Conversion circuits with the ideal voltage and current sources
So as a voltage source and current source are sources of energy, they can be converted into each other. So for the real voltage source (fig. 1.4,b) and real current source (Fig.1.6.b) if the same currents ‘’ I ’’ and voltages "u", expressing the current from the relation (1.32), we get
(4.58)
and equating it (1.33), we obtain
(4.59)
Expressing voltage from the ratio (1.33)
(4.60)
and equating it to (1.32), we obtain
(4.61)
There are also methods for transferring an ideal voltagу source and current source. Let us consider Fig. 4.9. Compile equations of Kirchhoff's laws for the network on Fig. 4.9, а
Fig. 4.9
Compile equations of Kirchhoff's laws for the network of Fig. 4.9.a
For the nodes A and b we get:
(4.62)
|
|
|
Or, in short we obtain
(4.63)
For the loops I – IV we get:
(4.64)
(4.65)
(4.66)
(4.67)
Make an equation for the circuit of Fig. 4.9.b
For the node A we get:
(4.68)
|
|
|
For the nodes I – IV we obtain:
(4.69)
(4.70)
(4.71)
(4.72)
Hence we see the equations (4.63) and (4.68), (4.64) - (4.67) and (4.69) - (4.72) are identical. That is the networks of Fig. 4.9.a and Fig. 4.9.b are equivalent. As a result we can conclude: if an ideal voltage source is included between two nodes, it can be transferred to all branches growing from one of the nodes.
Fig.4.10
Let us consider Fig. 4.10,a. According to the Kirchhoff's law for the current for nodes A, B, C we get
|
|
|
(4.73)
(4.74)
(4.75)
For the network of Fig. 4.10,b we can write down for the nodes A, B, C
(4.76)
|
|
|
(4.77)
(4.78)
Hence we see the equations (4.73) - (4.75 and (4.76) - (4.78) are identical. Therefore the networks of Fig. 4.10 and Fig.. 4.10,b are equivalent. As a result of a conclusion can be made: if an ideal current source is included between two nodes, it can be moved in parallel all branches, forming a path between these nodes.
The simplest harmonic current circuit
Harmonic current circuit with series connection of R , L , C elements
Let us consider the networks of Fig 1.9.a. Let voltage ”u” of source is changed according to the harmonic law. Let write down his in the complex form.
(4.79)
According to the Kirchhoffs low for the voltage we get
(4.80)
or
(4.81)
Integral-differential equation (4.81) is the equation of electric balance for the circuit Fig. 1.9.a. Let the current
(4.82)
Write down the equation image (4.81) in the complex form
(4.83)
where
(4.84)
- the complex impedance of a circuit;
(4.85)
- reactance impedance of a circuit;
(4.86)
- impedance circuit
(4.87)
- phase angle of the circuit - the angle of phase shift between the current and the voltage in the circuit.
Now from (4.83) we get
(4.88)
That is
(4.89)
The voltage on resistance R
(4.90)
That is, the voltage across the active resistance r, in according to (4.88), (4.90), is in phase with the current and lags behind on the angle φ an applied voltage to the circuit .
The voltage across the inductance L
(4.91)
That is, the voltage across the inductor L, in according to (4.88), (4.91), ahead of the current phase on the angle .
The voltage across the capacitance C
(4.92)
I.e. voltage across the capacitance C. in according to (4.88), (4.92), lags behind of the current phase on the angle π/2.
Passing on from the complex image to the original, we will obtain from the expressions (4.88), (4.90) - (4.92)
(4.93)
(4.94)
(4.95)
(4.96)
In Fig. 4.11 vector diagrams for the r, L, C - circuit in Fig. 1.9.a is shown. Here in Fig. 4.11.a the voltage U is ahead of the current I on the angle . The angle from the current to voltageis positive. The circuit as a whole has inductive nature. On Fig. 4.11.b voltage U is lagging from the current I . The angle is negative. The circuit as a whole has capacitive nature.
Fig. 4.11
Dividing all values of vector diagrams in Fig. 4.11 on the current I , we obtain the corresponding vector diagrams for resistance (Fig. 4.12). Here the angle is measured from the active resistance of the r to the complex impedance Z. Vector diagrams in Fig .4.12.a and b for inductive ( > 0) nature of the load are equivalent. Also vector diagrams in fig. 4.12.c and d for capacitive ( < 0) the nature of the load are equivalent. Triangles OAB in Fig. 4.12- triangles of resistance.
Fig. 4.12
Inductive reactance xL = ω L and capacitive reactance xC = 1/ ω C depend on the frequency ω. Diagrams of dependences for the resistance r, inductive reactance xL, capacitive reactance xC , reactivereactance x = ω L - 1/ ω C and impedance Z are depicted in Fig. 4.13
Fig. 4.13
It is visible, we get at the frequency
(4.97)
(4.98)
(4.99)
This mode is called voltage resonance and will be discussed in detail below.
Vector diagrams at the frequency are shown in Fig.4.14.a (foe currents and voltages) and Fig. 4.14.b (for resistances).
Reactive power at voltage resonance is equal to zero. Power in the circuit is of pure active.
Fig. 4.14
4.3.2. Harmonic current circuit with series connection of R, L – elements
The ratio for this circuit can be obtained from the expressions of section 4.3.1. in with C . Indeed we get
(4.100)
that is, the capacitance C in Fig. 1.9.a you can replace by the short-circuited jumper. Then, from (4.80) we get
(4.101)
or
(4.102)
From (4.83) we can write down
(4.103)
where
(4.104)
Reactance
(4.105)
Impedance
(4.106)
The phase angle
(4.107)
The current in the circuit and voltages across resistance r and inductance L are defined by the expressions (4.88) - (4.91). The expression for the instantaneous values of the current and voltages are identical with expressions (4.93) - (4.95). In Fig. 4.15 vector diagrams for the voltages and current are shown for the circuit (a) and for the resistances (b).
Fig. 4.15
4.3.3. Harmonic current circuit with series connection of R, C elements
The ratio for this circuit can be obtained from the expressions of section 4.3.1 when L = 0. Indeed we get
(4.108)
that is the inductance L in Fig. 1.9.a you can replace by the short-circuit jumper. Then, from (4.80) we get
(4.109)
or
(4.110)
From (4.83) we can write down
(4.111)
where
(4.112)
Reactance
(4.113)
Impedance
(4.114)
The phase angle
(4.115)
The current in the circuit and voltages across resistance R and capacitance C are defined by the expressions (4.88) - (4.90), (4.92). The expressions for the instantaneous values of the current and voltages are identical with expressions (4.93) – (4.94), (4.96). In Fig. 4.16 presents vector diagrams for the voltage and current are shown for the circuit (a) and resistances (b).
Fig. 4.16
4.3.4. Harmonic current circuit with a parallel connection of R, L, C elements
Let us consider the networks of Fig. 1.9.b, which is dual circuit of Fig. 1.9.a. Obviously, all of the ratio for this network may be obtained from the expressions of sections 4.3.1- 4.3.3 by the dual replacement. Let the energy source creates a current
(4.115,a)
According to the Kirchhoff law for the currents we get
(4.116)
or
(4.117)
Write the image to (4.117) in the complex form
(4.118)
where
(4.119)
- complex admittance of the circuit;
(4.120)
- susceptance of the circuit;
(4.121)
- admittance of the circuit;
(4.122)
phase angle of the circuit - the angle of phase shift between the current and the voltage in the circuit.
Now from (4.118) we get
(4.123)
That is
(4.124)
That is the phase angle φ corresponds to the same expression (4.89). The current in conductance g
(4.125)
That is current in the active conductance g in according to (4.123), (4.125) is in phase with the voltage and lags behind on the angle φ an current source.
The current in the capacitance C
(4.126)
That is current in capacitance C in according to the (4.123), (4.126) ahead of the phase voltage on the angle φ.
The current in the inductance L
(4.127)
That is the current in the inductance L in according to (4.123), (4.127), lags behind of the voltage on the angle φ .
Passing on from the complex image to the original, we will obtain from (4.123), (4.125) - (4.127)
(4.128)
(4.129)
(4.130)
(4.131)
In Fig. 4.17 shows vector diagrams for the r, L, C - circuit in Fig. 1.9,b. Here in Fig. 4.17,a the current I is ahead of the voltage Um. Angle φ from the current to voltage, is negative. In Fig. 4.17.b current lags behind voltage Um. The angle φ is positive. The circuit as a whole has inductive nature.
Fig. 4.17
Dividing all values of vector diagrams in Fig. 4.17 on the voltage Um , we get the corresponding vector diagrams for conductances (Fig. 4.18). Here the angle is measured from the admittance of Y. Vector diagrams Fig. 4.18.a and b for capacitance (φ < 0) nature of the load are equivalent. Also equivalent to vector diagrams in Fig. 4.18.c and d for inductive (φ > 0) nature of the load are equivalent. Triangles OAB in Fig. 4.18 - triangles of conductances.
4.3.5. Harmonic current circuit with a parallel connection of R, C elements.
The ratio for this circuit can be obtained from the expressions of section 4.3.4.при L → ∞. Indeed we get
(4.132)
Fig. 4.18
that is the inductance L in Fig. 1.9.b, you can replace break in a branch with inductance. Then from (4.11.b) we get
(4.133)
or
(4.134)
From (4.118) we obtain
(4.135)
where
(4.136)
Susceptance
(4.137)
Admittance
(4.138)
The phase angle
(4.139)
The voltage across the terminals of the circuit and the currents in the conductance g and capacitance C are defined by the expressions (4.123) - (4.126). The expressions for the instantaneous values of voltages and currents are identical with (4.128) - (4.130). Fig. 4.19 shows vector diagrams for voltages and currents in the circuit (a) and conductances (b).
Fig. 4.19
4.3.6.Harmonic current circuit with a parallel connection of R, L elements
The ratio for this circuit can be obtained from the expressions of section 4.3.4.при C = 0. Indeed we get
(4.140)
that is, the capacitance C in Fig. 1.9.b you can replace by the gap of branches with a capacity. Then from (4.116) we obtain
(4.141)
or
(4.142)
From (4.118) we receive
(4.143)
where
(4.144)
Susceptance
(4.145)
Admittance
(4.146)
The phase angle
(4.147)
The voltage on the terminals of the circuit and the currents in the conductance g and inductance L are defined by the expressions (4.123) - (4.125), (4.127). The expression for the instantaneous values of voltages and currents are identical with expressions (4.128) - (4.129), (4.131). In fig. 4.20 vector diagrams for voltage and currents in the circuit (a) and conductances (b) are given.
Fig.4.20
Inductive - coupled circuit
4.4.1. Circuit with magnetic coupling
Electrical circuits, processes which influence each other by means of general electric or magnetic fields, are called coupled. If a magnetic field is general, then these circuits with are magnetic or inductive coupling circuits.
Fig.4.21
Let us consider magnetic flux and magnetic-flux linkage in circuits with magnetic coupling. In Fig. 4.21 two inductive - coupled coils W1 and W2 schematically depicted (W1 , W2 are the number of turns of the first and second coils). Here current i1 of coil W1 creates the self-induction flux of the first coil
(4.148)
where: Фs1- the flux- leakage of the first coil - part of the flux , crossing only turns of the first coil; Ф21- the mutual induction flux of the second coil of part of the flux Ф11 , crossing the turns of the first coil.
Similarly current i2 of the coil W2 creates the self-induction flux of the second coil
(4.149)
where: - the flux-leakage of the second coil - part of the flux , crossing only turns of the second coil; - the mutual induction flux of the first coil - part of the flux , crossing the turns of the first coil.
The full magnetic fluxes of coils
(4.150)
(4.150,a)
The product of the flow by the number of turns is called magnetic flux-linkage . Then for the first and second coils we get
(4.151)
(4.152)
where: , flux- linkage of self-induction of the first and the second coils;
, flux-linkage of mutual induction of the first (from second) and the second (from first) coils.
The attitude of the linkages of mutual induction to self-induction of the coil is called the degree of coupling
(4.153)
Here - the degree of coupling of the second coil with the first coil - shows which part of the flux-linkage of the second coil associated with the first coil; - degree of соupling of the first coil with the second coil – shows which part of the flux- linkage of the first coil associated with the second coil.
The ratio of magnetic-flux linkage to the current, its creating, is called inductance L. There are distinguished inductances of self-induction (the first and the second coils)
(4.154)
inductances of the mutual induction (the first and the second coils)
(4.155)
inductances of the leakage (the first and the second coil)
(4.156)
Now degrees of coupling in according of (4.153) - (4.155)
(4.157)
The geometric mean of the degrees of coupling and are called the coefficients of coupling
(4.158)
for the linear circuits M = M = M. Therefore
(4.159)
In according to the electromagnetic induction law voltage across terminals of the inductive coil is the time derivative of its total flux linkage
(4.160)
(4.161)
Here
(4.162)
- voltage of self-induction of the first and second coils;
(4.163)
- voltage mutual induction of the first and second coils.
The expression (4.160), (4.161) can be represented in the complex form
(4.164)
In the expressions (4.150) - (4.152), (4.160), (4.161, (4.164) double sign indicates the flux, flux- linkage or the voltage of the mutual induction ( , , , , , ) of a given coil coincides with the flux, flux-linkage or a voltage of self-induction of the this coil (plus sign) or opposed them (minus sign). Hence aiding and opposed connection of inductive coils.
Aiding connection is called the inclusion of two coils, where their magnetic flux of self-induction and mutual induction coincide in the direction. Opposed - opposite in direction.
Here the same name and different name terminals are distinguished of the coils. The same name terminals are called such terminals of two coils, when the currents of these coils, equally directed to these terminals, create a magnetic flaxes of the one direction, that is the coils are aiding connected. Different name - when the coils are opposed connected.
The same name terminals identifies by points on the circuit diagram. In Fig. 4.21 points on the conclusions of the windings of the W and W point to the same name terminals.
Дата добавления: 2021-03-18; просмотров: 62; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!