Laws for connecting conductors. Series and parallel connection of conductors
All known species conductors have certain properties, including electrical resistance. This quality has found its application in resistors, which are circuit elements with a precisely set resistance. They allow you to adjust current and voltage with high precision in circuits. All such resistances have their own individual qualities. For example, the power for parallel and series connection of resistors will be different. Therefore, in practice, various calculation methods are often used, thanks to which it is possible to obtain accurate results.
Properties and technical characteristics of resistors
As already noted, resistors in electrical circuits and circuits perform a regulatory function. For this purpose, Ohm's law is used, expressed by the formula: I = U/R. Thus, with a decrease in resistance, a noticeable increase in current occurs. And, conversely, the higher the resistance, the lower the current. Due to this property, resistors are widely used in electrical engineering. On this basis, current dividers are created that are used in the designs of electrical devices.
In addition to the current regulation function, resistors are used in voltage divider circuits. In this case, Ohm's law will look slightly different: U = I x R. This means that as the resistance increases, the voltage increases. The entire operation of devices designed to divide voltage is based on this principle. For current dividers, a parallel connection of resistors is used, and for a serial connection.
In the diagrams, resistors are displayed in the form of a rectangle measuring 10x4 mm. The symbol R is used for designation, which can be supplemented with a power value of this element. For power above 2 W, the designation is made using Roman numerals. The corresponding inscription is placed on the diagram near the resistor icon. Power is also included in the composition applied to the element body. The units of resistance are ohm (1 ohm), kilohm (1000 ohm) and megaohm (1,000,000 ohm). The range of resistors ranges from fractions of an ohm to several hundred megaohms. Modern technologies make it possible to produce these elements with fairly accurate resistance values.
An important parameter of a resistor is the resistance deviation. It is measured as a percentage of the nominal value. The standard series of deviations represents values in the form: + 20, + 10, + 5, + 2, + 1% and so on up to the value + 0,001%.
The power of the resistor is of great importance. For each of them, during operation, electricity, causing heating. If permissible value the dissipated power exceeds the norm, this will lead to failure of the resistor. It should be taken into account that during the heating process the resistance of the element changes. Therefore, if devices operate over wide temperature ranges, a special value called the temperature coefficient of resistance is used.
To connect resistors in circuits, three different connection methods are used - parallel, series and mixed. Each method has individual qualities, which allows these elements to be used for a variety of purposes.
Power in series connection
When resistors are connected in series, electric current passes through each resistance in turn. The current value at any point in the circuit will be the same. This fact determined using Ohm's law. If you add up all the resistances shown in the diagram, you get the following result: R = 200+100+51+39 = 390 Ohms.
Considering the voltage in the circuit is 100 V, the current will be I = U/R = 100/390 = 0.256 A. Based on the data obtained, the power of the resistors in series connection can be calculated using the following formula: P = I 2 x R = 0.256 2 x 390 = 25.55 W.
- P 1 = I 2 x R 1 = 0.256 2 x 200 = 13.11 W;
- P 2 = I 2 x R 2 = 0.256 2 x 100 = 6.55 W;
- P 3 = I 2 x R 3 = 0.256 2 x 51 = 3.34 W;
- P 4 = I 2 x R 4 = 0.256 2 x 39 = 2.55 W.
If we add up the received power, then the total P will be: P = 13.11 + 6.55 + 3.34 + 2.55 = 25.55 W.
Power with parallel connection
With a parallel connection, all the beginnings of the resistors are connected to one node of the circuit, and the ends to another. In this case, the current branches out and it begins to flow through each element. According to Ohm's law, the current will be inversely proportional to all connected resistances, and the voltage value across all resistors will be the same.
Before calculating the current, it is necessary to calculate the admittance of all resistors using the following formula:
- 1/R = 1/R 1 +1/R 2 +1/R 3 +1/R 4 = 1/200+1/100+1/51+1/39 = 0.005+0.01+0.0196+ 0.0256 = 0.06024 1/Ohm.
- Since resistance is a quantity inversely proportional to conductivity, its value will be: R = 1/0.06024 = 16.6 Ohms.
- Using a voltage value of 100 V, Ohm's law calculates the current: I = U/R = 100 x 0.06024 = 6.024 A.
- Knowing the current strength, the power of resistors connected in parallel is determined as follows: P = I 2 x R = 6.024 2 x 16.6 = 602.3 W.
- The current strength for each resistor is calculated using the formulas: I 1 = U/R 1 = 100/200 = 0.5A; I 2 = U/R 2 = 100/100 = 1A; I 3 = U/R 3 = 100/51 = 1.96A; I 4 = U/R 4 = 100/39 = 2.56A. Using these resistances as an example, a pattern can be seen that as the resistance decreases, the current increases.
There is another formula that allows you to calculate the power when resistors are connected in parallel: P 1 = U 2 / R 1 = 100 2 / 200 = 50 W; P 2 = U 2 /R 2 = 100 2 /100 = 100 W; P 3 = U 2 /R 3 = 100 2 /51 = 195.9 W; P 4 = U 2 / R 4 = 100 2 / 39 = 256.4 W. By adding up the powers of individual resistors, you get their total power: P = P 1 + P 2 + P 3 + P 4 = 50 + 100 + 195.9 + 256.4 = 602.3 W.
Thus, the power for series and parallel connection of resistors is determined different ways, with which you can get the most accurate results.
Usually everyone finds it difficult to answer. But this riddle, when applied to electricity, is solved quite definitely.
Electricity begins with Ohm's law.
And if we consider the dilemma in the context of parallel or serial connections - considering one connection to be a chicken and the other to be an egg, then there is no doubt at all.
Because Ohm's law is the very original electrical circuit. And it can only be consistent.
Yes, they came up with a galvanic cell and didn’t know what to do with it, so they immediately came up with another light bulb. And this is what came out of it. Here, a voltage of 1.5 V immediately flowed as current, in strict compliance with Ohm's law, through the light bulb to the back of the same battery. And inside the battery itself, under the influence of the sorceress-chemistry, the charges again ended up at the original point of their journey. And therefore, where the voltage was 1.5 volts, it remains that way. That is, the voltage is always the same, and the charges are constantly moving and successively pass through the light bulb and the galvanic cell.
And it is usually drawn on the diagram like this:
According to Ohm's law I=U/R
Then the resistance of the light bulb (with the current and voltage that I wrote) will be
R= 1/U, WhereR = 1 Ohm
And the power will be released P = I * U , that is, P=2.25 Vm
IN series circuit, especially with such a simple and undoubted example, it is clear that the current that runs through it from beginning to end is the same all the time. And if we now take two light bulbs and make sure that the current runs first through one and then through the other, then the same thing will happen again - the current will be the same in both the light bulb and the other. Although different in size. The current now experiences the resistance of two light bulbs, but each of them has the same resistance as it was, and remains the same, because it is determined solely by the physical properties of the light bulb itself. We calculate the new current again using Ohm's law.
It will turn out to be equal to I=U/R+R, that is, 0.75A, exactly half of the current that was at first.
In this case, the current has to overcome two resistances, it becomes smaller. As can be seen from the glow of the light bulbs - they are now burning at full intensity. A total resistance a chain of two light bulbs will be equal to the sum of their resistances. Knowing arithmetic, in a particular case you can use the action of multiplication: if N identical light bulbs are connected in series, then their total resistance will be equal to N multiplied by R, where R is the resistance of one light bulb. The logic is impeccable.
And we will continue our experiments. Now let's do something similar to what we did with light bulbs, but only on the left side of the circuit: add another galvanic element, exactly the same as the first. As you can see, now our total voltage has doubled, and the current has returned to 1.5 A, which is signaled by the light bulbs, which light up again at full power.
We conclude:
- For serial connection electrical circuit the resistances and voltages of its elements are summed up, and the current on all elements remains unchanged.
It is easy to verify that this statement is true for both active components (galvanic cells) and passive ones (light bulbs, resistors).
That is, this means that the voltage measured across one resistor (it is called the voltage drop) can be safely summed up with the voltage measured across another resistor, and the total will be the same 3 V. And at each of the resistances it will be equal to half - then there is 1.5 V. And this is fair. Two galvanic cells produce their voltages, and two light bulbs consume them. Because in a voltage source, the energy of chemical processes is converted into electricity, which takes the form of voltage, and in light bulbs the same energy is converted from electrical into heat and light.
Let's return to the first circuit, connect another light bulb in it, but differently.
Now the voltage at the points connecting the two branches is the same as on galvanic cell- 1.5 V. But since the resistance of both bulbs is also the same as it was, then the current through each of them will flow 1.5 A - the “full glow” current.
The galvanic cell now supplies them with current at the same time, therefore, both of these currents flow out of it at once. That is, the total current from the voltage source will be 1.5 A + 1.5 A = 3.0 A.
What is the difference between this circuit and the circuit when the same light bulbs were connected in series? Only in the glow of light bulbs, that is, only in current.
Then the current was 0.75 A, but now it is immediately 3 A.
It turns out that if we compare it with the original circuit, then when connecting the light bulbs in series (scheme 2), there was more resistance to the current (which is why it decreased, and the light bulbs lost their luminosity), and a parallel connection has LESS resistance, although the resistance of the light bulbs remained unchanged. What's the matter?
But the fact is that we forget one interesting truth, that every sword is a double-edged sword.
When we say that a resistor resists current, we seem to forget that it still conducts current. And now that the light bulbs have been connected in parallel, their overall ability to conduct current rather than resist it has increased. Well, and, accordingly, a certain amount G, by analogy with resistance R and should be called conductivity. And it must be summed up in a parallel connection of conductors.
Well here she is
Ohm's law will then look like
I = U* G&
And in the case of a parallel connection, the current I will be equal to U*(G+G) = 2*U*G, which is exactly what we observe.
Replacement of circuit elements with a common equivalent element
Engineers often need to recognize currents and voltages in all parts of circuits. But real electrical circuits can be quite complex and branched and can contain many elements that actively consume electricity and are connected to each other in completely different combinations. It's called calculation electrical diagrams. It is done when designing the energy supply of houses, apartments, and organizations. In this case, it is very important what currents and voltages will act in the electrical circuit, if only in order to select appropriate wire sections, loads on the entire network or its parts, and so on. And how complicated they can be electronic circuits, containing thousands, or even millions of elements, I think everyone understands.
The very first thing that suggests itself is to use the knowledge of how voltage currents behave in such simple network connections as serial and parallel. They do this: instead of a serial connection found on the network of two or more active consumer devices (like our light bulbs), draw one, but so that its resistance is the same as both. Then the picture of currents and voltages in the rest of the circuit will not change. Similarly with parallel connections: instead of them, draw an element whose CONDUCTIVITY would be the same as both.
Now, if we redraw the circuit, replacing the serial and parallel connections with one element, we will get a circuit called an “equivalent equivalent circuit.”
This procedure can be continued until we are left with the simplest one - with which we illustrated Ohm’s law at the very beginning. Only instead of the light bulb there will be one resistance, which is called the equivalent load resistance.
This is the first task. It allows us to use Ohm's law to calculate the total current in the entire network, or the total load current.
This is the full calculation electrical network.
Examples
Let the chain contain 9 active resistances. It could be light bulbs or something else.
A voltage of 60 V is applied to its input terminals.
The resistance values for all elements are as follows:
Find all unknown currents and voltages.
It is necessary to follow the path of searching for parallel and serial sections of the network, calculating their equivalent resistances and gradually simplifying the circuit. We see that R 3, R 9 and R 6 are connected in series. Then their equivalent resistance R e 3, 6, 9 will be equal to their sum R e 3, 6, 9 = 1 + 4 + 1 Ohm = 6 Ohm.
Now we replace the parallel piece of resistance R 8 and R e 3, 6, 9, getting R e 8, 3, 6, 9. Only when connecting conductors in parallel will the conductivity have to be added.
Conductivity is measured in units called siemens, the reciprocal of ohms.
If we turn the fraction over, we get resistance R e 8, 3, 6, 9 = 2 Ohm
Exactly the same as in the first case, we combine resistances R 2, R e 8, 3, 6, 9 and R 5 connected in series, obtaining R e 2, 8, 3, 6, 9, 5 = 1 + 2 + 1 = 4 Ohm.
There are two steps left: obtain a resistance equivalent to two resistors for parallel connection of conductors R 7 and R e 2, 8, 3, 6, 9, 5.
It is equal to R e 7, 2, 8, 3, 6, 9, 5 = 1/(1/4+1/4)=1/(2/4)=4/2 = 2 Ohm
On last step sum up all the series-connected resistances R 1, R e 7, 2, 8, 3, 6, 9, 5 and R 4 and get a resistance equivalent to the resistance of the entire circuit R e and equal to the sum of these three resistances
R e = R 1 + R e 7, 2, 8, 3, 6, 9, 5 + R4 = 1 + 2 + 1 = 4 Ohm
Well, let’s remember in whose honor the unit of resistance we wrote in the last of these formulas was named, and use his law to calculate the total current in the entire circuit I
Now, moving in the opposite direction, towards increasing complexity of the network, we can obtain currents and voltages in all chains of our fairly simple circuit according to Ohm’s law.
This is how apartment power supply schemes are usually calculated, which consist of parallel and serial sections. Which, as a rule, is not suitable in electronics, because a lot of things work there differently, and everything is much more intricate. And such a circuit, for example, when you don’t understand whether the connection of conductors is parallel or serial, is calculated according to Kirchhoff’s laws.
Content:
As you know, the connection of any circuit element, regardless of its purpose, can be of two types - parallel connection and serial connection. A mixed, that is, series-parallel connection is also possible. It all depends on the purpose of the component and the function it performs. This means that resistors do not escape these rules. The series and parallel resistance of resistors is essentially the same as the parallel and series connection of light sources. IN parallel circuit The connection diagram implies input to all resistors from one point, and output from another. Let's try to figure out how it's done serial connection, and in what way - parallel. And most importantly, what is the difference between such connections and in which cases is a serial and in which parallel connection necessary? It is also interesting to calculate such parameters as the total voltage and total resistance of the circuit in cases of series or parallel connection. Let's start with definitions and rules.
Connection methods and their features
The types of connections of consumers or elements play a very important role, because the characteristics of the entire circuit, the parameters of individual circuits, and the like depend on this. First, let's try to figure out the serial connection of elements to the circuit.
Serial connection
A serial connection is a connection where resistors (as well as other consumers or circuit elements) are connected one after another, with the output of the previous one connected to the input of the next one. This type of switching of elements gives an indicator equal to the sum of the resistances of these circuit elements. That is, if r1 = 4 Ohms, and r2 = 6 Ohms, then when they are connected in a series circuit, the total resistance will be 10 Ohms. If we add another 5 ohm resistor in series, adding these numbers will give 15 ohms - this will be the total resistance of the series circuit. That is general values equal to the sum of all resistances. When calculating it for elements that are connected in series, no questions arise - everything is simple and clear. That is why there is no need to even dwell more seriously on this.
Completely different formulas and rules are used to calculate the total resistance of resistors at parallel connection, here it makes sense to dwell on it in more detail.
Parallel connection
A parallel connection is a connection in which all resistor inputs are combined at one point, and all outputs at the second. The main thing to understand here is that the total resistance with such a connection will always be lower than the same parameter of the resistor that has the smallest one.
It makes sense to analyze such a feature using an example, then it will be much easier to understand. There are two 16 ohm resistors, but only 8 ohms are required for proper installation of the circuit. IN in this case when you use both of them, when they are connected in parallel to the circuit, you will get the required 8 ohms. Let's try to understand by what formula calculations are possible. This parameter can be calculated as follows: 1/Rtotal = 1/R1+1/R2, and when adding elements, the sum can continue indefinitely.
Let's try another example. 2 resistors are connected in parallel, with a resistance of 4 and 10 ohms. Then the total will be 1/4 + 1/10, which will be equal to 1:(0.25 + 0.1) = 1:0.35 = 2.85 ohms. As you can see, although the resistors had significant resistance, when they were connected in parallel, the overall value became much lower.
You can also calculate the total resistance of four parallel connected resistors, with a nominal value of 4, 5, 2 and 10 ohms. The calculations, according to the formula, will be as follows: 1/Rtotal = 1/4+1/5+1/2+1/10, which will be equal to 1:(0.25+0.2+0.5+0.1)=1/1.5 = 0.7 Ohm.
As for the current flowing through parallel-connected resistors, here it is necessary to refer to Kirchhoff’s law, which states “the current strength in a parallel connection leaving the circuit is equal to the current entering the circuit.” Therefore, here the laws of physics decide everything for us. In this case, the total current indicators are divided into values that are inversely proportional to resistance branches. To put it simply, the higher the resistance value, the smaller the currents will pass through this resistor, but in general, the input current will still be at the output. In a parallel connection, the voltage at the output also remains the same as at the input. The parallel connection diagram is shown below.
Series-parallel connection
A series-parallel connection is when a series connection circuit contains parallel resistances. In this case, the general series resistance will be equal to the sum of individual common parallel ones. The calculation method is the same in the relevant cases.
Summarize
Summarizing all of the above, we can draw the following conclusions:
- When connecting resistors in series, no special formulas are required to calculate the total resistance. You just need to add up all the indicators of the resistors - the sum will be the total resistance.
- When connecting resistors in parallel, the total resistance is calculated using the formula 1/Rtot = 1/R1+1/R2…+Rn.
- The equivalent resistance in a parallel connection is always less than the minimum similar value of one of the resistors included in the circuit.
- The current, as well as the voltage, in a parallel connection remains unchanged, that is, the voltage in a series connection is the same at both the input and output.
- A serial-parallel connection during calculations is subject to the same laws.
In any case, whatever the connection, it is necessary to clearly calculate all the indicators of the elements, because the parameters play a very important role when installing circuits. And if you make a mistake in them, then either the circuit will not work, or its elements will simply burn out from overload. In fact, this rule applies to any circuit, even in electrical installations. After all, the cross-section of the wire is also selected based on power and voltage. And if you put a light bulb rated at 110 volts in a circuit with a voltage of 220, it’s easy to understand that it will burn out instantly. The same goes for radio electronics elements. Therefore, attentiveness and scrupulousness in calculations is the key proper operation scheme.
A sequential connection is a connection of circuit elements in which the same current I occurs in all elements included in the circuit (Fig. 1.4).
Based on Kirchhoff’s second law (1.5), the total voltage U of the entire circuit is equal to the sum of the voltages in individual sections:
U = U 1 + U 2 + U 3 or IR eq = IR 1 + IR 2 + IR 3,
whence follows
R eq = R 1 + R 2 + R 3.
Thus, when connecting circuit elements in series, the total equivalent resistance of the circuit is equal to the arithmetic sum of the resistances of the individual sections. Consequently, a circuit with any number of series-connected resistances can be replaced by a simple circuit with one equivalent resistance R eq (Fig. 1.5). After this, the calculation of the circuit is reduced to determining the current I of the entire circuit according to Ohm’s law
and using the above formulas, calculate the voltage drop U 1 , U 2 , U 3 in the corresponding sections of the electrical circuit (Fig. 1.4).
The disadvantage of sequential connection of elements is that if at least one element fails, the operation of all other elements of the circuit stops.
Electric circuit with parallel connection of elements
A parallel connection is a connection in which all consumers of electrical energy included in the circuit are under the same voltage (Fig. 1.6).
In this case, they are connected to two circuit nodes a and b, and based on Kirchhoff’s first law, we can write that the total current I of the entire circuit is equal to the algebraic sum of the currents of the individual branches:
I = I 1 + I 2 + I 3, i.e.
whence it follows that
.
In the case when two resistances R 1 and R 2 are connected in parallel, they are replaced by one equivalent resistance
.
From relation (1.6), it follows that the equivalent conductivity of the circuit is equal to the arithmetic sum of the conductivities of the individual branches:
g eq = g 1 + g 2 + g 3.
As the number of parallel-connected consumers increases, the conductivity of the circuit g eq increases, and vice versa, the total resistance R eq decreases.
Voltages in an electrical circuit with resistances connected in parallel (Fig. 1.6)
U = IR eq = I 1 R 1 = I 2 R 2 = I 3 R 3.
It follows that
those. The current in the circuit is distributed between parallel branches in inverse proportion to their resistance.
According to a parallel-connected circuit, consumers of any power, designed for the same voltage, operate in nominal mode. Moreover, turning on or off one or more consumers does not affect the operation of the others. Therefore, this circuit is the main circuit for connecting consumers to a source of electrical energy.
Electric circuit with a mixed connection of elements
A mixed connection is a connection in which the circuit contains groups of parallel and series-connected resistances.
For the circuit shown in Fig. 1.7, the calculation of equivalent resistance begins from the end of the circuit. To simplify the calculations, we assume that all resistances in this circuit are the same: R 1 =R 2 =R 3 =R 4 =R 5 =R. Resistances R 4 and R 5 are connected in parallel, then the resistance of the circuit section cd is equal to:
.
In this case, the original circuit (Fig. 1.7) can be represented in the following form (Fig. 1.8):
In the diagram (Fig. 1.8), resistance R 3 and R cd are connected in series, and then the resistance of the circuit section ad is equal to:
.
Then the diagram (Fig. 1.8) can be presented in an abbreviated version (Fig. 1.9):
In the diagram (Fig. 1.9) the resistance R 2 and R ad are connected in parallel, then the resistance of the circuit section ab is equal to
.
The circuit (Fig. 1.9) can be represented in a simplified version (Fig. 1.10), where resistances R 1 and R ab are connected in series.
Then the equivalent resistance of the original circuit (Fig. 1.7) will be equal to:
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Rice. 1.10
Rice. 1.11As a result of the transformations, the original circuit (Fig. 1.7) is presented in the form of a circuit (Fig. 1.11) with one resistance R eq. Calculation of currents and voltages for all elements of the circuit can be made according to Ohm's and Kirchhoff's laws.
LINEAR CIRCUITS OF SINGLE-PHASE SINEUSOIDAL CURRENT.
Obtaining sinusoidal EMF. . Basic characteristics of sinusoidal current
The main advantage of sinusoidal currents is that they allow the most economical production, transmission, distribution and use of electrical energy. The feasibility of their use is due to the fact that the efficiency of generators, electric motors, transformers and power lines in this case is the highest.
To obtain sinusoidally varying currents in linear circuits, it is necessary that e. d.s. also changed according to a sinusoidal law. Let us consider the process of occurrence of sinusoidal EMF. The simplest sinusoidal EMF generator can be a rectangular coil (frame), uniformly rotating in a uniform magnetic field with angular velocity ω (Fig. 2.1, b).
Magnetic flux passing through the coil as the coil rotates abcd induces (induces) in it based on the law of electromagnetic induction EMF e . The load is connected to the generator using brushes 1 , pressed against two slip rings 2 , which in turn are connected to the coil. Coil induced value abcd e. d.s. at each moment of time is proportional to the magnetic induction IN, the size of the active part of the coil l = ab + dc and the normal component of the speed of its movement relative to the field vn:
e = Blvn (2.1)
Where IN And l- constant values, a vn- a variable depending on the angle α. Expressing the speed v n through the linear speed of the coil v, we get
e = Blv·sinα (2.2)
In expression (2.2) the product Blv= const. Therefore, e. d.s. induced in a coil rotating in a magnetic field is a sinusoidal function of the angle α .
If the angle α = π/2, then the product Blv in formula (2.2) there is a maximum (amplitude) value of the induced e. d.s. E m = Blv. Therefore, expression (2.2) can be written in the form
e = Emsinα (2.3)
Because α is the angle of rotation in time t, then, expressing it in terms of angular velocity ω , we can write α = ωt, and rewrite formula (2.3) in the form
e = Emsinωt (2.4)
Where e- instantaneous value e. d.s. in a reel; α = ωt- phase characterizing the value of e. d.s. V this moment time.
It should be noted that instant e. d.s. over an infinitesimal period of time can be considered a constant value, therefore for instantaneous values of e. d.s. e, voltage And and currents i the laws of direct current are valid.
Sinusoidal quantities can be represented graphically by sinusoids and rotating vectors. When depicting them as sinusoids, instantaneous values of quantities are plotted on the ordinate on a certain scale, and time is plotted on the abscissa. If a sinusoidal quantity is represented by rotating vectors, then the length of the vector on the scale reflects the amplitude of the sinusoid, the angle formed with the positive direction of the abscissa axis at the initial time is equal to the initial phase, and the rotation speed of the vector is equal to the angular frequency. Instantaneous values of sinusoidal quantities are projections of the rotating vector onto the ordinate axis. It should be noted that the positive direction of rotation of the radius vector is considered to be the direction of rotation counterclockwise. In Fig. 2.2 graphs of instantaneous e values are plotted. d.s. e And e".
If the number of pairs of magnet poles p ≠ 1, then in one revolution of the coil (see Fig. 2.1) occurs p full cycles of change e. d.s. If the angular frequency of the coil (rotor) n revolutions per minute, then the period will decrease by pn once. Then the frequency e. d.s., i.e. the number of periods per second,
f = Pn / 60
From Fig. 2.2 it is clear that ωТ = 2π, where
ω = 2π / T = 2πf (2.5)
Size ω , proportional to the frequency f and equal to the angular velocity of rotation of the radius vector, is called the angular frequency. Angular frequency is expressed in radians per second (rad/s) or 1/s.
Graphically depicted in Fig. 2.2 e. d.s. e And e" can be described by expressions
e = Emsinωt; e" = E"msin(ωt + ψe") .
Here ωt And ωt + ψe"- phases characterizing the values of e. d.s. e And e" at a given point in time; ψ e"- the initial phase that determines the value of e. d.s. e" at t = 0. For e. d.s. e the initial phase is zero ( ψ e = 0 ). Corner ψ always counted from the zero value of the sinusoidal value when it passes from negative to positive values to the origin (t = 0). In this case, the positive initial phase ψ (Fig. 2.2) are laid to the left of the origin (towards negative values ωt), and the negative phase - to the right.
If two or more sinusoidal quantities that change with the same frequency do not have the same sinusoidal origins in time, then they are shifted relative to each other in phase, i.e., they are out of phase.
Angle difference φ , equal to the difference in the initial phases, is called the phase shift angle. Phase shift between sinusoidal quantities of the same name, for example between two e. d.s. or two currents, denote α . The phase shift angle between the current and voltage sinusoids or their maximum vectors is denoted by the letter φ (Fig. 2.3).
When for sinusoidal quantities the phase difference is equal to ±π , then they are opposite in phase, but if the phase difference is equal ±π/2, then they are said to be in quadrature. If the initial phases are the same for sinusoidal quantities of the same frequency, this means that they are in phase.
Sinusoidal voltage and current, the graphs of which are presented in Fig. 2.3 are described as follows:
u = Umsin(ω t+ψ u) ; i = Imsin(ω t+ψ i) , (2.6)
and the phase angle between current and voltage (see Fig. 2.3) in this case φ = ψ u - ψ i.
Equations (2.6) can be written differently:
u = Umsin(ωt + ψi + φ) ; i = Imsin(ωt + ψu - φ) ,
because the ψ u = ψ i + φ And ψ i = ψ u - φ .
From these expressions it follows that the voltage leads the current in phase by an angle φ (or the current is out of phase with the voltage by an angle φ ).
Forms of representation of sinusoidal electrical quantities.
Any sinusoidally varying electrical quantity (current, voltage, emf) can be presented in analytical, graphical and complex forms.
1). Analytical presentation form
I = I m sin( ω·t + ψ i), u = U m sin( ω·t + ψ u), e = E m sin( ω·t + ψ e),
Where I, u, e– instantaneous value of sinusoidal current, voltage, EMF, i.e. values at the considered moment in time;
I m , U m , E m– amplitudes of sinusoidal current, voltage, EMF;
(ω·t + ψ ) – phase angle, phase; ω = 2·π/ T– angular frequency, characterizing the rate of phase change;
ψ i, ψ u, ψ e – the initial phases of current, voltage, EMF are counted from the point of transition of the sinusoidal function through zero to a positive value before the start of time counting ( t= 0). The initial phase can have both positive and negative meanings.
Graphs of instantaneous current and voltage values are shown in Fig. 2.3
The initial phase of the voltage is shifted to the left from the origin and is positive ψ u > 0, the initial phase of the current is shifted to the right from the origin and is negative ψ i< 0. Алгебраическая величина, равная разности начальных фаз двух синусоид, называется сдвигом фаз φ . Phase shift between voltage and current
φ = ψ u – ψ i = ψ u – (- ψ i) = ψ u+ ψ i.
The use of an analytical form for calculating circuits is cumbersome and inconvenient.
In practice, one has to deal not with instantaneous values of sinusoidal quantities, but with actual ones. All calculations are carried out for effective values; the rating data of various electrical devices indicate effective values (current, voltage), most electrical measuring instruments show effective values. RMS current is the equivalent of direct current, which at the same time generates the same amount of heat in the resistor as alternating current. Effective value is related to the amplitude simple relation
2). Vector the form of representation of a sinusoidal electrical quantity is a vector rotating in a Cartesian coordinate system with a beginning at point 0, the length of which is equal to the amplitude of the sinusoidal quantity, the angle relative to the x-axis is its initial phase, and the rotation frequency is ω = 2πf. The projection of a given vector onto the y-axis at any time determines the instantaneous value of the quantity under consideration.
Rice. 2.4
A set of vectors depicting sinusoidal functions is called a vector diagram, Fig. 2.4
3). Complex The presentation of sinusoidal electrical quantities combines the clarity of vector diagrams with accurate analytical calculations of circuits.
Rice. 2.5
We depict current and voltage as vectors on the complex plane, Fig. 2.5 The abscissa axis is called the axis of real numbers and is designated +1 , the ordinate axis is called the axis of imaginary numbers and is denoted +j. (In some textbooks, the real number axis is denoted Re, and the axis of imaginary ones is Im). Let's consider the vectors U And I at a point in time t= 0. Each of these vectors corresponds to a complex number, which can be represented in three forms:
A). Algebraic
U = U’+ jU"
I = I’ – jI",
Where U", U", I", I" – projections of vectors on the axes of real and imaginary numbers.
b). Indicative
Where U,
I– modules (lengths) of vectors; e– the base of the natural logarithm; 
rotation factors, since multiplication by them corresponds to rotation of the vectors relative to the positive direction of the real axis by an angle equal to the initial phase.
V). Trigonometric
U = U·(cos ψ u+ j sin ψ u)
I = I·(cos ψ i – j sin ψ i).
When solving problems, they mainly use the algebraic form (for addition and subtraction operations) and the exponential form (for multiplication and division operations). The connection between them is established by Euler's formula
e jψ = cos ψ + j sin ψ .
Unbranched electrical circuits
Let's check the validity of the formulas shown here using a simple experiment.
Let's take two resistors MLT-2 on 3 And 47 Ohm and connect them in series. Then we measure the total resistance of the resulting circuit digital multimeter. As we can see, it is equal to the sum of the resistances of the resistors included in this chain.
Measuring total resistance in series connection
Now let's connect our resistors in parallel and measure their total resistance.
Resistance measurement in parallel connection
As you can see, the resulting resistance (2.9 Ohms) is less than the smallest (3 Ohms) included in the chain. This leads to another well-known rule that can be applied in practice:
When resistors are connected in parallel, the total resistance of the circuit will be less than the smallest resistance included in this circuit.
What else needs to be considered when connecting resistors?
Firstly, Necessarily their rated power is taken into account. For example, we need to select a replacement resistor for 100 Ohm and power 1 W. Let's take two resistors of 50 ohms each and connect them in series. How much power dissipation should these two resistors be rated for?
Since the same current flows through resistors connected in series D.C.(let's say 0.1 A), and the resistance of each of them is equal 50 ohm, then the dissipation power of each of them must be at least 0.5 W. As a result, on each of them there will be 0.5 W power. In total this will be the same 1 W.
This example is quite crude. Therefore, if in doubt, you should take resistors with a power reserve.
Read more about resistor power dissipation.
Secondly, when connecting, you should use resistors of the same type, for example, the MLT series. Of course, there is nothing wrong with taking different ones. This is just a recommendation.