When solving a number of practical problems, it is often necessary to obtain a certain phase shift, not only in magnitude, but also in a given direction. Such examples are described in the article “Transformer connection groups”.
Shift by 30 and 60°.
By connecting the windings in star and triangle, shifts are obtained that are multiples of 30°, depending on what (ends, beginnings) are connected to what and in what direction (from phase A to phase B or vice versa), a shift is obtained in one direction or another.
When connecting in a zigzag - star (see the article "Zigzag connection diagram"), the end of one section is connected to the end of another section and the angle changes by 30°. If you connect not the end to the end, but the end to the beginning, then the vectors will rotate by 60° (see Figure 4, in the article “Some errors in star, triangle, zigzag connections”) In other words, by reconnecting the windings, you can easily get a shift of 30 and 60°.
The following must be kept in mind. Firstly, when reconnecting the windings, not only the angle can change (which is required), but also the voltage (see Figure 4, V, in the article “Some errors when connecting in a star, triangle, zigzag”). Secondly, connecting the windings in opposite directions - an extreme case - or changing the angle between them can reduce the inductive reactance, and this will lead to an increase in current. An increase in current is dangerous for the winding and, in addition, can lead to saturation of the magnetic circuit. The matter is much more serious than it might seem at first glance, and therefore, without making sure that the current has not exceeded the specified value, reconnections cannot be performed.
Shift by 90°.
Let's look at a common example of obtaining a 90° shift. In Figure 1, A the activation of the reactive energy meter is shown. Note: the current winding (thick line) is in phase A, and the voltage winding is connected to the phases B And C. Referring to the vector diagram in Figure 1, b, it is easy to see that in this simplest way a shift of 90° is obtained, which is what is required in this case.
Figure 1. Obtaining a 90° phase shift.
Shift to any angle from 0 to 90°
easy to get with phase regulator– rotary three-phase transformer. It is an asynchronous machine with a locked rotor. By turning the rotor relative to the stator, the phase of the electromotive force (emf) of the rotor is smoothly changed without changing its value (magnitude).
It is necessary to distinguish a phase regulator from a potential regulator, also called an induction regulator. In a phase regulator, only the phase changes; In the potential regulator both voltage and phase change. In addition, the primary and secondary windings of the phase regulator are mutually isolated, while those of the potential regulator are connected.
Let us note in conclusion that any phase shifts can also be obtained by connecting active and inductive resistances and capacitances. Such converters are widely used and are called static.
The initial phases of electromagnetic sinusoidal oscillations of the primary and secondary voltage, with a frequency of the same value, can differ significantly by a certain phase shift angle (angle φ). Variable quantities can change repeatedly over a certain period of time with a certain frequency. If electrical processes are unchanged and the phase shift is zero, this indicates synchronism of sources of alternating voltage values, for example, transformers. Phase shift is a determining factor of power factor in AC electrical networks.
The phase shift angle is found if necessary, then if one of the signals is a reference signal, and the second signal with a phase at the very beginning coincides with the phase shift angle.
The phase shift angle is measured using a device that has a normalized error.
The phase meter can measure the shift angle within the range from 0 o to 360 o, in some cases from -180 o C to +180 o C, and the range of measured signal frequencies can range from 20 Hz to 20 GHz. The measurement is guaranteed if the input signal voltage is between 1 mV and 100 V, but if the input signal voltage exceeds these limits, the measurement accuracy is not guaranteed.
Methods for measuring phase angle
There are several ways to measure the phase angle, these are:
- Using a dual-beam or dual-channel oscilloscope.
- The compensation method is based on comparing the measured phase shift with the phase shift provided by a reference phase shifter.
- The sum-difference method consists of using harmonic or shaped square-wave signals.
- Conversion of phase shift in the time domain.
How to measure phase angle with an oscilloscope
The oscillographic method can be considered the simplest with an error of around 5 o. The shift is determined using oscillograms. There are four oscillographic methods:
- Application of linear sweep.
- Ellipse method.
- Circular scanning method.
- Using brightness marks.
Determination of the phase shift angle depends on the nature of the load. When determining the phase shift in the primary and secondary circuits of a transformer, the angles can be considered equal and practically do not differ from each other.
The phase angle of the voltages, measured using a reference frequency source and using a measuring element, makes it possible to ensure the accuracy of all subsequent measurements. Phase voltages and phase shift angle depend on the load, so a symmetrical load determines the equality of phase voltage, load currents and phase shift angle, and the load in terms of power consumption in all phases of the electrical installation will also be equal.
The phase angle between current and voltage in asymmetrical three-phase circuits is not equal to each other. In order to calculate the phase shift angle (angle φ), series-connected resistances (resistors), inductances and capacitors (capacitors) are included in the circuit.
From the results of the experiment, it can be determined that the phase shift between voltage and current serves to determine the load and cannot depend on the variable current and voltage in the electrical network.
As a conclusion, we can say that:
- The constituent elements of complex resistance, such as resistor and capacitance, as well as conductivity, will not be reciprocal quantities.
- The absence of one of the elements makes the resistive and reactive values, which are part of the complex resistance and conductivity, and makes them reciprocal quantities.
- Reactive quantities in complex resistance and conductivity are used with the opposite sign.
The phase angle between voltage and current is always expressed as the main reasoned factor in the complex resistance φ.
Ohm's law for alternating current
If the circuit contains not only active, but also reactive components (capacitance, inductance), and the current is sinusoidal with a cyclic frequency ω, then Ohm’s law is generalized; the quantities included in it become complex:
U = I Z
U = U 0 e iωt- voltage or potential difference,
I- current strength,
Z = Re -iδ- complex resistance (impedance),
R = (R a 2 +R r 2 ) 1/2 - total resistance,
R r = ωL - 1/ωC- reactance (difference between inductive and capacitive),
R A- active (ohmic) resistance, independent of frequency,
δ = -arctg R r /R a- phase shift between voltage and current.
In this case, the transition from complex variables in the values of current and voltage to real (measured) values can be made by taking the real or imaginary part (but in all elements of the circuit the same!) of the complex values of these quantities. Accordingly, the reverse transition is constructed for, for example, U = U 0 sin(ωt + φ) such a selection U = U 0 e iωt, What InU = U. Then all values of currents and voltages in the circuit must be considered as F = ImF.
If the current varies with time, but is not sinusoidal (or even periodic), then it can be represented as the sum of sinusoidal Fourier components. For linear circuits, the components of the Fourier expansion of the current can be considered to act independently.
It should also be noted that Ohm’s law is only the simplest approximation for describing the dependence of the current on the potential difference and for some structures it is valid only in a narrow range of values. To describe more complex (nonlinear) systems, when the dependence of resistance on current cannot be neglected, it is customary to discuss the current-voltage characteristic. Deviations from Ohm's law are also observed in cases where the rate of change of the electric field is so high that the inertia of charge carriers cannot be neglected.
2. What is the phase shift between voltage and current in a circuit containing a coil or capacitance?
Phase shift- the difference between the initial phases of two variable quantities that change periodically over time with the same frequency. The phase shift is a dimensionless quantity and can be measured in degrees, radians or fractions of a period. In electrical engineering, the phase shift between voltage and current determines the power factor in AC circuits.
In radio engineering, RC circuits are widely used, which shift the phase by approximately 60°. To shift the phase by 180°, you need to connect three RC chains in series. Used in RC generators.
The EMF induced in the secondary windings of a transformer for any current shape coincides in phase and shape with the EMF in the primary winding. When the windings are turned on in antiphase, the transformer changes the polarity of the instantaneous voltage to the opposite; in the case of sinusoidal voltage, it shifts the phase by 180°. Used in the Meissner generator, etc.
Fig. 305
Rice. 305. Experience in detecting phase shifts between current and voltage: on the left - a diagram of the experiment, on the right - the results gives the shape of the voltage between the plates of the capacitor (points a and b), because in this loop of the oscilloscope the current at each instant of time is proportional to the voltage. Experience shows that in this case the current and voltage curves are shifted in phase, with the current leading the voltage in phase by a quarter of a period (p/2). If we replaced the capacitor with a coil with high inductance (Fig. 305, b), it would turn out that the current is out of phase with the voltage by a quarter of a period (by p/2). Finally, in the same way it could be shown that in the case of active resistance, voltage and current are in phase (Fig. 305, c). In the general case, when a section of a circuit contains not only active, but also reactive (capacitive, inductive, or both) resistance, the voltage between the ends of this section is phase shifted relative to the current, and the phase shift ranges from +p/2 to -p/2 and is determined by the ratio between the active and reactance of a given section of the circuit. What is the physical reason for the observed phase shift between current and voltage? If the circuit does not include capacitors and coils, i.e., the capacitive and inductive resistance of the circuit can be neglected in comparison with the active one, then the current follows the voltage, passing simultaneously with it through maxima and zero values, as shown in Fig. 305, v. If the circuit has noticeable inductance L, then when alternating current passes through it, a EMF. self-induction. This EMF, according to Lenz's rule, is directed in such a way that it tends to interfere with those changes in the magnetic field (and, consequently, changes in the current that creates this field) that cause emf. d.s. induction. As the current increases, e. d.s. self-induction prevents this increase, and therefore the current reaches its maximum later than in the absence of self-induction. As the current decreases, e. d.s. self-induction tends to maintain the current and zero current values will be reached at a later point than in the absence of self-induction. Thus, in the presence of inductance, the current lags in phase with the outflow in the absence of inductance, and therefore lags in phase with its voltage. If the active resistance of the circuit R can be neglected compared to its inductive reactance XL=wL, then the time lag of the current and voltage 
equals T/4(phase shift is p/2), i.e. maximum u coincides with i=0, as shown in Fig. 305, b. Indeed, in this case the voltage across the active resistance Ri=0, because R=0, and therefore all external stress u is balanced by the induced emf, which is opposite to it in the direction: u=LDi/Dt. So the maximum u coincides with the maximum Di/Dt, i.e. occurs at the moment when i changes most quickly, and this happens when i=0. On the contrary, at the moment when i passes through the maximum value, the current change is the smallest ( Di/Dt=0), i.e. at this moment u=0. If the active resistance of the circuit R is not so small that it can be neglected, then part of the external voltage drops across the resistance R, and the rest is balanced by e. d.s. self-induction: u=Ri+LDi/Dt. In this case the maximum i distance from the maximum and in time less than T/4(phase shift less p/2) as shown 
Let's do the following experiment. Let's take the oscilloscope with two loops described in § 153 and connect it to the circuit so (Fig. 305, a) that loop 1 is connected to the circuit in series with the capacitor, and loop 2 is parallel to this capacitor. Obviously, the curve obtained from loop 1 depicts the shape of the current passing through the capacitor, and from loop 2 gives the shape of the voltage between the plates of the capacitor (points and ), because in this oscilloscope loop the current at each moment of time is proportional to the voltage. Experience shows that in this case the current and voltage curves are shifted in phase, with the current leading the voltage in phase by a quarter of a period (by ). If we were to replace the capacitor with a coil with high inductance (Fig. 305, b), it would turn out that the current is out of phase with the voltage by a quarter of a period (by ). Finally, in the same way it could be shown that in the case of active resistance, voltage and current are in phase (Fig. 305, c).
Rice. 305. Experience in detecting phase shifts between current and voltage: on the left - experimental diagram, on the right - results
In the general case, when a section of a circuit contains not only active, but also reactive (capacitive, inductive, or both) resistance, the voltage between the ends of this section is phase shifted relative to the current, and the phase shift lies in the range from to and is determined by the relationship between active and reactive resistance of a given section of the circuit.
What is the physical reason for the observed phase shift between current and voltage?
If the circuit does not include capacitors and coils, i.e., the capacitive and inductive resistance of the circuit can be neglected in comparison with the active one, then the current follows the voltage, passing simultaneously with it through maxima and zero values, as shown in Fig. 305, v.
If a circuit has a noticeable inductance, then when an alternating current passes through it, an emission occurs in the circuit. d.s. self-induction. This e. d.s. according to Lenz's rule, it is directed in such a way that it tends to prevent those changes in the magnetic field (and, consequently, changes in the current that creates this field) that cause e. d.s. induction. As the current increases, e. d.s. self-induction prevents this increase, and therefore the current reaches its maximum later than in the absence of self-induction. As the current decreases, e. d.s. self-induction tends to maintain the current and zero current values will be reached at a later point than in the absence of self-induction. Thus, in the presence of inductance, the current is out of phase with the current in the absence of inductance, and therefore out of phase with its voltage.
If the active resistance of the circuit can be neglected in comparison with its inductive resistance, then the time lag of the current from the voltage is equal (the phase shift is equal to), i.e., the maximum coincides with, as shown in Fig. 305, b. Indeed, in this case the voltage across the active resistance is , for , and, therefore, all external voltage is balanced by e. d.s. induction, which is opposite to it in direction: . Thus, the maximum coincides with the maximum, i.e., it occurs at the moment when it changes the fastest, and this happens when . On the contrary, at the moment when it passes through the maximum value, the current change is smallest, i.e. at this moment.
If the active resistance of the circuit is not so small that it can be neglected, then part of the external voltage drops across the resistance, and the rest is balanced by e. d.s. self-induction: 
. In this case, the maximum is separated from the maximum in time by less than (the phase shift is less), as shown in Fig. 306. Calculation shows that in this case the phase lag can be calculated using the formula
. (162.1)
When we have and , as explained above.
Rice. 306. Phase shift between current and voltage in a circuit containing active and inductive resistance
If the circuit consists of a capacitor and the active resistance can be neglected, then the plates of the capacitor connected to a current source with a voltage are charged and a voltage arises between them. The voltage on the capacitor follows the voltage of the current source almost instantly, that is, it reaches a maximum simultaneously with and goes to zero when.
The relationship between current and voltage in this case is shown in Fig. 307, a. In Fig. 307,b conventionally depicts the process of recharging a capacitor associated with the appearance of alternating current in the circuit.
Rice. 307. a) Phase shift between voltage and current in a circuit with capacitance in the absence of active resistance. b) The process of recharging a capacitor in an alternating current circuit
When the capacitor is charged to the maximum (i.e., and therefore have a maximum value), the current and all the energy of the circuit is the electrical energy of the charged capacitor (point in Fig. 307, a). As the voltage decreases, the capacitor begins to discharge and current appears in the circuit; it is directed from plate 1 to plate 2, i.e. towards the voltage. Therefore, in Fig. 307, and it is depicted as negative (the points lie below the time axis). By the moment of time, the capacitor is completely discharged (and), and the current reaches its maximum value (point); electrical energy is zero, and all energy is reduced to the energy of the magnetic field created by the current. Further, the voltage changes sign, and the current begins to weaken, maintaining the same direction. When (and) reaches its maximum, all energy will again become electrical, and the current (point). Subsequently (and) begins to decrease, the capacitor is discharged, the current increases, now having a direction from plate 2 to plate 1, i.e. positive; the current reaches its maximum at the moment when (point), etc. From Fig. 307, but it is clear that the current reaches a maximum earlier than the voltage and passes through zero, i.e. the current is ahead of the voltage in phase, as explained above.
Rice. 308. Phase shift between current and voltage in a circuit containing active and capacitive resistance
Phase characterizes the instantaneous value of a harmonic signal at a certain point in time. The unit of measurement for phase is electrical degree or radian. The phase shift is determined by two main methods: direct assessment and comparison.
Phase meters for direct assessment include analog electromechanical devices with a ratiometric mechanism, analog electronic phase meters and digital phase meters.
The comparison method is measured using an oscilloscope. This method is used in low-power circuits, with a small level of measured signals, when high accuracy is not required. For more accurate results, a compensation method is used, where the oscilloscope serves as an indicator of phase equality.
When measuring in the frequency range of signals from several tens to 6-8 kHz, ratiometric devices are used, which makes it possible to measure large amplitude signals with low accuracy and high internal consumption of the device.
Analog electronic phase meters. The operation of a two-channel circuit, an analog electronic phase meter, is based on the conversion of the shift angle between signals into time intervals between pulses T, followed by conversion to current difference ICP, the average value of which is proportional to this angle.
The formula expressing the dependence of the shift angle on the output current of the circuit is written as follows:
Ψ=(180*Icp)/Iм;
Where Ψ
– phase shift angle;
ICP– average value of the current difference at the output of the circuit;
Im– amplitude of output pulses.
Harmonic signals U1 And U2 are supplied respectively to the reference and signal input elements of the circuit. The input element is an amplifier-limiter of the input signal and is used to convert sinusoidal signals into a series of pulses with a constant edge slope.
Synchronized multivibrators, under the influence of the input signal, produce rectangular pulses (graph 3). The output signals of multivibrators have a constant duration T/2 and shifted relative to each other for a while ΔT, proportional to the angle ψ .
The output signal from the reference and signal parts of the circuit is fed to a special differentiating element, at the output of which peaked signals are generated. Positive impulses are converted into edges, negative impulses into cuts (graph 4).
The following signals are received at the output multivibrators. Day off MV reference channel: positive pulse of the reference channel and negative pulse of the measuring channel. Output MF of the measuring channel: positive pulse of the measuring channel and negative pulse of the reference channel.
At the same time, at the output of the reference MV a signal of duration is obtained (T/2+ΔT), and at the output of the measuring MV–(T/2-ΔT).
A measuring microammeter connected to the pulse difference of the output MVs shows the average value of the current difference:
Icp=(2ΔТ/Т)Iм;
If we substitute the formulas into this expression ψ=ωΔТ, ω=2π/Т, we get:
ψ=360ºΔT/T=(180ºIcp)/Im;
The ammeter scale is calibrated in phase angle units. The error when using this method depends on the accuracy class of the device.
Digital phase meters. The operating principle of these digital devices is based on the dependence ψ=360ºΔТ/Т, but instead of a multiplier ΔT/T the formula involves the value of the number of sample pulses N. The operation of the digital phase meter is illustrated in Figure 2.
The open time of the time selector depends on the measured period T. During this period of time, a reference frequency signal passes through the time selector fo and exemplary duration That, produced by the timestamp generator. Number of pulses N during the period T will be:
N=T/To;
Input signals U1 And U2 through a strobe pulse shaper are converted into a series of pulses shifted in time by ΔT, proportional to the phase shift of the signals. The open state time of the temporary selector is ΔT, and the number of missed pulses of reference frequency is equal to:
n=ΔT/To;
Then dependency ψ from the frequency and number of pulses of the reference frequency will be written as follows:
ψ=360ºn/N or ψ=360º(fo/f)n;
Such frequency meters are used provided that the reference frequency is more than 1000 times the signal frequency.
To measure the average phase shift, another time selector controlled by a voltage divider is added to the digital phase meter circuit. In this case, several groups of pulses proportional to the magnitude of the shift angle will pass through two sequentially connected time selectors.
Measurement by comparison method. To determine the phase shift by comparison, an electronic oscilloscope is used. Phase shift ψ
are found by the parameters of the figures shown on the screen of an oscilloscope operating in linear or circular scanning mode.
When using a dual-beam oscilloscope, two signals of the same frequency are supplied to the vertical deflection plates, between which the phase shift is measured. When the horizontal lines of two signals are combined, the diagram in Fig. 3 is observed on the oscilloscope screen. Based on the segments measured to scale ab And ac define:
ψ=360ºΔТ/Т=360º.
The error of this method lies in the inaccuracy of determining the segments ab And ac, inaccurate alignment of horizontal lines, and the thickness of the light beam on the screen.
When measuring ψ
According to Lissajous figures, the measured voltages are supplied to the horizontal and vertical inputs of the oscilloscope. An ellipse-shaped figure appears on the screen.
The center of the ellipse is aligned with the center of the coordinate system. Measuring the size of the segments on the screen A And IN, the phase shift is found by the formula:
ψ=arctg(A/B);
Measurement error ψ using the Lissajous figure method is 5-10%. Another disadvantage of the method is the measurement of the phase shift without determining the sign.
This drawback is solved as follows: voltage u2 fed simultaneously to the horizontal plates and to the cathode ray tube modulator with a 90° phase shift. Moreover, in the area of positive values ψ - the upper part of the ellipse glows brighter, and when negative, the lower part glows brighter.
The most accurate definitions ψ performed using the compensation method. For this, an exemplary phase shifter (RC-chain, bridge or transformer circuit) is used, connected to the circuit of one of the voltages. The phase shifter introduces a phase shift equal to, but opposite to, the measured one ψ .
When shifting ψ On the oscilloscope screen, the inclined line will be deviated to the right from the vertical. If the line is deviated to the left, the shift is equal to (180º-ψ).