Resonance in a chain with in parallel connected elements (a resonance of currents)
For a chain fig. 4 is had
Condition
of a resonance of currents
|
. Thus,
as appears from (8) and (9)
.
Thus, at a resonance of currents driving-point admittance of a chain
is minimum, and entrance resistance, on the contrary, as much as
possible. In particular at absence in a chain on fig. 4 resistors R
its entrance resistance in a resonance regime tends to infinity, i.e.
at a resonance of currents the current on a chain entry is minimum.
34. Circuits with mutual inductance. When two inductors are in close proximity (or wound on the same core), they become magnetically coupled and exhibit mutual inductance
.
M is the
mutual inductance between the two inductors
35. Wye and Delta Three-phase Circuits. Phase voltage and current, line voltage and current. The wye or star connection is made by connecting one end of each ofthe three-phase windings together as shown in Figure 12-5. The voltage measured across a single winding or phase is known as the phase voltage,as shown in Figure 12-6. The voltage measured between the lines is knownas the line-to-line voltage or simply as the line voltage. In a wye connected system, the line voltage is higher than thephase voltage by a factor of the square root of 3 (1.732). Two formulasused to compute the voltage in a wye connected system are: (ну знаете же формулу)
36.Three-phase
Circuits Analisis. Balanced Three-Phase Circuits. Phazor Diagrammes.
In a wye connected system, the line voltage is higher than t
hephase
voltage by a factor of the square root of 3 (1.732).
In a balanced system, each of the three instantaneous voltages have equal amplitudes but are separated from the other voltages by a phase angle of 120o
37. . Ideal Transformer. A Ideal Transformer is a unity Coupled, lossless transformer in which the primary and secondary coils have infinite self inductances. A Transformer is ideal if:1.)Large reactance coils;2.)Unity Coupling k=1.3.) Coils are lossless (R1=R2=0)
38. Kirchhoffs voltage law (KVL) •
The algebraic sum of all the voltages around any closed path in a circuitequals zero.
To use Kirchhoffs voltage law, we must assign an algebraic sign (reference direction) to each voltage in the loop. As we trace a closed path, a voltage will appear either as a rise or a drop in the tracing direction. Assigning a positive sign to a voltage rise requires assigning a negative sign to a voltage drop. Conversely, giving a negative sign to a voltage rise requires giving a positive sign to a voltage drop.
39. . Linear and nonlinear circuits. Examples of linear (usually a very good approximation) chains are chains containing onlyresistors, capacitors and inductors. Well as linear within a certain range can be consideredchains containing linear amplifiers and some other electronic devices that contain active elements, but having a certain range rather linear characteristics.
(
рисунки
катушки, резистора и конденсатора, и
графики)
40. Mesh current analysis in phazor domain
The mesh-current is analog of the node-voltage method. We solve for a new set of variables,
mesh currents, that automatically satisfy KCLs. As such, mesh-current method reduces
circuit solution to writing a bunch of KVLs.
41. . Node voltage analysis in phazor domain.
Node voltage analysis is a systematic method for finding the voltage at all nodes in an electrical circuit.
Basic Procedure:
а) Identify all nodes in the circuit
b) Assign a reference node
с) Assign node voltages
d) Identify unknown node voltages
e) Write KCL equations
f) Solve for unknown node voltages
g) Calculate currents
42.Ohm’s law in phazor domain.
The voltage v across a resistor is proportional to the current i travelling through it.
Z = V/I
43. Wye to delta conversion (Y-Δ).
The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances.
Equations for the transformation from Δ-load to Y-load 3-phase circuit:
Equations
for the transformation from Y-load to Δ-load
3-phase circuit:
44. Application of complex numbers to parallel a.c. networks.
The complex number has two main forms, the rectangular form and the polar form.
● Rectangular form: A =x + jy ( j=√-1)
where x is the real part and y is the imagery part of the complex number A; j is called the imagery unit.
● Polar form:
This is the abbreviated form of the exponential form , in which a is called modulus of the complex number, and the angle φ is called argument of the complex number.
Using the
phase notation to represent the sinusoidal function is based on
Euler’s formula
For a sinusoidal function
.,
replacing
with
in Euler’s formula gives
‘Re[ ]’ and ‘Jm[ ]’ stand for ‘real part’ and ‘imaginary part’ of complex numbers, respectively.
Therefore,
sine function
That is, a sinusoidal function is actually taking the imaginary part of the complex number
.
There are two terms in (8.3),
and
.
The second term
is called the rotating factor that varies with time t. The first term
is the phasor of the sinusoidal function
.
Since a phasor is a vector that can be represented by a complex
number, it can be presented with a rotating line in the complex plane
as shown in Figure 8.16. The length of the phasor is the peak value
Fm (or RMS value F). The angle between the rotating line and the
positive horizontal axis is the phase angle φ of he sinusoidal
function. This diagram is called the phasor diagram.
45. Power in a.c. circuits
46.Rms value, meanvalue and the form factor of a complex wave.The rms value of a periodic function is defined as the square root of the mean value of the squared function. Hence, if v = Vmcos(ωt + φ ), the rms value of v is
The quantity under the radical sign in Eq. 9.5 reduces to V2„/2.Hencethermsvalueofv is
47. Complex Power in a Balanced Wye Load.
Threelinecurrents
|
|
,
The line-to-neutral voltages are VAN, VBN, and VCN. We can now describe the line-to-line voltages in terms of the line-to-neutral voltages, using Kirchhoff s voltage law:
48.Average Power in a Balanced Wye Load.
|
|
,
The line-to-neutral voltages are VAN, VBN, and VCN. We can now describe the line-to-line voltages in terms of the line-to-neutral voltages, using Kirchhoff s voltage law:
49. Three-phase Circuits Analisis. Delta connection.In the delta connection, line voltage and phase voltage are the
same.
The line current of a delta connection is higher than the phase current by a factor of the square root of 3.
Formulas for
determining the current in a delta connection are:
50. Three-phase Circuits Analisis. Wye-Wye Circuit.
|
|
,
The line-to-neutral voltages are VAN, VBN, and VCN. We can now describe the line-to-line voltages in terms of the line-to-neutral voltages, using Kirchhoff s voltage law:
|
|
1. The magnitude of the line-to-line voltage is √3 times the magnitude of the line-to-neutral voltage.
2. The line-to-line voltages form a balanced three-phase set of voltages.
3. The set of line-to-line voltages leads the set of line-to-neutral voltages by 30°.
Line voltage refers to the voltage across any pair of lines; phase voltage refers to the voltage across a single phase. Line current refers to the current in a single line; phasecurrent refers to current in a single phase. Observe that in a Δ connection, line voltage and phase voltage are identical, and in a Y connection, line current and phase current are identical.