Let's write Ohm's law for the complex form on the minus element
LECTURE 7 ANALYSIS OF LANZUGU WITH THE LAST
Z'EDNANNYAM RECEIVER
Lecture plan
4. Voltage resonance
1. The main laws of the lanceugs of the snake strum
IN lances of the changing strum, Ohm's law wins over all meanings, Kirchhoff's laws are less mittevikh and complex, like the protection of phases of spivvіdnoshennia.
Kirchhoff's first law. Algebraic sum of mittev and strum values at node:
∑ i k = 0 ,
k=1
otherwise, the algebraic sum of the complex values of the streams at the nodes is equal to zero:
∑ I k = 0 .
k=1
Another law of Kirchhoff. The algebraic sum of the mittievs is the value of the voltages on the priymachas near the contour, the more algebraic sum of the mittevs is the value of the EPC, which is at that contour:
Equal, folded according to the laws of Kirchhoff, are called equal to the electric camp.
1. Basic Laws
The scheme of the replacement of the lancer from the last batches of primacs is shown in fig. 7.1.
For the analysis of processes, we quickly compare on the basis of another Kirchhoff's law in the complex form:
U = UR + UL + UC.
Let's represent the value of the voltage in terms of Ohm's law:
U = R I + j XL I - j XC I = [R + j (XL - XC)] I = Z I,
de Z - Complex opir of Lancjug.
Obviously what
Z = R+ j(XL − XC ) = R+ j X,
de R- active opir, X - Reactive opir.
Ohm's law for the complex form for the Frenchman with the last primaries:
U=ZI.
Reactive Opir X can be positive or negative.
Reactive reference X > 0, ie XL > X C . Chomu vipadku lanjug
may be inductive.
Reactive Opir X< 0 , еслиX L < X C . Тогда цепь имеет емкостный характер.
2. Pobudov vector diagrams
Sound when її pobudovі do not bind to the complex plane, only small amounts of value mutual roztashuvannya vector_v.
Pobudov's vector diagrams start from the vector of the magnitude, which is critical for the lance. With the last concatenation of the elements, such
LECTURE 7
2. Pobudov vector diagrams
size є strum. Type of diagrams to lie in the character of the lanceug. Pobudova with vector diagrams for Lanziug, which has an active-inductive character, so XL > X C and X > 0 is shown in Fig. 7.2.
The input voltage is added up from the voltage on three ideal elements with the shape of the phases. The voltage on the resistor is changed by the strum in phase. The voltage on the inductive element viperedzhaє strum by 90 °, on the mnіsny - vіdstaє 90 °.
Otrimany pіd h pobudovi vector diagrams trikutnik OAV images fig. 7.3.
Kut φ = ψu − ψi –
ka and again the voltage.
Trikutnik OAV gives the opportunity to operate with important meanings, for which Kirchhoff's laws do not count:
U = UR 2 + (UL − UC ) 2 ,
Arctg U L − U C ,
U R = Ucos ϕ, UL − UC = Usin ϕ.
UL-UC
O A UR
3. Tricks for tension support
How to divide all sides of the tricot of the voltage on the stream I, we take away the similar to the new tricot of the supports (Fig. 7.4), de Z - outside opir lanciug,R – active opir, X- Reactive opir-
nya, X L = L ω - inductive support, X C = | - єmnіsne op- |
|||||||||||||||
tivlenie. | ||||||||||||||||
U − U | ||||||||||||||||
−X C | ||||||||||||||||
Ohm's law for exploding values at the last call, I may look at:
LECTURE 7
3. Tricks of support and tightness
U=ZI.
From the authority of the tricoutnik of the support, it is necessary to spivvіdnoshennia:
Z = R 2 + X 2 = R 2 + (X L -X C) 2; ϕ =arctan | |||
R = Z cosϕ; X = Z sinϕ. | |||
Kut to deposit in the spіvvіdnoshnja opіv lansyug.
The alignment of the formulas of the complete and complex support allows the development of visnovok, which is the module of the complex. From the tricutnik of supports it is clear that the argument of the complex support is kut.
So you can write:
Z = R + jX = Z e j ϕ.
The latest opir be-like the number of consecutively taken priymachiv
Z = (∑ R) 2 + (∑ XL − ∑ XC ) 2 .
To multiples of all sides of the tricot, the tension on the strum is taken by the tricot of tightness (Fig. 7.5).
Active pressure
P = UR I = R I2 = U Icos ϕ
characterizes the energy that is transmitted in one straight line from the generator to the receiver. Vaughn is tied with resistive elements.
U I = S
UL − UC I= Q
UR I = P
Reactive pressure Q \u003d U L - U C I \u003d X I 2 \u003d U I sinϕ characteristic
part of the energy that circulates continuously in the lance and does not stop cordless roboti. Vaughn is tied with reactive elements.
Povna (given) tightness S \u003d U I \u003d P 2 + Q 2.
LECTURE 7
3. Tricks of support and tightness
Active tension is reduced by watts (W), reactive - by reactive voltamperes (var), exactly - by volt-amperes (VA).
4. Resonance voltage
An inductive coil and a capacitor are mutually inductive antipodes. If the stench is more than compensating for the deed one of the one, the Lanciuzi have
a resonant mode is observed.
Resonance of the voltage vinikaє at the last connection of inductive coils and capacitors. Umov to the resonance of the voltage: the input reactive support X is equal to zero.
Let's look at the Lanzug resonance mode, the substitution scheme for which is shown in Fig. 7.1.
At resonance
X = XL −X C =0 .
Stars X L = X C.
Oskilki X L \u003d L ω, and X C \u003d C 1 ω, then at resonance L ω0 \u003d C 1 ω 0. Todi LC ω0 2 = 1
Rice. 7.1 you can change the inductance L, frequency capacitance ω. Cyclic resonant frequency
ω 0= | ||||||||||
Same frequency | ||||||||||
f0= | ||||||||||
At resonance outside opir Z \u003d R 2 + X 2 \u003d R. Lanciug May |
||||||||||
Suto active character. | (ω= ω0 ) | X=0 | X L = X C, |
|||||||
resonant | ||||||||||
Z = R2 + X2 = R = Zmin, I = U | I max. | |||||||||
Let's have a vector diagram (Fig. 7.6). | ||||||||||
It is obvious that U = U R , | U L = − U C ,UL = U C , kut = 0 . | |||||||||
Lanzyug may be a daily active character.
Resonant voltage value:
1. At the electric power outbuildings in U L U C in most of the valleys in Nebazhane,
LECTURE 7
4. Voltage resonance
pov'yazane with unsustainable overvoltage.
2. In electrical engineering, communication (radio engineering, fire telephones), in automation, the resonance of the voltage is widely vicorated to adjust the lancet to a singing frequency.
Nutrition for self-verification
1. For some meanings electrical quantities what are the laws of Kirchhoff counting?
2. What is the integrated support module?
3. What is the argument for integrated support?
4. How does it appear to be active, reactive and complex opir?
5. How to designate a new opir of the scheme?
6. In what way to lay a kut ts mizh with a strain and a strum?
7. How tightness is relieved?
8. How is the energy characterized by active tension?
9. What energy is characterized by reactive tension?
10. In some loneliness they die actively, reactively and completely
11. Yaka umova resonance naprugi?
12. What is the significance of the resonance of the voltage?
U m = U m e j u ; I m = I m e j i = CU m e = CUm e j u e,
vrakhovuyuchi, scho e j \u003d j, -j \u003d take away: I m = .
Let's move on to the complexes of fluctuating meanings: I = U / X h ,
de X
h =
- Comprehensive єmnіsny opіr.
The vector diagram on the complex plane of the strum and the pressure of the terminal element is shown in fig. 1.10. 
Rice. 1.10.
1.6. A complex method for the analysis of linear electric lances with sinusoidal strums
Like you see, rozrahunok be-like electric lansyug you can work on the basis of the laws of Kirchhoff, squeezing and violating the system of equalities. Zastosuvannya laws of Kirchhoff for mittevih the value of sinusoidal streams and voltages to bring to differential levels. For example, for a lancer with successively included active and inductive elements, the equal of another to Kirchhoff's law can be seen:
.
Outside the solution i(t) of this linear differential equation, as it seems, is composed of a private solution, which is determined by the type of function u(t), wild decision homogeneous differential equalization, won at u(t)=0. Warehouse strum at u(t)=0 can only be used for the calculation of the energy reserves in the magnetic field of the inductive element and extinguish the aftermath of the energy supply on the active element. In this way, after a short interval of an hour after switching on, a strum is left in the lance, which is considered to be no more than a private decision of the lance. This strum is called the strum of the set mode. We have analyzed the same mode. Let's assume that the voltage applied to the end of the lancet changes according to the law: u(t)=U 0 sin( t+ u) .
As shown earlier (div. p. 1.5), in the active and inductive elements of the stream of the installed mode, it also changes according to the sinusoidal law: i(t)=I m sin( t+ ) .
The task is to adjust the amplitude of the bud phase of the strum at a given frequency. If necessary, the designation of the strums of the tails or the voltage on the plots of the lancet will require the summation of the sinusoidal functions of the hour. This operation is connected with cumbersome and laborious charges. The bulkiness of the scales of the viklikan is that the sinusoidal value at a given frequency is not one, but two values - the amplitude and the phase. Іstotne sproshchenya reach when depicting sinusoidal functions hour complex numbers. The possibility of such a manifestation for sinusoidal streams and voltages was shown earlier (div. p. 1.4.).
The method, based on the images of real sinusoidal functions for an hour with complex numbers, is called the complex method. It is also called the symbolic method, the shards of wine are based on the symbolic function of the hour, the frequency function. In a complex way of winning, the power of the exponential function is also important, which means that the differentiation of the complex exponent in the hour is equal to the multiplication of it by j, and integration - fit on j:
; 
.
As a result, all differential equations, folded according to Kirchhoff's laws, are replaced by the equations of algebra in a complex form. Virishyuchi tsі vnyannya algebra, we know complex strumi and in them we can pass to mittev znachen. In this rank, the complex method of suttєvo polegshuє rozrahunki to that, scho є by the method of algebraicization of differential equations.
1.7. Viraz laws of Ohm and Kirchhoff in the complex form
Looking at the active, inductive and ominous elements in Lanciuge sinusoidal struma, we introduced the concept of active and reactive (inductive chimney) supports. Uzagalnyuyuchi, we call the extension of the complex stress to the complex struma by the complex support of the lance Z:
.
Modulus and argument support equal to vіdpovіdno up to vіdnoshennia fiyuchih znachenya zsuvu phases mizh strum and narugoyu.
Speech and manifest parts Z called active and reactive supports. The value that returns to the complex support is called the complex conductivity:
.
Її modulus і argument for designations є return values Z і . Speech and visible parts of Y are called active and reactive conductances. Let's put a link between active and reactive supports and conductions.
stars 
.
The introduction of complex supports and conductivities means the introduction of Ohm's law in a complex form for the sinusoidal mode, which is: 
.
On the basis of Ohm's law for a constant strum, here it is safe, the value of the strum and the voltage, which is to blow, and the destruction of the phases between them.
Now let's write Kirchhoff's laws in the complex form.
The first Kirchhoff's law for knots in complex forms is written as: .
Another Kirchhoff law for contours in complex forms is written as: .
After the introduction to understand the complex support and the installation of the laws of Ohm and Kirchhoff for complex strums and stresses, there is no need in the forward folding of the systems of differential alignments of the lancet with further transformations of them into the alignment of algebra for the complexes of the stream and tensions. When analyzing the lancet in a complex way, the skin element of the lancet is represented by its complex support of conductivity, and the strumite and tension - by the main complexes of the resulting values. As a result, a complex scheme for the replacement of the lanceug comes out. On this scheme, a passive skin can be seen in a bipolar person with a complex support, and an active skin can be seen in a dzherel with a complex EPC and an internal support.
Such a scheme for substituting the matime looks like a resistive lance, only the replacement of speech values on the scheme will be the complex values of the struma, voltage, EPC and support.
Before 
The complex nature of the quantities reflects the need for the appearance of the sound of the phases between sinusoidal streams and voltages in the mode that is standing up. Rivnyannia will become complex replacement circuits are formed similarly to resistive lances on fast stream. Therefore, in the analysis of lanceg in a complex way, it is possible to zastosovuvat all those methods, which are valid on a constant stream:
Methods of equivalent transformation of schemes (parallel last day elements, transformation of the star - trikutnik and back, transformation of the tension and struma);
Method of proportional values;
node potential method;
Method of contour strums;
equivalent generator method;
The principle of overhead, reciprocity.
Formally, the analysis of the analysis in a complex way, in the analysis of resistive lances on a constant stream, will be less for everyone, which the coefficients of all equalities, and the change of changes will be complex quantities.
Shards of skin dodanki in a complex equalization can be shown by a vector, and the equalization itself can be shown by a sum of vectors, a complex method allows you to accompany analytical developments with primary graphic illustrations - vector diagrams.
Let's take a look at the complex method of rozrahunka of specific languages.
1.8. The coil of inductance in the lance of a sinusoidal struma is real
Here is the complete complex operation of the cat: Z
=R+j
L
The coil of inductance is real, the crim of inductance is real, the active support of the winding is possible, for which it is prepared. Therefore, a complex substitution scheme is formed from successively connected inductive and active supports, fig. 1.11.
According to another Kirchhoff law for complexes, the values of the voltage and the total voltage
U= U L+ U R =jL I+R I=(jL+R) I=Z I
it is composed of active and reactive (inductive) storage.
Rice. 1.11.
Module ta prop argument: Z=,
vynachayut vіdpovіdno spіvvіdnoshnja amplitudes i zsuv phases mizh naprugoyu and strum. Complex Struma Dorіvnyuє 
,
de u- Pochatkova phase of the applied voltage.
Otzhe, viraz for the mittevnoe meaning of the sinusoidal struma in the real coil of inductance may look:
.
Strum vidstaє in phase vіd applied to the lance of the voltage on the cut , scho to lie down in the spіvvіdnoshennia between the active and inductive supports of the coil.
Otrimanі complex spіvvіdnoshennia can be shown on the diagram, fig. 1.12. 
Rice. 1.12.
The vector of the struma, which is primal for successively included elements, is taken as a weekend and applied to a fairly straight line, sounding horizontal.
Vector U R vector straight line I the shards of the wines move in phase, and the vector U L, Viperedzhayuchi vector strum on 90 pro, being perpendicular to the strum protigodinnikovo's arrow. The geometric sum of two vectors is a vector. U voltage, which is added to the inductance coil. Vector U move ahead in phase vector I on kut . What is the cob phase of the voltage u is given, it is possible to plot the axes of the complex coordinate system and the way of geometrical variations i and other parameters for us.
It is necessary to prote the memory that ignition voltage on the back of a real coil of inductance, it looks like the sum of active and inductive storage, formally and in real lances, the stench is not known and is not measured by a voltmeter without a middle.
1.9. After switching on the real inductance coil of the capacitor without losses in the lances of the sinusoidal struma
The last coil of a variable stream with an inductive coil and a capacitor can be represented by a complex substitution circuit for R, L, C elements, fig. 1.13. 
Rice. 1.13.
The applied stress is written as the sum of the stresses on the elements of the lance:
u = u R + u L + u C
or in complex form: U = U R + U L + U C .
2.4.2. Law of Ohm and Kirchhoff in a complex view
Ohm's law for a complex view:
Ỉ=Ủ/ Z or Ỉ= Y∙Ủ, (2.26)
de Ỉ - strum that flows through the electric lance,
Ủ - voltage. dodane to the electric lansyug,
Y- Comprehensive conduction of the electric lance,
Z- Comprehensive opir of the electric lansyug.
Kirchhoff's first law. The sum of strums at the drots, which converge at the nodes of the electric lancet, is equal to zero:
Another law of Kirchhoff. The sum of complex e.f.s. or the voltages that are due to a closed circuit, more than the sum of the voltage drops on the elements of this circuit.
(2.28)
Laws of Ohm and Kirchhoff are just like mittevikh, so the meaning of e.m.s. voltage and strum.
Dіyuchі (effective or r.m.s. voltage) is determined by viraz:
,
(2.29)
de T - the period of the voltage drop, which is good 1/f,
f is the frequency of voltage surge.
With a strictly sinusoidal form, the voltage is choked, which is good, strong: U = Um / 
,
(2.30)
de Um is the maximum value of the stress u(t).
In a similar way, the fiduciary values of e.r.s. that strumiv.
2.4.3. Pobudov vector diagrams on the complex plane, what to wrap around.
To make it easier to make vector diagrams on the plane, which wraps around, it is necessary to remember the following basic positions:
a) In a lance with an active support, the jets and the voltage are phase-shifting.
b) In an idealized lance, only with an inductive support without voltage losses in phase, the stream is forwarded to the coil, which is 90 degrees
c) At the lance with a daily support without strum vtrat vperedzha in phase, the voltage on the cut is +90 degrees.
Fig. 2.1. Mnemonic diagram that explains the possible turns
radius-vectors with different inclusions of r-L-elements.
When prompted vector diagrams it is necessary to start pobudovu from the voltage vector or the struma of a wild one for the entire analyzed lance. Zocrema, with the subsequent inclusion of the elements of the lancet, it is necessary to start a vector struma that flows through all the elements of the lancet. When the elements of the lancet are connected in parallel, the vector diagrams need to start from the vector of the main voltage, and then the vectors and strums that flow through the skin from the neck of the electric lancet. Possible failure of the phases of the voltage vectors in electric lances, which are formed from different combinations of r-L-C elements, pointed at the mnemonic diagrams (div. Fig. 2.1.).
The radius vectors in the scheme and below are shown in bold or dots above them.
2.4.4. The resonance of the voltage in the lance, which is formed from the successive inclusion of the inductance coil and the capacitor
Let's look at such an analysis in case of allowances, that the magnitude of the support, the capacitance and inductance do not change at the hour and not lie down in the form of applied voltage and streams (div. Fig. 2.2).
Fig.2.2.Electric circuit of sequential inclusions of r-L-C-elements.
Processes that are included in the doslidzhuvanom lance (according to another Kirchhoff law) are described (with the remaining values of the elements in the case of the independence of their magnitudes, which flow) by linear integral-differential rivnyan:
u(t)=ri(t)+Ldi(t)/dt+1/C ∫i(t)dt, (2.31)
de u(t) - changing voltage, which is fed from the dzherel to the kolyvalny circuit,
i(t) - change strum that flows through the lances,
L - inductance,
r – active support of the inductance coil,
З - capacity of the capacitor.
Opіr (r), inductance (L) and capacitance (C) make the coil circuit, in which the resonance of the voltage is possible. The term “voltage resonance” can be applied to the voltage, so when X l = Xc is equal, the voltage on the elements of the circuit L and C is changed in Q times at the same time as the voltage, which is applied to the circuit. Under the value of Q, the quality factor of the circuit is understood, which is good Q \u003d Xc / r.
With accepted allowances, equalization (2.31) can be filed with such a look:
u(t)=i(t)*(r+j). (2.32)
Zvіdki follow viraz for a complex support to the contour
Z= r + j (Xl -Xc).
At resonance, the voltage, if X l \u003d Xc, Z\u003d r, then the circuit support is active, and the stream that flows through the circuit reaches the maximum value, which is i (t) max \u003d u (t) / r.
In this case, using vector diagrams, it is necessary to start the struma (Ỉ) vector from the lanky lancet, then the voltage vectors will be generated. In the case of the last connection of the coil of inductance and capacity, the igneous reactive support of the lancer X is more algebraic difference of the inductive and the impulsive support Xl and Xc. The voltage added to such a lance can be represented by looking at the vector sum of the vector of the voltage drop on the active support (U r), which changes in phase with the stream vector; the vector of the voltage drop on the inductance (U l), which transduces the strum in phase at a cut of 90°; With what possible onset of depression:
a) The inductive opir is greater than the minus one (Х l>Х С). In this direction, the voltage of the strum viperejatime in phase per cut φ is input (div. Fig. 2.3.).
b) Emnisny opіr larger for inductive (Х l<Х с). При этом ток опережает напряжение на угол φ. Векторная диаграмма тока и напряжений показана на рис. 2.4.
Rice. 2.3 Small 2.4
in). The inductive opir is more similar to the seminal one (Х l = Xс). Apparently the latest reactive opir of the lancer (X) is closer to zero, and the latest opir of the lancer Z=r, then. reach its minimum value. For whom, the strum in phase zbіgatisya with tension, tobto. kut = 0. The vector diagram of the streak and voltage for the th wave is shown in fig. 2.5.
The resonance of the voltage is also observed in quartz resonators, as they are widely used in oscillators.
