Measurement of phase shift between current and voltage. What determines the phase angle of voltage and current in a circuit?

The phase shift is a dimensionless quantity and can be measured in radians (degrees) or fractions of a period. With a constant, in particular zero phase shift, they say about synchronicity two processes, or about the completed synchronization of two sources of variables.

Phase (phase angle) is the angle $\backslash varphi\; =\; 2\; \backslash pi\; \backslash frac\; (t)\; (T)\; ,$ Where $T$- period, $t$- the fraction of the phase shift period when sinusoids are superimposed on each other. So if the curves (variable quantities - sinusoids: oscillations, currents) are shifted relative to each other by a quarter of a period, then we say that they are shifted in phase by $\backslash frac\; (\backslash pi)\; (2)\; ~\; (90^\backslash circ)\; ,$ if for an eighth part (share) of a period, then it means for $\backslash frac\; (\backslash pi)\; (4)$ etc.
When talking about several sinusoids out of phase, technicians talk about current or voltage vectors. The length of the vector corresponds to the amplitude of the sinusoid, and the angle between the vectors corresponds to the phase shift. Many technical devices give us not a simple sinusoidal current, but one whose curve is the sum of several sinusoids (respectively, shifted in phase).

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.

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An excerpt characterizing the Phase Shift

The units of measurement for phase shift are radian and degree:

1° = π/180 rad.

In the catalog classification, electronic meters of phase difference and group delay time are designated as follows: F1 - standard devices, F2 - phase meters, FZ - measuring phase shifters, F4 - group delay time meters, F5 - correlation meters.

Electromechanical phase meters on the front panel have the sign ∆φ.

The phase characterizes the state of the harmonic process in this moment time:

u(t) = Um sin (ωt+ φ).

The phase is the entire argument of the sine function ( ωt+ φ). Typically, ∆φ is measured for oscillations of the same frequency:

u 1(t) =Um sin( ωt+ φ 1);

u 2(t) =Um sin( ωt+ φ 2).

In this case, the phase shift

∆φ = ( ωt+ φ 1) - ( ωt- φ 2) = φ 1 - φ 2 (5.10)

To simplify, we take the initial phase of one oscillation as zero (for example, φ 2 = 0), then ∆φ = φ 1.

The above concept of phase shift applies only to harmonic signals. For non-harmonic (pulse) signals, the concept of time shift (delay time) is applicable t 3), diagrams of which are shown in Fig. 5.6.

Rice. 5.6. Stress diagrams with time shift

Phase shift measurement is widely used at industrial and ultra-high frequencies, i.e. over the entire frequency range.

The phase shift occurs, for example, between the input and output voltages of a four-terminal network, as well as in AC power circuits between current and voltage and determines the power factor (cos φ), and therefore the power in the circuit under study.

To measure the phase shift at industrial frequencies, electromechanical phase meters of electrodynamic and ferrodynamic systems are widely used. The disadvantages of such phase meters are the relatively large power consumption from the signal source and the dependence of the readings on frequency. The relative reduced error of electromechanical phase meters is no more than ±0.5%.

Depending on the required accuracy of measuring the phase shift and signal frequency, one of the following methods is used: oscillographic (one of three), compensation, electronic discrete counting method, method of converting phase shift into current pulses, measurement method using phase meters based on a microprocessor system, method signal frequency conversion.

Oscillographic methods, in turn, are divided into three: linear scan, sinusoidal scan (ellipse) and circular scan.

For implementation linear sweep method use a two-channel or two-beam oscilloscope (or a single-beam oscilloscope with an electronic switch). The screen produces an image of sinusoidal signals (Fig. 5.7).

Rice. 5.7. Oscillograms of two sinusoidal signals when measuring phase shift using the linear sweep method

Signals u 1(t)And u 2(t) are supplied to the inputs Y1 and Y2 of the oscilloscope. To ensure the motionlessness of the oscillograms, it is necessary to synchronize the sweep with one of the signals being studied.

By measured segments 0 a and 0 b the phase shift is calculated from the relation

(5.11)

The linear sweep method allows you to determine the sign of the phase shift and covers the full range of its measurement - 0...360°. The error of the method is ± (5...7°) and is determined by the nonlinearity of the unfolding voltage, the inaccuracy of measuring the linear dimensions of segments 0 A and 0 b, the quality of focusing and brightness of the beam (i.e., the skill of the operator).

Sine sweep method implemented using one; beam oscilloscope. Tested voltage signals u 1 (t) And u2(t) are supplied to the X and Y inputs of the oscilloscope when the internal linear scan generator is turned off. A figure in the form of an ellipse will appear on the screen (Fig. 5.8), the shape of which depends on the phase shift between the two voltages and their amplitudes. The phase shift is determined by the formula

(5.12)

Rice. 5.8. The resulting oscillogram when measuring the phase shift using the sinusoidal sweep method

To reduce the error, the amplitudes are equalized before measurement X t And Ym their smooth regulation along channels Y and X.

The sinusoidal sweep method allows you to measure the phase shift in the range from 0...180° without determining the sign.

The error in measuring ∆φ by the sinusoidal sweep method (ellipse method) depends on the accuracy of measuring the segments included in equation (5.12), on the quality of focusing and the brightness of the beam on the CRT screen. These reasons have a noticeable effect at a phase shift close to zero and 90°.

Both methods considered are indirect and quite labor-intensive.

Circular scanning method - the most convenient oscillographic method for measuring phase shift. In this case, the sign of the phase shift is determined over the entire angle measurement range (0...360°). The measurement error is constant over the entire range.

The block diagram of the oscilloscope for measuring the phase shift using the circular scanning method is shown in Fig. 5.9, A.

Rice. 5.9. Block diagram of the implementation of the circular scanning method (a), angle reading (b) and diagrams of sinusoidal signals (V) when measuring phase shift

The X and Y inputs of the oscilloscope are supplied with sinusoidal voltage signals U 1 And U 3, shifted relative to each other by 90° using a phase shifter consisting of a resistor and capacitor. If the arm resistances are equal, the voltage amplitudes U 1 And U 3 are also equal and an oscillogram in the form of a circle will be observed on the screen (Fig. 5.9, b).

Comparable signals u 1 (t) And u2(t) are supplied to the inputs of two identical shapers, which convert sinusoidal voltages into a sequence of short unipolar pulses with voltage U 4 And U 5(Fig. 5.9 , V) with steep fronts. The beginning of the pulses coincides with the moment of transition of the sinusoids across the time axis as they increase. Signals with voltage U 4 And U 5 arrive at the OR logic circuit, where they are summed up, and a sequence of pulses with voltage appears at the output U6, which are fed to the control electrode (modulator) of the tube, controlling the brightness of the beam at points 1 and 2, and points of increased brightness are observed on the circle at points 1 and 2.

The phase shift between signals occurs as follows (see Fig. 5.9, b). When measuring, the center of the transparent protractor is aligned with the center of the circle, the total circumference of which corresponds to 360°. During the period T signals under study with voltage U 1 And U 2 the electron beam describes a circle. The beam describes an arc between points 1 and 2, the length of which is equal to a certain angle α, during the delay time of these signals: ∆ t =∆φ T/ 360°, from where α= ∆φ.

The absolute measurement error using the circular scanning method reaches 2...5° and depends on the accuracy of determining the center of the circle, the accuracy of measuring the phase shift using a protractor and the degree of identity of the response threshold of both shapers.

Compensation method(overlay method) is implemented using an oscilloscope. The method diagram is shown in Fig. 5.10, A.

Rice. 5.10. Scheme for implementing the compensation method ( A) and oscillogram (6) when measuring phase shift

Signals with voltage U 1 And U 2 are supplied to the Y and X inputs of the oscilloscope, and to the Y input - through a graduated phase shifter, and to the X input directly.

Phase shift between test voltages U 1 And U 2 determined by changing the phase of the signal with voltage U 3 phase shifter until a straight inclined line appears on the screen (Fig. 5.10, b), which indicates that the phases of both signals are equal. The determined phase shift ∆φ is counted on the phase shifter scale relative to the primary position, corresponding to a phase rotation of 180°. To reduce the measurement error, it is necessary to correct the phase shifts created by the amplifiers of the vertical and horizontal deflection channels of the oscilloscope beam. This procedure is carried out in the same sequence as when measuring the phase shift using a sinusoidal sweep method (see Fig. 5.8). An electronic voltmeter can be used as a zero indicator.

The measurement error using the compensation method is small (0.2...0.5°) and is determined mainly by the quality of the phase shifter calibration.

The compensation method is also used in the microwave range when measuring the phase shift introduced by any element additionally included in the microwave path (filter, waveguide section). The block diagram of measuring the phase shift using the compensation method is shown in Fig. 5.11.

Rice. 5.11. Block diagram of measuring phase shift in the microwave range using a compensation method

The measurement process is carried out in the following order. When the test element Z is disconnected, the microwave path at the output of the phase shifter is short-circuited with a plug. When the generator is turned on, a standing wave is established in the path. Since the minimum standing wave is more pronounced than the maximum, then by adjusting the phase shifter, the standing wave node is moved relative to the transverse plane of the probe so that the rectifier device (milliammeter) shows the minimum, and the readings of φ 1, the phase shifter, are noted. Then, between the phase shifter and the plug, the test element Z is connected, creating a displacement of the standing wave voltage node, and again, using the phase shifter, the minimum indicator reading is achieved, which will be φ 2 when counted on the phase shifter scale.

The phase shift introduced by the element Z under study into the microwave path is determined by the formula

Instead of a phase shifter and a probe, a measuring line can be used in the circuit under consideration. The described compensation method is indirect.

A two-channel phase meter allows you to measure phase shift directly. The operating principle of a two-channel phase meter is based on converting the phase shift into rectangular pulses. The block diagram of a two-channel phase meter, timing diagrams of signals explaining its operation, and a graph of the readings of the relative ∆φ indicator are presented in Fig. 5.12.

Rice. 5.12. Block diagram of a two-channel phase meter ( A), signal timing diagrams explaining its operation (6) and a graph of indicator readings relative to ∆φ ( V)

The phase meter consists of a converter ∆φ into a time shift ∆ t, equal to the desired phase shift ∆φ and the measuring indicator. The converter consists of two identical signal conditioners and an adder, which is used as a trigger.

Tested voltage signals U 1 And U 2 with a phase shift ∆φ are fed to the inputs of two identical shapers, which convert the incoming sinusoidal signals into a sequence of short pulses with voltage U 3 And U4. Pulses with voltage U 3 trigger the trigger, and the voltage pulses U 4 set it to its original position. As a result, a periodic sequence of pulses is formed at the output, the repetition period and duration of which are equal to the repetition period T and time shift ∆ t studied signals with amplitude I m.

The microammeter of the magnetoelectric system is most often used as a measuring indicator, the readings of which are proportional to the average current value over the signal repetition period T.

As can be seen from the timing diagram I = f (t) ( see fig. 5.12, b), in the circuit of the measuring device, rectangular pulses with a duration of ∆ are obtained t. Consequently, the average value of the current flowing through the devices over the period is proportional to twice the relative time interval:

From the graph (see Fig. 5.12, b) it follows that the phase shift between the studied signals with voltage U 1 And U 2 corresponds to the time shift ∆ t and can be expressed by the formula

from which it follows that the phase angle depends linearly on the ratio ∆ t/t:

Substituting equation (5.15) into expression (5.14), we obtain

(5.16)

At a constant value of the amplitude of the output pulses, the scale of the indicator measuring the average value of the current I 0 , graduated in ∆φ values. In this case, the phase meter indicator scale will be linear. The advantage of a two-channel phase meter is the direct measurement of ∆φ in the range of ±180°.

Electronic discrete counting method is the basis for the operation of a digital phase meter and consists of two main stages: converting the phase shift into the corresponding time interval and measuring this time interval using the discrete counting method.

A simplified block diagram of a digital phase meter and timing diagrams explaining its operation are presented in Fig. 5.13.

Rice. 5.13. Block diagram of a phase meter when measuring the phase shift using the discrete counting method (a), and timing diagrams of signals explaining its operation (b)

The sinusoidal signal generated by the quartz oscillator is fed to the formation block, at the output of which counting pulses are generated, arriving at one input of the time selector. Its other input receives a converted sequence of pulses of duration ∆ t with repetition period of the studied signals T. The selector opens only for a time equal to the duration ∆ t pulses with voltage U 3 and passes voltage pulses to the counter U 4 from the generator. The time selector generates pulse packets with voltage U 5 ( without changing the period T), arriving at the counter in one package.

Where T 0 - repetition period of counting pulses of a quartz oscillator.

Substituting the relation for ∆ into formula (5.17) t from formula (5.16), we determine ∆φ for signals with voltage U 1 And U 2

(5.18)

The overall measurement error by this method depends on the discreteness error, which is due to the fact that the interval ∆ t measured accurate to one period T 0, and from the instability of the response time of the converter.

Great opportunities have phase meters with a built-in microprocessor, which can measure the phase shift between two periodic signals for any selected period.

Figure 5.14 shows a block diagram of a phase meter with a built-in microprocessor and timing diagrams of signals explaining its operation.

After the input device, sinusoidal signals with voltage U 1 And U 2 arrive at the inputs of a pulse converter, in which they are converted into short pulses with voltage U" 1 and U" 2 Using the first pair of these pulses, driver 1 generates a pulse with voltage U 3 duration ∆ t, which is equal to the time shift of signals with voltage U 1 And U2. This pulse opens time selector 1, and during its operation, counting pulses with a repetition period pass to the input of counter 1 T 0, which are generated by the microprocessor. 1 packet of voltage pulses passed to the counter input U 4 shown in Fig. 5.14, b. The number of pulses in a packet is expressed by the formula

At the same time, driver 2 generates pulses with voltage U5, with a duration equal to the repetition period of the studied signals with voltage U 1 And U2. This pulse opens selector 2 (for the duration of its action) and passes a packet of voltage pulses from the microprocessor to counter 2 U 6 and with period T0, the number of which in the package is

Rice. 5.14. Block diagram of a phase meter with a built-in microprocessor ( A) and signal timing diagrams explaining its operation (b)

To determine the desired value of the phase shift ∆φ for the selected signal repetition period T it is necessary to find the ratio of quantities (5.19) and (5.20), equal to

then, taking into account the basic formula ∆φ = 360° ∆ t/T multiply this ratio by 360°:

(5.21)

This calculation is performed by a microprocessor, to which the number codes generated by counters 1 and 2 are transmitted P And N. With the appropriate microprocessor program, the display shows the value of the phase shift ∆φ for any selected period T. By comparing such shifts in different periods it becomes possible to observe fluctuations of ∆φ and estimate their static parameters, which include expected value, dispersion, standard deviation, measured average phase shift.

When measuring with a phase meter with a built-in microprocessor the average value of the phase shift ∆φ for a given amount TO periods T counters 1 and 2 accumulate codes for the number of pulses received at their inputs during TO periods, i.e. number codes PC And N.K. respectively, transmitted to the microprocessor.

A small error in measuring ∆φ with this phase meter can only be obtained at a sufficiently low frequency of the signals being studied. Preliminary (heterodyne) signal conversion allows you to expand the frequency range.

The main metrological characteristics of phase meters that you need to know when choosing a device include the following:

· purpose of the device;

· phase shift measurement range;

· frequency range;

· permissible measurement error.

But because the turns are shifted in space, then the EMF induced in them will not reach amplitude and zero values ​​at the same time.

At the initial moment of time, the EMF of the turn will be:

In these expressions the angles are called phase , or phase . The angles are called initial phase . The phase angle determines the value of the emf at any time, and the initial phase determines the value of the emf at the initial time.

The difference in the initial phases of two sinusoidal quantities of the same frequency and amplitude is called phase angle

Dividing the phase angle by the angular frequency, we obtain the time elapsed since the beginning of the period:

Graphic representation of sinusoidal quantities

U = (U 2 a + (U L - U c) 2)

Thus, due to the presence of a phase angle, the voltage U is always less algebraic sum U a + U L + U C . The difference U L - U C = U p is called reactive voltage component.

Let's consider how current and voltage change in a series circuit alternating current.

Impedance and phase angle. If we substitute the values ​​U a = IR into formula (71); U L = lL and U C =I/(C), then we will have: U = ((IR) 2 + 2), from which we obtain the formula for Ohm’s law for a series alternating current circuit:

I = U / ((R 2 + 2)) = U / Z (72)

Where Z = (R 2 + 2) = (R 2 + (X L - X c) 2)

The Z value is called circuit impedance, it is measured in ohms. The difference L - l/(C) is called circuit reactance and is denoted by the letter X. Therefore, the total resistance of the circuit

Z = (R 2 + X 2)

The relationship between active, reactive and impedance of an alternating current circuit can also be obtained using the Pythagorean theorem from the resistance triangle (Fig. 193). The resistance triangle A'B'C' can be obtained from the voltage triangle ABC (see Fig. 192,b) if we divide all its sides by the current I.

The phase shift angle is determined by the relationship between the individual resistances included in a given circuit. From triangle A’B’C (see Fig. 193) we have:

sin? = X/Z; cos? = R/Z; tg? = X/R

For example, if the active resistance R is significantly greater than the reactance X, the angle is relatively small. If the circuit has a large inductive or large capacitive reactance, then the phase shift angle increases and approaches 90°. Wherein, if the inductive reactance is greater than the capacitive reactance, the voltage and leads the current i by an angle; if the capacitive reactance is greater than the inductive reactance, then the voltage lags behind the current i by an angle.

An ideal inductor, a real coil and a capacitor in an alternating current circuit.

A real coil, unlike an ideal one, has not only inductance, but also active resistance, therefore, when alternating current flows in it, it is accompanied not only by a change in energy in the magnetic field, but also by a transformation electrical energy into a different form. Specifically, in the coil wire, electrical energy is converted into heat in accordance with the Lenz-Joule law.

It was previously found that in an alternating current circuit the process of converting electrical energy into another form is characterized by active power of the circuit P , and the change in energy in the magnetic field is reactive power Q .

In a real coil, both processes take place, i.e. its active and reactive powers are different from zero. Therefore, one real coil in the equivalent circuit must be represented by active and reactive elements.

On the generator armature there are two identical turns 1 and 2, shifted in space (Fig. 5-6). When the armature rotates, e will be induced in the turns. d.s. the same frequency and with the same amplitudes; since the turns rotate with the same angular velocity in the same magnetic field.

Due to the shift of the turns in space, the turns do not simultaneously pass under the middles of the poles and e. i.e. they do not simultaneously reach amplitude values.

When the armature rotates at an angular speed and in the direction opposite to the clockwise direction, at the moment the time begins to count, the turns are located at angles to the neutral plane (Fig. 5-6).

Rice. 5-6. Two turns of the generator armature winding.

Rice. 5-7. Graphs of two variables e. d.s.

Induced in turns e. d.s.

where the angle is called the phase angle or simply phase, so that the instantaneous value of a sinusoidal quantity is determined by the amplitude and phase.

Graphs of these e. d.s. are plotted in Fig. 5-7.

At the initial moment of time, the e.g. induced in turns. d.s.

In Fig. 5-7 they are depicted by the initial ordinates. Electrical angles, determining the values ​​of e. d.s. at the initial moment of time are called initial phase angles or simply initial phases.

Thus, a sinusoidal quantity is characterized by: 1) amplitude, 2) frequency or period, and 3) initial phase.

The difference in the initial phases of two sinusoidal quantities of the same frequency is called the phase angle (phase shift):

The phase shift shows by what part of the period or what period of time one sinusoidal quantity reaches the beginning of the period before another quantity.

The beginning of the period is considered to be the moment in time at which the sinusoidal value passes through the zero value, after which it is positive. The value for which the beginning of the period is reached earlier than the other is considered to be leading in phase, and the one for which the same value is reached later is considered to be lagging in phase.

Two sinusoidal quantities that have the same initial phases are in phase. Two sinusoidal quantities, the phase angle of which is equal to 180°, change in antiphase.

Example 5-3. Two e. d.s. given by equations

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. The phase shift is the determining factor for the power factor in electrical networks alternating current.

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:

1. Using a dual-beam or dual-channel oscilloscope.
2. The compensation method is based on comparing the measured phase shift with the phase shift provided by a reference phase shifter.
3. The sum-difference method consists of using harmonic or shaped square-wave signals.
4. 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:

1. Application of linear sweep.
2. Ellipse method.
3. Circular scanning method.
4. 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 experimental results it can be determined that the phase shift between voltage and current serves to determine the load and cannot depend on variable quantities current and voltage in the electrical network.

As a conclusion, we can say that:

1. The constituent elements of complex resistance, such as resistor and capacitance, as well as conductivity, will not be reciprocal quantities.
2. 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.
3. 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.