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

Transient processes in oscillatory circuits are described by 2nd order differential equations. Let's consider the case of a capacitance discharge on an RL circuit (Fig. 2). Let's compose the chain equation according to Kirchhoff's first law:

After differentiation (1) we get

Rice. 2.

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

U c =U St +U Ave. (3)

U pr depends on E, and U st (t) is determined by solving 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 R/2L = 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 = 2c, Q = 0.5;

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

Here is the characteristic impedance, Q = c/R is the quality factor of the circuit.

In the diagram 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 a system of equations

From the solution of the system we have

As a result, for the current and voltage 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)=i st (t)+i pr

Let us determine the initial conditions.

IL(0)=E/R=19.799A

Let's write down the solution to the differential. equations for the free component.

i st (t)=A*e bt *sin(wt+i)

Z input =2R+jwL+1/jwC

p=-883.833-7.016i*10 3

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

T=2р/w=8.956*10 -4

Let's determine the forced components at t=?

Let us determine the constant of integration Ai and

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

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

LAb*sin and+ LAw*cosand =0

p Acos u=2.494

tg and=19.799/Acos and=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 use of the symbolic method for calculating alternating current circuits becomes impossible. For linear electrical circuits, the calculation problem can be solved based on the superposition method using the spectral decomposition of non-sinusoidal voltages and currents into 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 u 1 = 2p/T of a periodic current or voltage with period T, and a set of higher harmonics with frequencies u n = n 1, multiples of the fundamental frequency u 1. For most periodic functions, the Fourier series contains an infinite number of terms. In practice, they are limited to a finite number of terms of the series. In this case, the original periodic function will be represented using a Fourier series with some error.

Let there be a periodic emf with period T. e(t)=e(t±nT), satisfying the Dirichlet conditions (a 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 u n =n 1 and initial phases u n in the form of a Fourier series

The Fourier series can be represented in another form:

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

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

The diagram showing the dependence of the amplitude of the harmonics E n on the frequency u n = n u 1 is called a spectrum.

Using the superposition method and the spectral representation of the periodic emf. in the form of a Fourier series, the electrical 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 emf harmonic are calculated.

3. The required 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 , well-known methods for calculating circuits of direct and alternating sinusoidal currents are used to determine them.

Laboratory work No. 4

Purpose of the work: study of transient processes in RLC circuits under the influence of rectangular voltage pulses.

One of the methods for studying transient processes in electrical circuits is the operator method /1,2/. In this case, the Laplace transform is used:

defining image F(p) from the known original f(t) .

The solution of the integro-differential equation of the chain with respect to the desired time function (original) is reduced to the solution of the algebraic equation for the image.

1. RC - circuit

Let a rectangular voltage pulse be applied to the input of the circuit, the diagram of which is shown in Fig. 1, a. It is required to find the voltage shape at the input of the circuit. To do this, it is necessary to perform the following calculation steps:

1) write down the analytical expression of the input signal;

2) compose an integro-differential equation of the circuit;

3) go to the operator equation;

4) having solved the operator equation, find the image of the desired function;

5) go to the original of the required function.

We write the analytical expression for an ideal rectangular voltage pulse of amplitude E in the form.

where l(t) is a unit function determined by the conditions:

l(t)=0 if t<0 и l(t)=1, если t>=0.

Expression (2) is presented graphically in Fig. 1, b. For t>t u the difference of the unit functions gives zero. The chain equation is

where the input effect U(t) is determined by expression (2), U R (t) and i(t) are the voltage on the capacitor and the current in the circuit at an arbitrary point in time. The output voltage U R =i(t)R coincides with i(t) up to a factor R, so let’s choose i(t) as the desired function and take into account that i(t)=dq(t)/dt=CdU C ( t)/dt. Then (3), taking into account (2), will take the form

Let's introduce the image of the current I(p)=a and apply the Laplace transform (1) to both parts (4). Taking into account the image of the unit function and the integration theorem of the original, the operator equation takes the form

Solving it

The transition to the original is also carried out using Table 1:

Table 1

Some properties of the Laplace transform

No. Property

Graphically, dependence (7) is presented in Fig. 1c for the case t<

Consider the circuit in Fig. 2, a. To obtain the dependence U c (t) under input action (2), we present equation (3) as follows:

Introducing the voltage image U c (p) = a, using Table 1 we proceed to the operator equation:

where it is taken into account that U c (0)=0. Solving (9) for U c (p) and passing to the original, we obtain

This dependence is presented graphically in Fig. 2c.

Thus, as follows from expressions (7) and (10) (see Fig. 1, c; 1, d; 2, c), the leading and trailing edges of the input P-voltage pulse cause a transient process in the RC circuit. At the leading edge, a capacitor is charged over time (an increase in U c (t)), and the current i(t) decreases to zero as the capacitor is charged. When exposed to the trailing edge of the pulse, the capacitor begins to charge through the resistor and the input signal source. The current flows in the opposite direction and gradually decreases in absolute value. This is associated with the appearance of a negative surge U R (t) on the oscillogram. Transition time, i.e. the time it takes for the capacitor to charge to the source voltage E is theoretically infinite. In practice, the duration of the transient process in RC circuits is characterized by the time constant t=RC, which shows over what period of time the current in the circuit decreases by e times (from (7) at t=t i=0.367(E/R)) or - for what period of time will the voltage on the capacitor reach 0.633 E (from (10)) at t=t U c =(1-e -1)E=0.633E). When estimating t from the oscillogram U c (t), the condition t must be met<

oscillograms U R (t) and U C (t) will have the form shown in Fig. 1, e and 2, d.

Let's consider an RL circuit, the circuit of which is shown in Fig. 3a, for which the input voltage

U(t)=i(t)R+U L (t) (11)

Or taking into account (2) and U L (t)=L di(t)/dt

Comparing (12) and (4), we note that these equations coincide with the mutual replacement of the sought functions and the introduction of a time constant t=R/L for the RL circuit, therefore we write the solution to (12) by analogy with (7):

where t=L/R. The voltage shape U L (t) for the RL circuit repeats the voltage shape U R (t) for the RL circuit (Fig. 3). Similarly, it can be shown that the shape of U R (t) for the RL circuit repeats the shape of U C (t) for the RC circuit (Fig. 4). To do this, it is enough to obtain an equation for l(t) from (11) and compare it with (8).

The transient process in the RL circuit at the leading and trailing edges of the input pulse is determined by the extent of the process of accumulation and dissipation of magnetic field energy in the coil.

In radio electronics, circuits are used whose input voltage is proportional to the derivative or integral of the input voltage. Such chains are called differentiating or integrating, respectively. The circuits whose circuits are shown in Fig. 1 and 3 are differentiating if their time constants are sufficiently small (compared to the duration of the input signal). Integrating circuits are the circuits of which are shown in Fig. 2. And 4, if their time constants are large enough (compared to the integration interval). To do this, the output voltage must be chosen significantly less than the output voltage.

3. RLC circuit.

Let's consider the circuit, the diagram of which is shown in Fig. 5, a. To simplify the calculation, consider the effect of a positive voltage step on the circuit, i.e. We choose the input action in the form U(t)=E l(t). Then the equation U(t)=U R (t)+U L (t)+U C (t), written relative to U C (t), will take the form

Passing to the operator equation for the image and solving it, we find

Roots P 1,2 =

The equations p 2 +(r/L)p+1/LC=0 can be complex, real (equal in a particular case), therefore, they distinguish between oscillatory, aperiodic and critical modes of operation of the circuit. Provided (l/LC)>R 2 /4L 2 we have an oscillatory circuit. Then, assuming p 1 = -s ± jw, where s = R/2L is the damping coefficient of the circuit, is the circular frequency of free (natural) oscillations, is the resonant frequency of the circuit, we rewrite (15) as follows:

The roots of the denominator in (16) are simple, therefore, applying the expansion theorem (see Table 1) and considering the damping to be small, i.e. w=w 0, we have

This shows that the current in the circuit and the voltage on the capacitor oscillate, and the amplitude of the oscillations monotonically decreases, which is typical for a transient process in an oscillatory circuit.

4. Practical part

1. Familiarize yourself with the equipment (rectangular voltage pulse generator, oscilloscope, breadboard).

2.Assemble the RC circuit. Using an oscilloscope, view and sketch the waveforms of the input voltage pulse and the voltage pulses across the resistor and capacitor. Using the oscillograms, estimate the circuit time constant t and compare it with the product RC, where R C are the nominal values ​​of the parameters of the elements.

3. Complete task step 2 for cases when the same RC circuit is acted upon by rectangular voltage pulses of different durations and a pulse with t u =const acts on an RC circuit whose time constant changes due to changes in both R and C .Consider cases t<t u . For the case t<

4. Complete the tasks of points 2 and 3 applicable to RL circuits. For the case t<

5. Assemble a serial RLC circuit. Using an oscilloscope, view and sketch the shapes of the input voltage pulse and voltage pulses on the circuit elements. Using voltage oscillograms on the circuit elements, observe the transition from aperiodic to oscillatory when the attenuation coefficient changes

In the oscillatory mode, estimate the oscillation period T and compare it with the calculated value. Register the dependence of T on capacitance C at .

6. Discuss the results obtained.

5. Test questions

1. What is a transient process in an electrical circuit?

2. How do we estimate the duration of the transition process?

3. What is the time constant of an electrical circuit?

4. What expressions describe the dependence of the voltages on the elements of the RC and RL circuits on time, if the input action is a rectangular voltage pulse?

5. How to estimate the time constant of an electrical circuit from a voltage oscillogram at a circuit element?

6. Is it possible to estimate t from the oscillogram in Fig. 2d using the transition edge of the pulse?

7. Are the circuit time constants estimated from the leading and falling edges of the pulse always the same?

8. What physical processes occur in RC and RL circuits when exposed to a rectangular voltage pulse?

9. Why does an oscillatory process occur in an RLC circuit with a rectangular pulse at the input?

10. How can the oscillograms l(t) and U c (t) in Fig. 5 be qualitatively explained?

11. How do the oscillograms i(t) and U c (t) in Fig. 5 change when the parameters of the oscillatory circuit change?

Ginzburg S.G. Methods for solving problems on transient processes in electrical circuits. – M.: Higher school, 1967.-388 p.

Mathanov P.N. Basics of electrical circuit analysis. Linear circuits. – M.: Higher school, 1981. – 334 p.

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

The equation of state of the circuit satisfies Kirchhoff's second 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 chain capable of storing energy. Dividing both sides of the equation by the coefficient L.C. with a higher order derivative, we find the equation of the transition process:

, (7.17)

the general solution of which consists of the sum of two terms:

The forced component is determined by the type of applied voltage. When the circuit is turned on for 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 using the symbolic method, and then we move 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 possible transition process options:

- aperiodic, when transient currents and voltages 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: ; The 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 limit mode corresponds to the general solution of the homogeneous equation (7.18) in the form

; (7.23)

- periodic, or oscillatory , when transient currents and voltages approach the final steady state, periodically changing sign and decaying in time along a sinusoid. Occurrence condition:

. (7.24)

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

Where α - attenuation coefficient:

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

. (7.26)

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

The general solution to equation (7.18) for an oscillatory transient process has the form

Where A And γ - integration constants determined from the initial conditions.

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

From the initial conditions

(7.30)

let's define integration constants A 1 and A 2 .

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

. (7.31)

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

(7.32)

Solving system (7.32) gives:

; (7.33)

. (7.34)

As a result of substituting forced components and constant A 1 and A 2V expressions for transient voltages uC(t) (7.28) and current i(t) (7.29) we get:

; (7.35)

since according to Vieta’s theorem .

Knowing the transition current, we write the transition voltages:

;

. (7.37)

Depending on the type of roots, three options for the transition process are possible.

1. During the transient process - aperiodic, Then

In Fig. 7.14, A, b curves and their components are shown; in Fig. 7.14, V curves , , are presented on one graph.

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

The voltage at the moment of switching increases abruptly 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 section of current growth (), the self-induction emf, which prevents growth, is negative. The voltage spent by the source to overcome the EMF is . In the section where the current decreases (), the emf is , and the voltage that balances the emf is .

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

3. When in the circuit occurs periodic(oscillatory)transition process when

Where - resonant frequency, on which in RLC- the circuit will resonate.

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

Substituting the conjugated 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 construct dependencies , , it is necessary to know the period of natural oscillations and time constant .

In Fig. 7.15 shows the curves for a sufficiently large constant. The construction order is as follows: first, envelope curves are constructed (dashed curves in Fig. 7.15) on both sides of the final steady state. Considering the initial phase on the same scale as t, The quarter periods in which the sine wave reaches a maximum or goes to zero are set aside. 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), capacitive voltage lags behind the current in phase by a quarter of a period, and 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./The 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 value, voltage u C(t) – about steady 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 .

Corresponds to an ideal oscillatory circuit.

Transient processes in the circuit R, L, C are described by a 2nd order differential equation. The steady-state components of currents and voltages are determined by the type of energy source and are determined by known methods for calculating steady-state conditions. The free components are of greatest theoretical interest, since the nature of the free process turns out to be significantly different depending on whether the roots of the characteristic equation are real or complex conjugate.

Let us analyze the transient process in the circuit R, L, C when it is connected to a source of constant EMF (Fig. 70.1).

General form of the solution for current: i(t)=iy(t)+iсв(t)=Iy+A1ep2t+A2ep2t

Steady-state component: Iy=0

Characteristic equation and its roots:

Differential equation:

Independent initial conditions: i(0)=0; uc(0)=0.

Dependent initial condition:

The integration constants are determined from the simultaneous solution of the system of equations:

Final solution for current:

Let us study the form of the function i(t) for different values ​​of the roots of the characteristic equation.

a) The roots of the characteristic equation are real and not equal to each other.

This is the case provided:

As t changes from 0 to ∞, the individual functions ep1t and ep2t decrease exponentially from 1 to 0, and the second of them decreases faster, while their difference ep1t - ep2t ≥ 0. From this it follows that the desired stream function i(t ) at the extreme points at t = 0 and at t = ∞ is equal to zero, and in the time interval 0< t < ∞ - всегда положительна, достигая при некотором значении времени tm своего максимального значения Imax. Найдем этот момент времени:

A graphical diagram of the function i(t) for the case of real roots of the characteristic equation is shown in Fig. 70.2.

The duration of the transition process in this case is determined by the smaller root: Tп=4/|pmin|.

The nature of the transition process with real roots of the characteristic equation is called damped or aperiodic.

b) The roots of the characteristic equation are complex conjugate.

This occurs when the parameters are:

attenuation coefficient:

angular frequency of natural vibrations:

The solution for the original function can be converted to another form:

Thus, in the case of complex conjugate roots of the characteristic equation, the desired function i(t) changes in time according to the harmonic law Imsinω0t with a damped amplitude Im(t)=A·e-bt. A graphical diagram of the function is shown in Fig. 70.3.

The oscillation period is T0=2π/ω0, the duration of the transition process is determined by the attenuation coefficient: Tп=4/b.

The nature of the transition process with complex conjugate roots of the characteristic equation is called oscillatory or periodic.

In the case of complex conjugate roots, a particular form is used to determine the free component:

where the coefficients A and ψ or B and C are new integration constants, which are determined through the initial conditions for the desired function.

c) The roots of the characteristic equation are real and equal to each other.

This is the case provided:

The previously obtained solution for the desired function i(t) in this case becomes uncertain, since the numerator and denominator of the fraction become zero. Let's reveal this uncertainty using L'Hopital's rule, considering p2=p=const, and p1=var, which tends to p. Then we get:

The nature of the transition process with equal roots of the characteristic equation is called critical. The critical nature of the transition process is borderline between damped and oscillatory and in form is no different from a damped one. Duration of the transition process Tп=4/p. When changing only the resistance of the resistor R=var=0…∞, the damped nature of the transient process corresponds to the range of values ​​Rvar (Rkp< Rvar < ∞), колебательный характер - также области значений (0 < Rvar < Rkp), а критический характер – одной точке Rvar = Rкр. Поэтому на практике случай равных корней характеристического уравнения встречается крайне редко.

In the case of equal roots, a particular form is used to determine the free component:

where coefficients A1 and A2 are new integration constants, which are determined through the initial conditions for the desired function.

The critical mode of the transition process is characterized by the fact that its duration is minimal. This property is used in electrical engineering.

Let us consider two cases of transient processes in a sequential RLC circuits:

sequential RLC circuit connects to a source of constant E.M.F. E;

The pre-charged capacitor is discharged by RLC circuit.

1) When connecting serial RLC circuits brush of constant E.M.F. E(Fig. 6.3.a) the equation of electrical equilibrium of the circuit according to Kirchhoff’s second 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 =L (di/dt)=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 coefficient,

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. The transition process is aperiodic in nature (Fig. 6.3.b).

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

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

For the first two cases, the solution to 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 reduced to the form:

, (6.14)

- frequency of damped oscillations.

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

To determine the time constant τ you can use the envelope of the oscillatory curve U c (t), having an exponential form:

exp(-δt)=exp(-t/τ).

For the third case δ=ω 0 solution to the equation (6.11) has the form:

. (6.15)

The peculiarity of this mode is that when decreasing R Below this value the transient process becomes oscillatory.

2. When the capacitor discharges to RL circuit(Figure 6.4.a) all three modes are possible, discussed above and determined by the ratio of quantities 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 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 attenuation coefficient δ=1/τ.