Time and frequency characteristics of linear electrical circuits. Calculation of time characteristics of linear electrical circuits Calculation of response to a given input influence

The time characteristic of a circuit is a function of time, the values of which are numerically determined by the response of the circuit to a typical impact. The reaction of a circuit to a given typical impact depends only on the circuit diagram and the parameters of its elements and, therefore, can serve as its characteristic. Time characteristics are determined for linear circuits, not containing independent energy sources, and under zero initial conditions. Temporary characteristics depend on the type of specified typical impact. Due With This divides them into two groups: transient and impulse time characteristics.

Transition characteristic or transition function, is determined by the response of the circuit to the influence of a single step function. It has several varieties (Table 14.1).

If the action is given in the form of a single voltage jump and the reaction is also voltage, then the transient characteristic turns out to be dimensionless, numerically equal to the voltage at the output of the circuit and is called the transient function or transfer coefficient KU(t) by voltage. If the output quantity is current, then the transition characteristic has the dimension of conductivity, is numerically equal to this current and is called transition conductivity Y(t). Similarly, when acting in the form of a current and reacting in the form of a voltage, the transition function has the dimension of resistance and is called the transition resistance Z(t). If the output quantity is current, then the transition characteristic is dimensionless and is called the transition function or transfer coefficient K I (t)no current

In general, a transition characteristic of any type is denoted by h(t). Transient characteristics are easily determined by calculating the circuit's response to a single step action, i.e., calculating the transient process when the circuit is switched on constant pressure 1 V or per D.C. 1 A.

Example 14.2.

Find temporary crossings O These characteristics of a simple rC circuit (Fig. 14.9, a), if in O The effects are stresses.

1. To determine the transient characteristics, we calculate the transient process when a voltage is applied to the input of the circuit u(t) - 1 (t). This corresponds to the switching on of the circuit at the moment t=0 to a source of constant e. d.s. e 0 =1 IN(Fig. 14.9,6). Wherein:

a) the current in the circuit is determined by the expression

therefore the transition conductivity is

b) voltage across the capacitance

therefore the voltage transition function

Pulse the characteristic, or impulse transient function, is determined by the response of the circuit to the influence of the δ(t) function. Like the transient characteristic, it has several varieties, determined by the type of impact and reaction - voltage or current. In general, the impulse response is denoted by a(t).

Let us establish a connection between the impulse response and the transient response of a linear circuit. To do this, we first determine the response of the circuit to a pulsed action of short duration t И =Δt, representing it by superimposing two step functions:

In accordance with the superposition principle, the response of the circuit to such an impact is determined using transient characteristics:

For small Δt we can write

Where S and =U m Δƒ- impulse area.

At Δt 0 and Um the resulting expression describes the reaction of the chain to the δ(t)-function, t . e, determines the impulse response of the circuit:

Taking this into account, the response of a linear circuit to a pulse of short duration can be found as the product of the pulse function and the pulse area:

This equality underlies the experimental determination of the impulse function. The shorter the pulse duration, the more accurate it is.

Thus, the impulse response is the derivative of the step response:

It is taken into account here that h(t)δ(t)=h(0)δ(t), and multiplication h(t) on l(t) is equivalent to indicating that the value of the function h(t) at t<0 равно нулю.

By integrating the resulting expressions, it is easy to verify that

Equalities (14.17) and (14.19) are a consequence of equalities (14.14) and (14.15). Since impulse characteristics have the dimension of the corresponding transient response divided by time. To calculate the impulse response, you can use expression (14.19), i.e., calculate it using the transient response.

Example 14.3.

Find the impulse characteristics of a simple rC circuit (see Fig. 14.9, a). Solution.

Using the expressions for the transient characteristics obtained in Example 14.2, using O Using expression (14.19) we find the impulse characteristics;

The timing characteristics of typical links are given in Table. 14.2.

Calculation of timing characteristics is usually carried out in the following order:

the points of application of the external influence and its type (current or voltage), as well as the output value of interest - the reaction of the circuit (current or voltage in some section of it) are determined; the required time characteristic is calculated as the response of the circuit to the corresponding typical impact: 1(t) or δ(t),

MINISTRY OF EDUCATION OF UKRAINE

Kharkov State Technical University of Radio Electronics

Settlement and explanatory note

for course work

in the course “Fundamentals of Radio Electronics”

Topic: Calculation of frequency and time characteristics of linear circuits

Option No. 34

INTRODUCTION	3
EXERCISE	4
1 CALCULATION OF THE COMPLEX INPUT RESISTANCE OF THE CIRCUIT	5
1.1 Determination of the complex input impedance of a circuit	5
1.2 Determination of the active component of the complex input resistance of the circuit	6
1.3 Determination of the reactive component of the complex input resistance of the circuit	7
1.4 Determination of the module of the complex input impedance of the circuit	9
1.5 Determination of the argument of the complex input resistance of the circuit	10
2 CALCULATION OF CIRCUIT FREQUENCY CHARACTERISTICS	12
2.1 Determination of the complex transmission coefficient of the circuit	12
2.2 Determination of the amplitude-frequency response of the circuit	12
2.3 Determination of the phase-frequency characteristics of the circuit	14
3 CALCULATION OF CIRCUIT TIMING CHARACTERISTICS	16
3.1 Determination of the transient response of a circuit	16
3.2 Determination of the impulse response of a circuit	19
3.3 Calculation of the circuit response to a given impact using the Duhamel integral method	22
CONCLUSIONS	27
LIST OF SOURCES USED	28

INTRODUCTION

Knowledge of fundamental basic disciplines in the preparation and formation of a future design engineer is very great.

The discipline “Fundamentals of Radio Electronics” (FRE) is one of the basic disciplines. By studying this course, you acquire theoretical knowledge and practical skills in using this knowledge to calculate specific electrical circuits.

primary goal course work– consolidation and deepening of knowledge in the following sections of the electronics training course:

calculation of linear electrical circuits under harmonic influence using the complex amplitude method;

frequency characteristics of linear electrical circuits;

timing characteristics of circuits;

methods for analyzing transient processes in linear circuits (classical, superposition integrals).

Course work consolidates knowledge in the relevant field, and those who do not have any knowledge are encouraged to obtain it by a practical method - by solving assigned problems.

Option No. 34

R1, Ohm	4,5	t1, μs	30
R2, Ohm	1590	I1, A	7
R3, Ohm	1100
L, µH	43
C, pF	18,8
Reaction

1. Determine the complex input resistance of the circuit.

2. Find the module, argument, active and reactive components of the complex resistance of the circuit.

3. Calculation and construction of frequency dependences of the module, argument, active and reactive components of the complex input resistance.

4. Determine the complex transmission coefficient of the circuit, plot graphs of amplitude-frequency (AFC) and phase-frequency (PFC) characteristics.

5. Determine the transient response of the circuit using the classical method and construct its graph.

6. Find the impulse response of the circuit and plot it.

1 CALCULATION OF THE COMPLEX INPUT RESISTANCE OF THE CIRCUIT

1.1 Determination of the complex input impedance of a circuit

(1)

After substituting the numerical values we get:

(2)

Specialists who design electronic equipment. Course work in this discipline is one of the stages of independent work, which allows you to determine and study the frequency and time characteristics of election circuits, establish a connection between the limiting values of these characteristics, and also consolidate knowledge of spectral and time methods for calculating the response of the circuit. 1. Calculation...

T, μs m=100 1.982*10-4 19.82 m=100000 1.98*10-4 19.82 The timing characteristics of the circuit under study are shown in Fig. 6, Fig. 7. Frequency characteristics are shown in Fig. 4, fig. 5. TIME METHOD OF ANALYSIS 7. DETERMINING THE RESPONSE OF A CIRCUIT TO AN IMPULSE Using the Duhamel integral, you can determine the response of a circuit to a given impact even in the case when an external impact on...

MILITARY
ACADEMY
CONNECTIONS
2 department
PRACTICAL LESSON
by academic discipline
"Electronics, electrical engineering and circuit engineering"
Topic No. 4 Mode of non-harmonic influences in
linear electrical circuits
Lesson No. 17 “Calculation of time characteristics
linear electrical circuits"
Saint Petersburg

STUDY QUESTIONS:
1. Analysis of time characteristics of linear
electrical circuits.
2. Monitoring the assimilation of the studied material.
LITERATURE:
Babkova L.A., Kiselev O.N. Methodological recommendations for
practical exercises and guidance for laboratory work on
discipline “Fundamentals of Circuit Theory”: Textbook. – St. Petersburg: VAS, 2011.
2. Ulakhovich D.A. Fundamentals of the theory of linear electrical circuits:
Textbook. – St. Petersburg: BHV-Petersburg, 2009.
1.

Problem 1

1. Analysis of time characteristics of linear
electrical circuits.
Problem 1
Find the impulse and transient characteristics of the electrical
low-pass filter with the most flat frequency response, if known
Transmission function:
1
H(p)2
.
p 2 p 1

1
h(p)H(p).
p
h(p)
1
p(p 2 p 1)
2
.

2. Define the image of the impulse response:
g(p)H(p).
Thus the impulse response image will be
look like:
g(p)
1
p 2 p 1
2
.
Using the correspondence table, we determine the graphic
image of transient and impulse characteristics:

Step response
h(p)
1
p(p 2 2 p 1)
Fig1. Graph f(t)
A
p(p 2 α1 p α2)

Impulse response

g(p)
1
p2 2 p 1
A
p 2 α1 p α2

Problem 2

Find the impulse and transient characteristics of the circuit, if known
its transfer function:
181.8p
H(p)2
p 1091 p 1,818 106
1. Define the image of the transient response
1
h(p) H(p)
p
2. Define the image of the impulse response:
g(p)H(p).
181.8p
g(p)2
p 1091 p 1,818 106

Step response
181,1
h(p) 2
p 1091 p 1,818 106
A
2
p α1 p α2

Impulse response

181.8p
g(p)2
6
p 1091 p 1.818 10
Ap
p 2 α1 p α2

Task 3 Determine the transient and impulse characteristics of a circuit consisting of elements R and C connected in series.

1. Let's find the transfer functions of this circuit for
presented reactions:
uc(p)
H1(p)
;
u1(p)
uR(p)
H 2 (p)
.
u1(p)

2. Let’s find the value of the reaction on elements C and R.

1
u1(p)
1
u1(p)
uc(p)i(p)
;
pC R 1 pC pRC 1
pC
u1(p)
u1(p)pRC
uR (p) i(p) R
R
.
1
pRC
1
R
pC

3. Transfer function in operator form:

1
H1(p)
;
pRC 1
pRC
H2(p)
.
pRC 1
4. Find images of transient characteristics:
H1(p)
1
hC(p)
p
p(pRC 1)
1
R.C.
1
p p
R.C.
H2(p)
R.C.
1
hR(p)
.
p
pRC 1 p 1
R.C.
;

4. The image of the impulse characteristics is found by the relation:

g(p)H(p)
1
1
g C (p) H1 (p)
RC ;
pRC 1 p 1
R.C.
1
pRC
1
g R (p) H 2 (p)
1
1 RC.
1
pRC 1
pRC 1
p
R.C.

Thank you for your attention!

Let us assume that a step action is applied to the circuit, the image of which is the function

Let us assume that a step action is applied to the circuit
whose image is function A
p
x(t) A 1(t)
.
x(t)
0 at t 0;
x(t)
A at t 0.
A
t
0
Rice. 1. Stepped impact
Then the operator transfer function will have the form:
y (p) y (p)
y(p)
H(p)
p
.
A
x(p)
A
p
(10)
,

Carrying out the L-transformation of expression (7), i.e. Let's find the L-image of the transition response. Due to the linearity property

Carrying out the L-transformation of expression (7), i.e. Let's find the L image of the transient response. Due to the linearity property
Laplace transformation we get:
1
h (p) T (p).
p
(11)
This expression coincides with the second factor on the right side of (10)
and therefore between the operator transfer function and
image of the transition characteristic h (p) there is the following
relationship:
H(p)ph(p);
1
h (p) T (p).
p
(12)
(13)
Similarly, we establish a connection between H (p) and the image
impulse response g(p):
y(t)
g(p)
;
Si

If a pulse action is applied to the circuit, the image of which is equal, then the operator transfer function,

If a pulse action x(t) S и (t) is applied to the circuit,
whose image x(p) is equal to
, then the operator transfer
And
the function corresponding to this effect has the form:
S
y (p) y (p)
H(p)
.
x(p)
Si
(14)
This expression coincides with the pulse image function
circuit characteristics. Hence,
g(p)H(p).
(15)

Let's consider the relationship between transient and impulse characteristics
chains. It is not difficult to notice that their images are related by the relation
g (p) ph (p).
Carrying out the identical transformation of the last equality
(adding
h(0) h(0)) we get:
g (p) ph (p) h(0) h(0).
ph(p) h(p)
Because the
is an image
arbitrary transition characteristic, then the initial equality
can be represented in the form
g (p) h(0) L h / (t) .
Moving to the area of originals, we obtain a formula that allows
determine the impulse response of the circuit using a known
her
transition characteristic, g (t) h(0) (t) h (t).
g
t
h
(t).
If h(0) 0 then
The inverse relationship between these characteristics is
t
view:
h(t)g(t)dt.
0
(15)

3. Relationship between time and frequency
circuit characteristics
e t
For a given circuit, determine the operator
transfer function and find expressions
for its frequency characteristics
C
C
R
u1(t)R
u2(t)
u2(p)
H(p)
.
e(p)
Rice. 5. RC circuit diagram
We determine the image of the reaction u2 (p) from the system of nodal
equations compiled for L-images of nodal stresses
u1(p); u2(p) :
(2 pC G)u1 (p) pCu2 (p) pCe(p);
pCu1 (p) (pC G)u2 (p) 0.

From here

e (p) p 2
u2(p)
;
2
G G
2
p 3p 2
C C
2
p
H(p)2
2
p 3 p
where to simplify notation the notation is introduced
G
.
C
To find the complex transfer function, let us put in
last expression p j . Then
H(j)2
.
2
() j3
2

The frequency response is determined by the module of the obtained function, and the phase response is found
as an argument
H(j).
H(j)
2
(2 2) 9 2 2
Hj
3
() arctan 2
(2)
1
0
A
0
b
Rice. 6. Graphs of frequency characteristics of an RC circuit: a – frequency response, b – phase response

CONCLUSIONS:
1. The transfer function is the L-image of the impulse response.
2. Gear
function
is
fractional-rational
function
With
real coefficients.
3. The poles of the stable transfer function lie in the left p-half-plane.
4. The degrees of polynomials of the numerators of the transfer function and the square of the frequency response are not
exceed the degrees of polynomials of the denominators; if this is not done
properties of the frequency response at infinitely high frequencies (ω → ∞) should take
infinitely large value, since the numerator in this case increases
faster than the denominator.
5. The frequency characteristics of the circuit are calculated using the transfer function at
p = jω.
6. The squared frequency response is an even rational function of a variable with
real coefficients: H(jω) 2 = H(–jω) 2 .
7. Using the transfer function, you can draw a circuit diagram.

.
Question No. 1 a. Free vibrations in
series oscillatory circuit.
At the moment t=0 commutation occurred,
those. the key (Kl.) went from position 1 to
position 2.
The charged capacity turned out to be
connected to the RL circuit.
Let us consider the processes occurring in the presented circuit before switching
Before switching, capacitance C was connected
parallel to the constant voltage source E,
(key (key) was in position 1).
The voltage on the capacitors was equal to E.
uC(+0) = uC(-0) = E;
iL(+0) = iL(-0) = 0.

Let's consider the processes occurring in the circuit after switching
Considering that the voltage across the capacitance
cannot change abruptly, in accordance with the commutation law we have:
uC(+0) = uC(-0) = E
Initial conditions are NON-ZERO
Let's consider the equivalent circuit for the moment in time
According to Ohm's law in operator form,
Let's define the reaction image:
E
p
E
E
L
L
i(p)
2
,
2
1
R
1
p 2 p 0
pL R
p2 p
pC
L
L.C.
Where:
0
R

2L
1
L.C.
- circular frequency of natural oscillations of the circuit without losses.

When analyzing free and transient oscillations in complex circuits
the image of the reaction y(p) is a fractional rational function
variable p with real coefficients, which can be written in
in the form of a ratio of two polynomials:
M (p) bm p m bm 1 p m 1 bm 2 p m 2 ... b0
y(p)
N(p)
p n a n 1 p n 1 a n 2 p n 2 ... a 0
By the fundamental theorem of algebra, a polynomial of degree n can be decomposed into n
simple factors, i.e.:
N(p) = (p-p1) (p-p2),…, (p-pn),
where p1, p2, p3,…,pn are the roots of the polynomial N(p) or the poles of the function y (p).
The polynomial can also be represented as a product of m factors:
M(p) = (p-p01) (p-p02) (p-p03),…,(p-p0m).
where p01, p02, p03,…,p0m are the roots of the polynomial M(p) or the zeros of the function y (p).
Due to the reality of the coefficients ai and bi, the zeros and poles of the image y (p)
can be real and (or) complex conjugate.
It is clear that the dislocation of the poles y (p) determines the nature of the free and
transient oscillations in the analyzed circuit.

Consider the equation:
p 2 2 p 02
It has two roots (image poles):
p1.2 2 02
Due to the reality of the coefficients of this equation (δ, ω), the poles
can be real and complex conjugate.
Therefore, when analyzing free oscillations in a series circuit
Three oscillation modes are possible.

The roots of the equation are complex conjugate:
p1,2 j 1
Where:
1 02 2 .
this type of roots occurs at 0
or R 2
L
.
C
Original for current in
in this case it will be:
Et
i(t)
e sin 1t,
1 L

The amplitude of the oscillation decreases with time according to an exponential law,
therefore the process is called damped. Amplitude decay rate
free oscillations is determined by the value of the damping coefficient δ.
2
Frequency: 1 02 2 0 1 is called the natural frequency
0
damped oscillations of the circuit. It, as can be seen from the formula, is always less
frequency of natural undamped oscillations of the circuit w0 and depends not only on
values of inductance and capacitance of the circuit, but also on the value of its resistive
resistance.
Period of damped oscillations:
T
2
2
0
2
.
The attenuation coefficient is related to the quality factor of the circuit as follows:
where: Q
R0
.
2L 2Q
0 L
- quality factor of the series circuit.
R
Thus, the oscillations in the circuit decrease more slowly, the higher it is
quality factor

2. Critical mode of harmonic oscillations.

p1 p2 ,
.e. 0 ; R 2
T
L
.
C
Oscillation mode in a circuit corresponding to multiple roots
characteristic equation (image poles), can
be considered as a limiting case of oscillatory mode,
when the frequency of natural damped oscillations in the circuit
zero, and the oscillation period becomes
1 02 2 is equal
infinitely large.

has the form:
E0 t
i(t)
te
L

The roots of the equation are real multiples:
p1,2 ,
where: 2 02 ; .
Primary
options
contour
must
satisfy the inequality:
L
R 2
.
C
The original i(t), corresponding to a given arrangement of image poles,
has the form:
E
E
i(t)
L(p1 p2)
e p1t
L(p1 p2)
e p2t

Question No. 1 b. Transient oscillations in series
oscillatory circuit.
Initial conditions are ZERO
E
E
E
p
L
L
i(p)
2
;
2
1
R
1
p
2
p
0
pL R
p2 p C
pC
L
L
uC (p) i(p)
According to the correspondence table:
uC (t) E Ee (cos 1t sin 1t).
1
t
Voltage across the circuit capacitance
as t→∞ tends to a steady value equal to
source voltage. Consequently, the capacitance at t→∞ is charged to voltage E. The process
charge at complex conjugate poles of the image
has an oscillatory character.
1
L.C.
.
2
2
pC p(p 2 p 0)

The value of uC(t) at certain moments of time exceeds the voltage value; at a high quality factor, it can almost double the emf of the source.
At t→∞, the values of the current in the circuit, voltages on the resistive element and on
the inductance of the circuit tends to zero, and the voltage across the capacitance tends to the EMF
source. Consequently, the circuit switches to constant current mode. Process
the establishment of oscillations occurs the slower, the higher the quality factor
contour. To estimate the settling time, you can use the obtained
previously by the formula:
ty
3 4, 6
,
which corresponds to the period of time after which the voltage amplitude uC(t) deviates from the steady-state value by no more than 0.05 or 0.01.
Question No. 2 Free and transient vibrations in
parallel oscillatory circuit.
2.1 Free vibrations in PrKK
Initial conditions are NON-ZERO
iL(+0) = iL(-0) = I0
uC(+0) = uC(-0) = u0

I0
Cu0
p
I0
u0 p
C,
u(p)
2
2
1
p
2
p
0
pC G
pL
G
- circuit attenuation coefficient;
2C
1
0
- frequency of natural oscillations of the circuit without losses.
L.C.
Where:
1. Mode of damped harmonic oscillations.
The primary parameters of the contour in this case must satisfy the inequality:
G
2C
1
L.C.
The law of voltage change on the circuit in accordance with the correspondence table is determined by the expression:
I0
u
0
t
C
u (t) e u0 cos 1t
sin 1t
1

Analysis of the obtained solution shows that
the oscillations are damped, and
amplitude
fluctuations
decreases
By
exponential law. The more
damping coefficient, the faster they fade
fluctuations. As in a series circuit,
free oscillation frequency:
1 0 1
0
2
0
2
2
always less than the frequency of the circuit’s own undamped oscillations
2. Critical mode of harmonic oscillations.
This nature of the roots occurs at δ=ω0, when the following relation is satisfied between the primary parameters of the contour:
G
2C
1
L.C.
I0
t
u (t) u0 u0 t e
C

3. Aperiodic mode of harmonic oscillations.
This case is possible under the condition δ=ω0, which corresponds to the following
the relationship between the primary parameters of the circuit:
G 2
C
.
L
I0
I0
u 0 p1
u0 p2
u(t)C
e p1t C
e p2t
p 2 p1
p 2 p1
It should be noted that at G = 0 the oscillations in the circuit are undamped,
since the circuit does not dissipate energy.

2.2 Transient vibrations in PrKK
Using Ohm's law in operator form, we find images for all
reactions:
I
p
I
I
C
u(p)
2 C
;
2
1
G
1
p 2 p 0
pC G
p2 p
L.C.
C
L.C.
I
G
C
iG (p) u (p)G 2
;
2
p 2 p 0
I
u(p)
L.C.
iL(p)
;
2
2
pL
p (p 2 p 0)
iC(p)u(p)pC
IP
.
2
2
p 2 p 0

Law of voltage change in parallel
vibrational
contour
similar
law
changes in current in a series circuit.
Let us determine the time dependence of the current iC(t).
iC(t)Ie
p
(cos 1t sin 1t).
1
Since at t=0 the voltage on the capacitor was equal to zero, then for this moment
time, the container terminals should be considered short-circuited. Hence,
at the moment t=+0 the entire current I flowed through the capacitance (iC(+0))=I. At t→∞ the circuit
switch to direct current mode, in which u(∞)=0, iL(∞)=I, iG(∞)=iC(∞)=0.
The lower the quality factor (the greater the attenuation) of the circuit, the faster it ends
transition process.

The temporary characteristics of the electrical circuit are transient h(l) and pulse k(t) characteristics. Time characteristic of an electrical circuit is the response of the circuit to a typical influence at zero initial conditions.

Step response electrical circuit is the response (reaction) of the circuit to a unit function under zero initial conditions (Fig. 13.7, a, b), those. if the input value is /(/)= 1(/), then the output value will be /?(/) = X(1 ).

Since the impact begins at time / = 0, then the response /?(/) = 0 at /c). In this case, the transition characteristic

will be written in the form h(t- t) or L(/-t)- 1(g-t).

The transient response has several varieties (Table 13.1).

Type of impact	Type of reaction	Step response
Single voltage surge	Voltage	^?/(0 U (G)
Single current surge	Voltage	2(0 TO,( 0

If the action is given in the form of a single voltage jump and the reaction is also voltage, then the transient response turns out to be dimensionless and is a transfer coefficient Kts(1) by voltage. If the output quantity is current, then the transient characteristic has the dimension of conductivity, is numerically equal to this current" and is the transient conductivity ?(1 ). Similarly, when exposed to a current step and a voltage response, the transient response is the transient resistance 1(1). If the output quantity is current, then the transient response is dimensionless and is a transfer coefficient K/(g) by current.

There are two ways to determine the transient response - calculated and experimental. To determine the transient response by calculation it is necessary: using the classical method to determine the response of the circuit to a constant impact; divide the resulting response by the magnitude of the constant impact and thereby determine the transient response. When experimentally determining the transient response, it is necessary to: apply a constant voltage to the input of the circuit at time / = O and take an oscillogram of the circuit’s response; normalize the obtained values relative to the input voltage - this is the transient response.

Let us consider, using the example of the simplest circuit (Fig. 13.8), the calculation of transient characteristics. For this circuit in Chap. 12 it was found that the reaction of the circuit to constant impact is determined by the expressions:

Dividing “s(G) and /(/) by the influence?, we obtain the transient characteristics for the voltage across the capacitor and the current in the circuit, respectively:

Graphs of transient characteristics are shown in Fig. 13.9, A, b.

To obtain the voltage transient characteristic across the resistance, the current transient characteristic should be multiplied by /- (Fig. 13.9, c):

Impulse response (weight function) is the response of the circuit to the delta function at zero initial conditions (Fig. 13.10, A - V):

If the delta function is shifted relative to zero by m, then the reaction of the chain will be shifted by the same amount (Fig. 13.10, d); in this case, the impulse response is written in the form /s(/-t) or hp(/-t)? 1 (/-t).

The impulse response describes a free process in a circuit, since an effect of the type 5(/) exists at the moment / = 0, and for Г*0 the delta function is equal to zero.

Since the delta function is the first derivative of the unit function, then between /;(/) and k(I) there is the following connection:

At zero initial conditions

Physically, both terms in expression (13.3) reflect two stages of the transient process in an electrical circuit when exposed to a voltage (current) pulse in the form of a delta function: the first stage is the accumulation of some finite energy (electric field in capacitors C or magnetic field in inductances?) for pulse action time (Dg -> 0); the second stage is the dissipation of this energy in the circuit after the end of the pulse.

From expression (13.3) it follows that the impulse response is equal to the transient response divided by a second. By calculation, the impulse response is calculated from the transition response. Thus, for the previously given circuit (see Fig. 13.8), the impulse characteristics in accordance with expression (13.3) will have the form:

The impulse response graphs are shown in Fig. 13.11, a-c.

To determine the impulse response experimentally, it is necessary to apply, for example, a rectangular pulse with a duration of

. The output of the circuit is a transient curve, which is then normalized relative to the area of the input process. The normalized oscillogram of the response of a linear electrical circuit will be the impulse response.

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COURSE WORK

Time and frequency characteristics of linear electrical circuits

Initial data

Diagram of the circuit under study:

Element parameter values:

External influence:

u 1 (t)=(1+e - bt) 1 (t) (B)

As a result of completing the course work, you need to find:

1. Expression for the primary parameters of a given two-port network as a function of frequency.

2. Find an expression for the complex voltage transfer coefficient K 21 (j w) quadripole in idle mode on terminals 2 - 2".

3. Amplitude-frequency K 21 (j w) and phase-frequency Ф 21 (j w

4. Operator voltage transfer coefficient K 21 (p) of a four-terminal network in no-load mode on clamps 2-2".

5. Transient response h(t), impulse response g(t).

6. Response u 2 (t) to a given input influence in the form u 1 (t)=(1+e - bt) 1 (t) (B)

1. Let's defineYparameters for a given quadripole

I1=Y11*U1+Y12*U2

I2=Y21*U1+Y22*U2

To make it easier to find Y22, let’s find A11 and A12 and express Y22 through them.

Experiment 1. XX on clamps 2-2"

Let's make the replacement 1/jwС=Z1, R=Z2, jwL=Z3, R=Z4

Let's make an equivalent circuit

Z11=(Z4*Z2)/(Z2+Z3+Z4)

Z33=(Z2*Z3)/(Z2+Z3+Z4)

U2=(U1*Z11)/(Z11+Z33+Z1)

Experiment 2: Short circuit on 2-2" terminals

Using the loop current method, we will compose equations.

a) I1 (Z1+Z2) - I2*Z2=U1

b) I2 (Z2+Z3) - I1*Z2=0

From equation b) we express I1 and substitute it into equation a).

I1=I2 (1+Z3/Z2)*(Z1+Z2) - I2*Z2=U1

A12=Z1+Z3+(Z1*Z3)/Z2

From here we get that

Experiment 2: Short circuit on 2-2" terminals

Let's create an equation using the loop current method:

I1*(Z1+Z2) - I2*Z2=U1

I2 (Z2+Z3) - I1*Z2=0

Let's express I2 from the second equation and substitute it into the first:

From the second equation we express I1 and substitute it into the first:

For a mutual quadripole Y12=Y21

Matrix A of the parameters of the quadripole under consideration

2 . Let's find the complex voltage transfer coefficientTO 21 (jw ) quadrupole in idle mode at terminals 2-2 ".

Complex voltage transfer coefficient K 21 (j w) is determined by the relation:

It can be found from the system of standard basic equations for Y parameters:

I1=Y11*U1+Y12*U2

I2=Y21*U1+Y22*U2

So, according to the condition for idle speed I2=0, we can write

We get the expression:

K 21 (j w)=-Y21/Y22

Let's replace Z1=1/(j*w*C), Z2=1/R, Z3=1/(j*w*C), Z4=R, and obtain an expression for the complex voltage transfer coefficient K 21 (j w) in idle mode on clamps 2-2"

Let's find the complex voltage transfer coefficient K 21 (j w) quadripole in idle mode at 2-2" terminals in numerical form, substituting the parameter values:

Let's find the amplitude-frequency K 21 (j w) and phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient.

Let's write down the expression for K 21 (j w) in numerical form:

Let's find the calculation formula for phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient as arctg of the imaginary part to the real one.

As a result we get:

Let's write down the expression for phase-frequency Ф 21 (j w) characteristics of the voltage transfer coefficient in numerical form:

Resonant frequency w0=7*10 5 rad/s

Let's build graphs of the frequency response (Appendix 1) and phase response (Appendix 2)

3. Let's find the operator voltage transfer coefficientK 21 x (p) quadrupole in idle mode at terminals 2-2 "

operator voltage pulse circuit

The operator equivalent circuit of the circuit does not differ in appearance from the complex equivalent circuit, since the analysis of the electrical circuit is carried out under zero initial conditions. In this case, to obtain the operator voltage transmission coefficient, it is sufficient to replace jw in the expression for the complex transmission coefficient with the operator R:

Let us write the expression for the operator voltage transfer coefficient K21x(p) in numerical form:

Let's find the value of the argument p n for which M(p)=0, i.e. poles of the function K21x(p).

Let us find the values of the argument p k for which N(p)=0, i.e. zeros of the function K21x(p).

Let's create a pole-zero diagram:

Such a pole-zero diagram indicates the oscillatory damped nature of transient processes.

This pole-null diagram contains two poles and one zero.

4. Timing calculation

Let's find the transition g(t) and impulse h(t) characteristics of the circuit.

The operator expression K21 (p) allows you to obtain an image of the transition and impulse characteristics

g(t)hK21 (p)/р h(t)hK21 (p)

Let us transform the image of the transition and impulse characteristics to the form:

Let us now determine the transition characteristic g(t).

Thus, the image is reduced to the following operator function, the original of which is in the table:

Thus we find the transition characteristic:

Let's find the impulse response:

Thus, the image is reduced to the following operator function, the original of which is in the table:

From here we have

Let's calculate a series of values of g(t) and h(t) for t=0h10 (μs). And we will build graphs of the transition (Appendix 3) and impulse (Appendix 4) characteristics.

To qualitatively explain the type of transient and impulse characteristics of the circuit, we connect an independent voltage source e(t) = u1 (t) to the input terminals 1-1". The transient response of the circuit numerically coincides with the voltage at the output terminals 2-2" when the circuit is exposed to a single voltage jump e(t)=1 (t) (V) at zero initial conditions. At the initial moment of time after switching, the voltage on the capacitor is zero, because According to the laws of switching, at a finite value of the amplitude of the input step, the voltage across the capacitor cannot change. Therefore, looking at our circuit it is clear that u2 (0) = 0 i.e. g(0)=0. Over time, with t tending to infinity, only direct currents will flow through the circuit, which means that the capacitor can be replaced with a break, and the coil with a short-circuited section, and looking at our diagram it is clear that u2 (t) = 0.

The pulse characteristic of the circuit numerically coincides with the output voltage when a single voltage pulse e(t) = 1d(t) V is applied to the input. During the action of a single pulse, the input voltage is applied to the inductance, the current in the inductance jumps from zero to 1/L, and the voltage across the capacitance does not change and is equal to zero. At t>=0, the voltage source can be replaced by a short-circuited jumper, and a damped oscillatory process of energy exchange between inductance and capacitance occurs in the circuit. At the initial stage, the inductance current smoothly decreases to zero, charging the capacitance to the maximum voltage value. Subsequently, the capacitance is discharged, and the inductive current gradually increases, but in the opposite direction, reaching its greatest negative value at Uc=0. As t tends to infinity, all currents and voltages in the circuit tend to zero. Thus, the oscillatory nature of the voltage across the capacitor, which decays over time, explains the type of impulse response, with h(?) equal to 0

6. Calculation of response to a given input influence

Using the superposition theorem, impacts can be represented as partial impacts.

U 1 (t)=U 1 1 +U 1 2 = 1 (t)+e - bt 1 (t)

The response U 2 1 (t) coincides with the transient response

The operator response U 2 2 (t) to the second partial impact is equal to the product of the operator transmission coefficient of the circuit and the Laplace exponential image:

Let's find the original U22 (p) according to the Laplace transform table:

Let's define a, w, b, K:

Finally we get the original response:

Let's calculate a series of values and draw a graph (Appendix 5)

Conclusion

During the work, the frequency and time characteristics of the circuit were calculated. Expressions are found for the response of the circuit to harmonic influence, as well as the main parameters of the circuit.

The complex conjugate poles of the operator voltage coefficient indicate the damped nature of transient processes in the circuit.

Bibliography

1. Popov V.P. Fundamentals of circuit theory: Textbook for universities - 4th ed., revised, M. Higher. school, 2003. - 575 p.: ill.

2. Biryukov V.N., Popov V.P., Sementsov V.I. Collection of problems in circuit theory / ed. V.P. Popova. M.: Higher. school: 2009, 269 p.

3. Korn G., Korn T., Handbook of mathematics for engineers and university students. M.: Nauka, 2003, 831 p.

4. Biryukov V.N., Dedyulin K.A., Methodological manual No. 1321. Methodological instructions for completing course work on the course Fundamentals of Circuit Theory, Taganrog, 1993, 40 p.

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Time and frequency characteristics of linear electrical circuits. Calculation of time characteristics of linear electrical circuits Calculation of response to a given input influence

Problem 1

Impulse response

Problem 2

Impulse response

Task 3 Determine the transient and impulse characteristics of a circuit consisting of elements R and C connected in series.

2. Let’s find the value of the reaction on elements C and R.

3. Transfer function in operator form:

4. The image of the impulse characteristics is found by the relation:

Let us assume that a step action is applied to the circuit, the image of which is the function

Carrying out the L-transformation of expression (7), i.e. Let's find the L-image of the transition response. Due to the linearity property

If a pulse action is applied to the circuit, the image of which is equal, then the operator transfer function,

From here

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