1) active resistance
2) capacitor
3) coil
Solution.
An alternating current generator to which some unknown element of the electrical circuit is connected X X.
The graph shows that the amplitude of the current increases linearly with increasing frequency. This is how a capacitor behaves. Indeed, the voltage across the capacitor is related to the charge on its plates by the relation According to Ohm's law, which means, From this we obtain (using the relations for the oscillatory circuit) that the amplitude of the current fluctuations is equal to
Correct answer: 2.
Answer: 2
If, when connecting an unknown element of the electrical circuit to the output of an alternator with a variable frequency of harmonic oscillations at a constant amplitude of voltage oscillations,
the dependence of the amplitude of the current fluctuations on the frequency is found, shown in the figure, then this element of the electrical circuit is
1) active resistance
2) capacitor
3) coil
4) series-connected capacitor and coil
Solution.
X, excites forced electromagnetic oscillations in this element. By the nature of the dependence of the amplitude of current fluctuations on frequency with a constant amplitude of voltage fluctuations, it is possible to establish qualitatively what the element is X... It can be seen from the graph that the amplitude of the current strength decreases with increasing frequency, as this is how the inductor behaves. There are several ways to verify this (in fact, both methods are very close to each other).
The coil has a reactance associated with the frequency of current oscillations in it and its inductance by the ratio The generator creates an alternating voltage and supplies it to the coil. According to Ohm's law, the amplitudes of voltage and current fluctuations are related to the value of the reactance by the ratio It is this dependence on the frequency that we need.
The voltage on the coil, according to the law of electromagnetic induction, is related to the rate of change of the current through it by the relation According to Ohm's law, which means that the rate of change of current magnitude) that the amplitude of current oscillations is equal to
Correct answer: 3.
Answer: 3
If, when connecting an unknown element of the electrical circuit to the output of an alternator with a variable frequency of harmonic oscillations at a constant amplitude of voltage oscillations,
the dependence of the amplitude of the current fluctuations on the frequency is found, shown in the figure, then this element of the electrical circuit is
1) active resistance
2) capacitor
3) coil
4) series-connected capacitor and coil
Solution.
An alternator to which some unknown element of the electrical circuit is connected X, excites forced electromagnetic oscillations in this element. By the nature of the dependence of the amplitude of current fluctuations on frequency with a constant amplitude of voltage fluctuations, it is possible to establish qualitatively what the element is X... It can be seen from the figure that the amplitude of the current strength has a rather sharp maximum at a certain certain value of the frequency. This behavior resembles resonance. From this we conclude that the unknown element is an oscillatory circuit, that is, a capacitor with a coil connected in series. Resonance occurs when the frequency of the alternator matches the natural frequency of the oscillating circuit.
Correct answer: 4.
Answer: 4
If, when connecting an unknown element of the electrical circuit to the output of an alternator with a variable frequency of harmonic oscillations at a constant amplitude of voltage oscillations,
the dependence of the amplitude of the current fluctuations on the frequency is found, shown in the figure, then this element of the electrical circuit is
1) active resistance
2) capacitor
3) coil
4) series-connected capacitor and coil
Solution.
An alternator to which some unknown element of the electrical circuit is connected X, excites forced electromagnetic oscillations in this element. By the nature of the dependence of the amplitude of current fluctuations on frequency with a constant amplitude of voltage fluctuations, it is possible to establish qualitatively what the element is X... It can be seen from the figure that the amplitude of current fluctuations does not change with increasing frequency. This is how active resistance behaves. Indeed, the voltage across the active resistance is related to the strength of the current flowing through it by the ratio According to Ohm's law,
So,
Consequently, the amplitude of the current oscillations does not depend on the frequency and is equal to
Correct answer: 1.
Answer: 1
How will the inductive reactance of the coil change when the frequency of the alternating current decreases by 4 times?
1) will not change
2) will increase by 4 times
3) will decrease by 2 times
4) will decrease by 4 times
Solution.
The inductive resistance of the coil is proportional to the cyclic frequency of the alternating current flowing through it: Therefore, a decrease in the frequency of the alternating current by 4 times will lead to a decrease in the inductive resistance also by 4 times.
Correct answer: 4.
Answer: 4
With an increase in the frequency of the alternating current by 4 times, the inductive reactance of the coil
1) will not change
2) will increase by 4 times
3) will decrease by 2 times
4) will decrease by 4 times
Solution.
The inductive resistance of the coil is proportional to the cyclic frequency of the alternating current flowing through it: Therefore, an increase in the frequency of the alternating current by 4 times will lead to an increase in the inductive resistance also by 4 times.
Correct answer: 2.
Answer: 2
Along the section of the circuit with resistance R alternating current flows. How will the AC power change in this section of the circuit if the effective value of the current strength on it is doubled, and its resistance is halved?
1) will not change
2) will increase 2 times
3) will decrease by 3 times
4) will increase by 4 times
Solution.
The AC power in the section of the circuit with resistance is proportional to the product of the square of the effective current value and the resistance value. Consequently, an increase in the effective current value by 2 times and a decrease in resistance by 2 times will lead to an increase in the current power in this section of the circuit by 2 times.
Correct answer: 2.
Answer: 2
The figure shows oscillograms of voltages on two different elements of an alternating current electrical circuit. Oscillations of these voltages have
1) the same periods, but different amplitudes
2) different periods, but the same amplitudes
3) different periods and different amplitudes
4) the same periods and the same amplitudes
Solution.
The amplitude is the value of the maximum deviation from the equilibrium position (this is half the swing range). The period is called the minimum time after which the oscillation repeats. It can be seen from the graph that the amplitudes of the oscillations differ three times, and the periods of the oscillations coincide.
Answer: 1
Solution.
The oscillation period is related to the frequency by the ratio Therefore, the voltage oscillation period on the desired graph should be equal to
The effective voltage value is called a constant voltage, the action of which produces an equivalent work as the considered alternating voltage during one period. For a harmonic alternating current, the values of the effective voltage and the amplitude of the oscillation are related by the ratio: Therefore, for a current with an effective voltage of about 380 V, the amplitude of the oscillation should be of the order (since the value of the effective voltage has long been with some error, the value of the amplitude is also obtained with the same relative error). Thus, the industrial alternating voltage corresponds to graph 3.
Correct answer: 3.
Answer: 3
Which of the following voltage versus time plots is for commercial AC voltage (50 Hz rms)?
§ 50. Basic quantities characterizing alternating current
Variable e. etc. with., alternating voltage, as well as alternating current are characterized by period, frequency, instantaneous, maximum and effective values.
Period. The time during which the variable e. etc. with. (voltage or current) makes one complete change in magnitude and direction (one cycle), called period... The period is indicated by the letter T and is measured in seconds.
If one complete change in the variable e. etc. with. done in 1/50 sec, then the period of this e. etc. with. equal to 1/50 sec.
Frequency. The number of complete changes in the variable e. etc. with. (voltage or current) performed in one second is called frequency... The frequency is indicated by the letter f and is measured in hertz ( hz). When measuring high frequencies, use the units of kilohertz ( kHz) and megahertz ( MHz); 1 kHz = 1000 hz, 1 MHz = 1000 kHz, 1 MHz = 1 000 000 hz = 10 6 hz... The higher the AC frequency, the shorter the period. Thus, the frequency is the reciprocal of the period.
Example. The duration of one period of alternating current is 1/500 sec... Determine the frequency of the current.
Solution . One full AC change occurs in 1/500 sec... Consequently, 500 such changes will take place in one second. Based on this, the frequency
The longer the period of the alternating current, the lower its frequency. Thus, the period is the reciprocal of the frequency, i.e.
Example. The frequency of the current is 2000 hz (2 kHz). Determine the period of this alternating current.
Solution . For 1 sec there are 2000 total alternating current changes. Consequently, one complete change in current - one period occurs in 1/2000 of a second. But on the basis of this period
Angular frequency. When a loop rotates in a magnetic field, one loop of it corresponds to 360 °, or 2π radians. (one glad= 57 ° 17 '44 ″; π = 3.14.) If, for example, a loop in time T = 3 sec makes one revolution, then the angular velocity of its rotation in one second
Accordingly, the angular velocity of rotation of this loop is expressed in rad / sec and is determined by the ratio This quantity is called angular frequency and is denoted by the letter ω.
In this way,
Since the frequency of the alternating current then, substituting this value f into the expression for the angular frequency, we get:
Angular frequency ω, expressed in rad / sec, more current frequency f, expressed in hertz, by a factor of 2π.
If AC frequency f = 50 hz, then the angular frequency
ω = 2π f= 2 3.14 50 = 314 rad / sec
In various fields of technology, alternating currents of various frequencies are used. At the power plants of the USSR, generators are installed that generate a variable electromotive force, the frequency of which f = 50 hz... In radio engineering and electronics, alternating currents with a frequency of tens to many millions of hertz are used.
Instantaneous and maximum values. The magnitude of the variable electromotive force, current, voltage and power at any time is called instantaneous values these values and denote, respectively, in lowercase letters ( e, i, u, p).
Maximum value(amplitude) variable e. etc. with. (or voltage or current) is the highest value that it reaches in one period. The maximum value of the electromotive force is indicated by E m, voltage - U m, current - I m.
In fig. 51 shows that the variable e. etc. with. reaches its value twice in one period.
Effective value. Electric current flowing through the wires heats them up regardless of its direction. In this regard, heat is released not only in direct current circuits, but also in electrical circuits through which alternating current flows.
If the resistance across the conductor r ohm an alternating electric current flows, then a certain amount of heat is released every second. This amount of heat is directly proportional to the maximum AC value.
You can choose a direct current that, flowing through the same resistance as an alternating current, would give off an equal amount of heat. In this case, we can say that, on average, the action (efficiency) of the alternating current in terms of the amount of heat released is equal to the action of the direct current.
The effective (or effective) value of an alternating current is a direct current that, flowing through an equal resistance and for the same time as an alternating current, emits the same amount of heat.
Electrical measuring instruments (ammeter, voltmeter) connected to the alternating current circuit measure the rms value of current and voltage, respectively.
For sinusoidal alternating current, the effective value is less than the maximum by 1.41 times, that is, by a factor.
Similarly, the effective values of the variable electromotive force and voltage are also 1.41 times less than their maximum values.
By the magnitude of the measured rms values of the alternating current, voltage or electromotive force, their maximum values can be calculated:
E m = E 1.41; U m = U 1.41; I m = I 1.41; (55)
Example. A voltmeter connected to the circuit terminals shows the effective voltage U = 127 v... Calculate the maximum value (amplitude) of this alternating voltage.
Solution . The maximum voltage value is times greater than the effective one, therefore
U m = U· = 127 · 1.41 = 179.07 v
To characterize each variable electromotive force, alternating voltage or alternating current, it is not enough to know the period, frequency and maximum value.
Phase. Phase shift. When comparing two or more variable sinusoidal quantities (emf, voltage or current), it is also necessary to take into account that they can vary in time unequally and reach their maximum value at different points in time. If in an electrical circuit the current changes in time in the same way as the e. etc. with., that is, when the electromotive force is zero and the current in the circuit is zero, and with an increase in e. etc. with. to a positive maximum value simultaneously increases and reaches a positive maximum value and the current in the circuit, and then, when e. etc. with. decreases to zero and the current strength simultaneously becomes zero, etc., then in such a circuit the alternating electromotive force and the alternating current coincide in phase.
In fig. 52 shows the moments of rotation of two conductors in a magnetic field and graphs of changes in e. etc. with. in the wires. The wire 1
and wire 2
shifted by an angle φ = 90 °. When the magnetic flux is crossed, a variable e is generated in each of the wires. etc. with. When on the wire 2
electromotive force is zero, in the wire 1
it will be maximum. In the wire 2
e. etc. with. gradually increases and reaches its maximum value at the moment t 1, and in the wire 1
induced e. etc. with. gradually decreases and at the same moment of time is equal to zero. Thus, the e. etc. with. do not coincide in phase, but are shifted one relative to the other in phase by 1/4 of the period or by an angle φ = 90 °. Also, e. etc. with. in the wire 1
reaches a maximum earlier than e. etc. with. in the wire 2
, and therefore consider that the electromotive force e 1 is ahead of the phase e. etc. with. e 2 or e. etc. with. e 2 lags behind in phase from e. etc. with. e one . When calculating AC circuits, the phase shift between alternating voltage and current is of great practical importance.
Resonant frequency measurement method.
Frequency comparison method;
The discrete counting method is based on counting pulses of the required frequency for a specific period of time. It is most often used by digital frequency meters, and it is thanks to this simple method that fairly accurate data can be obtained.
You can learn more about the frequency of alternating current from the video:
The method of overcharging a capacitor also does not involve complex calculations. In this case, the average value of the recharge current is proportionally related to the frequency, and is measured using a magnetoelectric ammeter. The scale of the device, in this case, is graduated in Hertz.
The error of such frequency meters is within 2%, and therefore such measurements are quite suitable for domestic use.
The measurement method is based on electrical resonance arising in a circuit with adjustable elements. The frequency to be measured is determined by a special scale of the tuning mechanism itself.
This method gives a very low error, but only applies for frequencies above 50 kHz.
The frequency comparison method is used in oscilloscopes, and is based on mixing the reference frequency with the measured one. In this case, beats of a certain frequency occur. When these beats reach zero, then the measured one becomes equal to the reference one. Further, according to the figure obtained on the screen, using the formulas, you can calculate the desired frequency of the electric current.
Another interesting video about AC frequency:
The time during which one cycle of oscillation (complete change in the EMF) or one complete revolution of the radius vector takes place is called alternating current oscillation period
The period is measured in seconds and denoted by a Latin letter T... Smaller units of the period are also used, it is the millisecond. (ms)- one thousandth of a second and a microsecond (μs)- one millionth of a second.
1 ms = 0.001 sec = 10 -3 sec.
1 μs = 0.001 ms = 0.000001 sec = 10 -6 sec.
1000 μs = 1 ms.
The faster the EMF change, the shorter the oscillation period and the higher the frequency. Therefore, the frequency and period of the current are quantities that are inversely proportional to each other. The mathematical relationship between period and frequency is described by formulas.
The frequency is denoted by a Latin letter f and is expressed in periods per second or in hertz... One thousand hertz is called a kilohertz (kHz) and a million hertz is megahertz (MHz)... The physical unit is also used gigahertz (GHz) equal to one thousand megahertz.
1000 Hz = 10 3 Hz = 1 kHz;
1,000,000 Hz = 10 6 Hz = 1,000 kHz = 1 MHz;
1,000,000,000 Hz = 10 9 Hz = 1,000,000 kHz = 1,000 MHz = 1 GHz;
f = 1 / T or T = 1 / f
For example, it is known that the frequency of the current in the AC electrical network is 50 Hz, then the period will be 0.02 seconds
Frequencies from 20 to 20,000 Hz are called sound frequencies, since they can be perceived by the human ear. Next are ultrasonic frequencies, these are elastic waves of the range slightly higher than the sound one from 20 kHz or more, high frequency, perfectly demonstrates the work of ultrasound. But for example, some radio transmitters or mobile phones operate at frequencies already MHz and even GHz. Therefore, high frequencies are called radio frequencies. In addition, higher frequencies are used, for example, in antennas of radar stations, satellite communications, GLONASS, GPS, the frequency range is from 40 GHz and even higher.
The maximum value that the EMF or current reaches during a period is called amplitude EMF or AC power. It is easy to see from the figure that the scale amplitude is determined by the length of the radius vector. Amplitudes of current, EMF and voltage are indicated respectively by Latin symbols Im, Em and Um.
AC angular frequencyThe speed of rotation of the radius vector, or the change in the value of the angle of rotation within one second, is called the angular frequency of alternating current and is denoted by the Greek symbol ω (omega). The angle of rotation of the radius vector at any moment relative to its initial location is measured not in degrees, but in special units - radians... A radian is the angular value of a circular arc whose length corresponds to the radius of that circle. The entire 360 ° circumference is 6.28 radians, which is 2π.
Then, 1 rad = 360 ° / 2π
This means that the end of the radius vector travels a path equal to 6.28 radians (2π) during one period. Since within a second the radius vector will make a number of revolutions corresponding to the frequency of the alternating current f, then in a second its end will pass a path equal to 6.28 × f radian. This expression, which speaks about the speed of rotation of the radius vector, is the angular frequency of the alternating current ω.
ω = 6.28 × f = 2fπ
The angle of rotation of the radius vector at any possible moment relative to its initial position is called AC phase... The phase characterizes the magnitude of the EMF or current at some arbitrary specific instant or, as they say, the instantaneous value of the EMF, its direction in the circuit and the direction of its change; phase indicates whether the EMF decreases or increases, at an arbitrary moment in time
The full cycle (revolution) of the radius vector is 360 ° degrees. With the beginning of a new cycle of the radius vector, the EMF is changed in the same order as during the first revolution. Therefore, all phases of the EMF will go in the same order. For example, the EMF phase when the radius vector is rotated by an angle of 370 degrees will be the same as when the radius vector is rotated by ten degrees. In both cases, the radius vector will take the same position, and, therefore, the instantaneous EMF values will be the same in phase in both cases.
The time during which one complete change in the EMF occurs, that is, one cycle of oscillation or one complete revolution of the radius vector, is called alternating current oscillation period(picture 1).
Picture 1. The period and amplitude of the sinusoidal oscillation. Period is the time of one oscillation; Aplitude is its greatest instantaneous value.
The period is expressed in seconds and denoted by the letter T.
Smaller units of the period are also used: millisecond (ms) - one thousandth of a second and microsecond (μs) - one millionth of a second.
1 ms = 0.001 sec = 10 -3 sec.
1 μs = 0.001 ms = 0.000001 sec = 10 -6 sec.
1000 μs = 1 ms.
The number of complete changes in the EMF or the number of revolutions of the radius vector, that is, in other words, the number of complete cycles of oscillations performed by an alternating current during one second is called alternating current oscillation frequency.
The frequency is indicated by the letter f and is expressed in periods per second or in hertz.
One thousand hertz is called kilohertz (kHz) and a million hertz is called megahertz (MHz). There is also a unit of gigahertz (GHz) equal to one thousand megahertz.
1000 Hz = 10 3 Hz = 1 kHz;
1,000,000 Hz = 10 6 Hz = 1,000 kHz = 1 MHz;
1,000,000,000 Hz = 10 9 Hz = 1,000,000 kHz = 1,000 MHz = 1 GHz;
The faster the EMF changes, that is, the faster the radius vector rotates, the shorter the oscillation period. The faster the radius vector rotates, the higher the frequency. Thus, the frequency and period of alternating current are quantities that are inversely proportional to each other. The larger one of them, the smaller the other.
The mathematical relationship between the period and frequency of alternating current and voltage is expressed by the formulas
For example, if the current frequency is 50 Hz, then the period will be:
T = 1 / f = 1/50 = 0.02 sec.
And vice versa, if it is known that the period of the current is 0.02 sec, (T = 0.02 sec.), Then the frequency will be equal to:
f = 1 / T = 1 / 0.02 = 100/2 = 50 Hz
The frequency of alternating current used for lighting and industrial purposes is exactly 50 Hz.
Frequencies between 20 and 20,000 Hz are called audio frequencies. The currents in the antennas of radio stations fluctuate with frequencies up to 1,500,000,000 Hz, or, in other words, up to 1,500 MHz or 1.5 GHz. These high frequencies are called radio frequencies or high frequency vibrations.
Finally, the currents in the antennas of radar stations, satellite communication stations, and other special systems (for example, GLANASS, GPS) fluctuate at frequencies up to 40,000 MHz (40 GHz) and higher.
AC Amplitude
The largest value that the EMF or current reaches in one period is called amplitude of EMF or AC current... It is easy to see that the scale amplitude is equal to the length of the radius vector. Amplitudes of current, EMF and voltage are indicated respectively by letters Im, Em and Um (picture 1).
Angular (cyclic) AC frequency.
The speed of rotation of the radius vector, that is, the change in the value of the angle of rotation within one second, is called the angular (cyclic) frequency of the alternating current and is denoted by the Greek letter ? (omega). The angle of rotation of the radius vector at any given moment relative to its initial position is usually measured not in degrees, but in special units - radians.
A radian is the angular value of an arc of a circle, the length of which is equal to the radius of this circle (Figure 2). The entire 360 ° circumference is 6.28 radians, which is 2.
Figure 2.
1rad = 360 ° / 2
Therefore, the end of the radius vector in one period cover a path equal to 6.28 radians (2). Since within one second the radius vector makes a number of revolutions equal to the frequency of the alternating current f, then in one second its end runs a path equal to 6.28 * f radian. This expression characterizing the speed of rotation of the radius vector, and will be the angular frequency of the alternating current -? ...
? = 6.28 * f = 2f
The angle of rotation of the radius vector at any given instant relative to its initial position is called AC phase... The phase characterizes the magnitude of the EMF (or current) at a given moment or, as they say, the instantaneous value of the EMF, its direction in the circuit and the direction of its change; the phase shows whether the EMF decreases or increases.
Figure 3.
A complete revolution of the radius vector is 360 °. With the beginning of a new revolution of the radius vector, the EMF changes in the same order as during the first revolution. Consequently, all phases of the EMF will repeat in the same order. For example, the EMF phase when the radius vector is rotated by an angle of 370 ° will be the same as when the radius vector is rotated by 10 °. In both of these cases, the radius vector occupies the same position, and, therefore, the instantaneous values of the EMF will be the same in phase in both of these cases.
