = R i=R C (dU C /dt);

U L \u003d L (di / dt) \u003d L C (d 2 U C / dt 2)

the equation (6.10) can be written as:

L C (d 2 U C /dt 2) + R C (dU C /dt) + U C = E (6.11)

a	b	v
Rice. 6.3

Solution of an inhomogeneous differential equation (6.11) is determined by the characteristic equation: LCp 2 +RCp+1=0,

which has roots

δ=R/2L - attenuation factor,

resonant frequency.

Depending on the ratio δ2 and ω 2, three main types of transient processes are possible:

a) δ 2 > ω 2 or The roots of the characteristic equation are negative real ones. The transient process has an aperiodic character (Fig. 6.3.b).

b) δ2< ω 2 or The roots of the characteristic equation are complex and conjugate. The nature of the transient process is oscillatory and damped (Fig. 6.3.c)

v) δ 2 \u003d ω 2 or The roots of the characteristic equation are real and equal p 1 \u003d p 2 \u003d -R / 2L. The nature of the transient process is aperiodic and fading (critical case). The transition process time is minimal.

For the first two cases, the solution of the equation has the form:

(6.13)

V=U C (0) - voltage across the capacitor at the moment of switching.

For the occasion δ2< ω 2 the equation (6.13) is brought to the form:

, (6.14)

- frequency of damped oscillations.

From the equation (6.14) it follows that the transition process U c (t) has the character of oscillations with an angular frequency ω and period T=2π/ω, which decay with the time constant τ=2L/R=1/δ.

To determine the value of the time constant τ you can use the envelope of the oscillatory curve Uc(t), having the form of an exponent:

exp(-δt)=exp(-t/τ).

For the third case δ=ω 0 solution of the equation (6.11) looks like:

. (6.15)

The peculiarity of this mode is that with a decrease R below the value, the transient process becomes oscillatory.

2. When the capacitor is discharged RL circuit(Figure 6.4.a) all three modes considered above are possible and are determined by the ratio of values δ and ω 0 . Transient processes in these modes are described by the equations (6.13), (6.14), (6.15) at E=0. For example, for the case δ<ω 0 the equation (6.14) with an oscillatory discharge of a capacitor, it has the form:

(6.16)

Transient Curve U c (t) shown in (Fig. 6. 4.b). envelope curve U c (t) is a function exp(-δt)=exp(-t/τ), which can be used to determine the time constant τ and damping factor δ=1/τ.

Laboratory work

Communication, communication, radio electronics and digital devices

The solution of such an equation depends on the form of the roots of the characteristic equation. The roots of the equation are determined only by the parameters of the circuit. Calculation part For the electrical circuit shown in fig. Connecting the RLC circuit to a constant voltage source U at time t = 0 Determine: at what values ​​of R the transient process is aperiodic; at what values ​​of R the transient process is oscillatory; frequency ωС of natural damped oscillations for those values ​​of R for which the transient process is oscillatory ...

Lab #14

study of transient processes in rcL chains

If there are two independent energy storage devices in the circuit, transient processes are described by second-order equations of the type

The solution of such an equation depends on the form of the roots of the characteristic equation

The roots of the equation are determined only by the parameters of the circuit

The value α is called the damping coefficient of the circuit, and ω 0 - resonant frequency of the circuit.

The nature of the transition process essentially depends on the type of roots R 1 and R 2 , which can be:

real and different ( R > 2p);

real and equal ( R = 2p);

complex conjugate ( R< 2 ρ ).

Here is the characteristic impedance of the circuit.

Settlement part

For the electrical circuit shown in fig. 1, given:

coil inductance L;

capacitor capacitance C;

resistor resistance R.

Rice. 1. Connection RLC -circuit to a constant voltage source U

at the time t = 0

Define:

at what values R , the transient process is aperiodic;

at what values R , the transient process is oscillatory;

frequency ω С own damped oscillations for those values R , for which the transient process is oscillatory

quasi-period Т С own damped oscillations

Table 1

Determining the nature of the transition process in RLC circuits

Combination elements	C, nF	L , mH	R , Ohm	2ρ, Ohm	Character process	T C , µs
			1000
			2000
			5000

experimental part

In the experimental part, you must:

• observe voltage waveforms on the elements RLC - circuits in the process of charging and discharging a capacitor with different ratings of circuit elements;
• determine the influence of the values ​​of the circuit elements on the nature of the transient process.
• compare experimental results with calculated ones.

Prepare the laboratory setup for observing capacitor voltage waveforms. The schematic diagram of measurements is shown in fig. 2.

Rice. 2. Schematic diagram of voltage oscilloscope

on the RLC capacitor

In laboratory work, the transient process is studied using an electronic oscilloscope, so the process is periodically repeated. This is achieved by the fact that not a single voltage jump is applied to the input of the circuit from the output of the generator, but a periodic sequence of positive pulses (see "Technical description of the laboratory installation"). With a positive voltage jump (positive pulse), the capacitor is charged. With a negative voltage jump (pause between pulses), the capacitor is discharged.

The connection diagram of the installation elements for the combination of elements No. 1 is shown in fig. 3.

Rice. 3. Connection diagram of the elements of the installation for oscillography

voltage on the capacitor (C \u003d 10 nF; L = 10 mH; R = 200 Ohm)

Turn the output voltage regulator of the pulse generator counterclockwise until it stops. Present the completed diagram to the teacher. After checking the assembled circuit by the teacher, turn on the installation.

Power on the oscilloscope. Oscilloscope operation mode:

• two-channel with simultaneous voltage indication of both channels;
• entrance 1 – open; sensitivity 0.2 V / division;
• entrance 2 – open; 0.2 V / division;
• synchronization - external (connection to sockets on the left side surface of the laboratory module)
• sweep duration 0.2 ms/div.

When initially setting up the zero voltage lines of both channels, align and install in the center of the screen.

Turn on the pulse generator. Set the pulse amplitude regulator to the middle position. Obtain a stable image of the voltage waveform at the output of the pulse generator on the oscilloscope screen.

By adjusting the duration, set the duration of the positive pulses to 500 µs (pulse repetition period 1000 µs). Set the pulse amplitude to 1 volt. In the future, keep this value unchanged.

Sketch in the common axes the voltage oscillograms ("osc. No. 1") at the output of the generator and on the capacitor. Determine the nature of the transition process. If the transient process is oscillatory, determine the quasi-period T WITH own damped oscillations. Compare with the result obtained in the calculation part of the laboratory work. If necessary, adjust the sensitivity of the oscilloscope inputs.

Turn on the pulse generator. Sketch in the common axes the voltage oscillograms ("osc. No. 2") at the output of the generator and on the capacitor. Determine the nature of the transition process. If the transient process is oscillatory, determine the quasi-period T WITH

Prepare the laboratory setup for observing transient current waveforms in RLC circuits.

The schematic diagram of measurements is shown in fig. 4.

Rice. 4 . Schematic diagram of current oscilloscope

transition process in RLC circuits

The connection diagram of the installation elements for the combination of elements No. 1 is shown in fig. 5.

Rice. 5 . Connection diagram of the elements of the installation for oscillography

current in the circuit (C \u003d 10 nF; L = 10 mH; R = 200 Ohm)

Turn on the pulse generator. Draw waveforms of the current in the circuit. Draw the figure in the same axes as waveforms No. 1 of the voltages at the output of the generator and on the capacitor. Determine the nature of the transition process. If the transient process is oscillatory, determine the quasi-period T WITH own damped oscillations. Compare with the result obtained in the calculation part of the laboratory work.

Turn off the pulse generator. Replace the elements on the panel of the laboratory module (see combination No. 2 according to table 1).

Turn on the pulse generator. Draw waveforms of the current in the circuit. Draw the figure in the same axes as waveforms No. 2 of the voltage at the output of the generator and on the capacitor. Determine the nature of the transition process. If the transient process is oscillatory, determine the quasi-period T WITH own damped oscillations. Compare with the result obtained in the calculation part of the laboratory work.

And so on. Make observations and record the results of the experiment for combinations Nos. 3-7.

Turn off the pulse generator.

Turn off the laboratory setup.

Control questions

1. What are the causes of transient processes?
2. What mode of operation is called steady state?
3. What is a transitional process?
4. What is the physical meaning of the time constant τ?
5. What process in a circuit is called aperiodic?
6. What process in a circuit is called oscillatory?
7. How are the frequency and period of free oscillations determined?
8. Why does the amplitude of free oscillations of the circuit decrease?
9. What is the logarithmic damping factor?
10. What is the maximum voltage across the capacitor during charging?
11. Formulate the laws of commutation.
12. What are zero and non-zero initial conditions?
13. What form does the free component of transients have in second-order circuits?
14. What is the forced component?

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64158.		Statistical processing modules of the Tenzotrem analyzer	5.01MB
	The purpose of the work is the research and development of software modules for statistical processing of measuring information of a tensometric tremorograph. The object of study is a tensometric tremorograph. Tensometric tremorograph is designed to assess the activity of the human motor system...
64159.		Development of test tasks and an automated testing system for rechecking and evaluating current knowledge of students in the disciplines “Informatics. Calculating Mathematics and Programming" and "Computer Merezhі"	1.44MB
	The choice of computers for knowledge control is economically viable and ensures the improvement of the efficiency of the initial process. Yak means I. Bulakh, computer testing of success makes it possible to implement the main didactic principles of learning control: the principle of the individual nature of the re-verification and assessment of knowledge...
64160.		Development and research of an accelerated algorithm for calibrating large network models by the clustering coefficient	1.56MB
	The aim of the work is to study algorithms for generating random graphs, develop a new algorithm, implement it, and conduct the necessary tests. The paper outlines the necessary concepts from the theory of random graphs, analyzes in detail the methods for generating Barabashi-Albert, Erdős-Renyi, Wats-Strogats graphs...