Internal resistance of the current source formula. How to calculate the internal resistance of a power supply. Two-terminal network and its equivalent circuit

Two-terminal network and its equivalent circuit

The internal resistance of a two-terminal network is the impedance in the equivalent circuit of a two-terminal network, consisting of a voltage generator and impedance connected in series (see figure). The concept is used in circuit theory when replacing a real source with ideal elements, that is, when moving to an equivalent circuit.

Introduction

Let's look at an example. We will power it in a passenger car on-board network not from a standard lead-acid battery with a voltage of 12 volts and a capacity of 55 Ah, but from eight batteries connected in series (for example, AA size, with a capacity of about 1 Ah). Let's try to start the engine. Experience shows that when powered by batteries, the starter shaft will not turn a single degree. Moreover, even the solenoid relay will not work.

It is intuitively clear that the battery is “not powerful enough” for such an application, however, consideration of its declared electrical characteristics- voltage and charge (capacitance) - does not provide a quantitative description of this phenomenon. The voltage is the same in both cases:

Battery: 12 volts

Galvanic cells: 8·1.5 volts = 12 volts

The capacity is also quite sufficient: one ampere hour in the battery should be enough to rotate the starter for 14 seconds (at a current of 250 amperes).

It would seem that, in accordance with Ohm's law, the current in the same load with electrically identical sources should also be the same. However, in reality this is not entirely true. The sources would behave the same if they were ideal voltage generators. To describe the degree of difference real sources from ideal generators and the concept of internal resistance is used.

Resistance and internal resistance

The main characteristic of a two-terminal network is its resistance (or impedance). However, it is not always possible to characterize a two-terminal network with resistance alone. The fact is that the term resistance is applicable only to purely passive elements, that is, those that do not contain energy sources. If a two-terminal network contains an energy source, then the concept of “resistance” is simply not applicable to it, since Ohm’s law in the formulation U=Ir is not satisfied.

Thus, for two-terminal networks containing sources (that is, voltage generators and current generators), it is necessary to talk specifically about internal resistance (or impedance). If a two-terminal network does not contain sources, then “internal resistance” for such a two-terminal network means the same thing as simply “resistance”.

Related terms

If in any system it is possible to distinguish an input and/or an output, then the following terms are often used:

Input resistance is the internal resistance of the two-terminal network, which is the input of the system.

Output resistance is the internal resistance of the two-terminal network, which is the output of the system.

Physical principles

Despite the fact that in the equivalent circuit the internal resistance is presented as one passive element (and active resistance, that is, a resistor is necessarily present in it), the internal resistance is not concentrated in any one element. The two-terminal network only outwardly behaves as if it had a concentrated internal impedance and a voltage generator. In reality, internal resistance is an external manifestation of a set of physical effects:

If in a two-terminal network there is only an energy source without any electrical circuit (for example, a galvanic cell), then the internal resistance is purely active, it is caused by physical effects that do not allow the power supplied by this source to the load to exceed a certain limit. The simplest example of such an effect is the non-zero resistance of the conductors of an electrical circuit. But, as a rule, the greatest contribution to power limitation comes from non-electrical effects. So, for example, in a chemical source, the power can be limited by the contact area of ​​the substances participating in the reaction, in a hydroelectric generator - by limited water pressure, etc.

In the case of a two-terminal network containing inside electrical diagram, the internal resistance is “dispersed” in the circuit elements (in addition to the mechanisms listed above in the source).

This also implies some features of internal resistance:

Internal resistance impossible to remove from a two-terminal network

Internal resistance is not a stable value: it can change when any external conditions change.

The influence of internal resistance on the properties of a two-terminal network

The effect of internal resistance is an integral property of any two-terminal network. The main result of the presence of internal resistance is the limitation electrical power, which can be obtained in a load powered from this two-terminal network.

If a load with resistance R is connected to a source with an emf of a voltage generator E and an active internal resistance r, then the current, voltage and power in the load are expressed as follows.

Calculation

The concept of calculation applies to a circuit (but not to a real device). The calculation is given for the case of purely active internal resistance (differences in reactance will be discussed below).

Let there be a two-terminal network, which can be described by the above equivalent circuit. The two-terminal network has two unknown parameters that need to be found:

EMF voltage generator U

Internal resistance r

IN general case, to determine two unknowns, it is necessary to make two measurements: measure the voltage at the output of the two-terminal network (that is, the potential difference Uout = φ2 − φ1) at two different load currents. Then the unknown parameters can be found from the system of equations:

where Uout1 - output voltage at current I1, Uout2 - output voltage at current I2. By solving the system of equations, we find the unknown unknowns:

Typically, more than simple technique: voltage is in idle mode and current is in mode short circuit bipolar network. In this case, system (1) is written as follows:

where Uoc is the output voltage in open circuit mode, that is, at zero load current; Isc - load current in short circuit mode (eng. short circuit), that is, under a load with zero resistance. It is taken into account here that the output current in no-load mode and the output voltage in short-circuit mode are zero. From the last equations we immediately get:

Measurement

The concept of measurement applies to a real device (but not to a circuit). Direct measurement with an ohmmeter is impossible, since it is impossible to connect the probes of the device to the internal resistance terminals. Therefore, an indirect measurement is necessary, which is not fundamentally different from calculation - voltages across the load are also required at two different current values. However, it is not always possible to use the simplified formula (2), since not every real two-terminal network allows operation in short circuit mode.

The following simple measurement method that does not require calculations is often used:

Open circuit voltage is measured

A variable resistor is connected as a load and its resistance is selected so that the voltage across it is half the open circuit voltage.

After the described procedures, the resistance of the load resistor must be measured with an ohmmeter - it will be equal to the internal resistance of the two-terminal network.

Whatever measurement method is used, you should be wary of overloading the two-terminal network excessive current, that is, the current should not exceed the maximum permissible values ​​for a given two-terminal network.

Reactive internal resistance

If the equivalent circuit of a two-terminal network contains reactive elements - capacitors and/or inductors, then the calculation of the reactive internal resistance is performed in the same way as the active one, but instead of the resistances of resistors, the complex impedances of the elements included in the circuit are taken, and instead of voltages and currents, their complex amplitudes are taken, that is, the calculation is performed by the complex amplitude method.

The internal reactance measurement has some special features because it is a complex-valued function rather than a scalar value:

You can search various parameters complex value: module, argument, only the real or imaginary part, as well as a completely complex number. Accordingly, the measurement technique will depend on what we want to obtain.

In the age of electricity, there is probably no such person who would not know about the existence electric current. But few people remember school course physics is more than the name of quantities: current, voltage, resistance, Ohm's law. And only very few remember what the meaning of these words is.

In this article, we will discuss how electric current occurs, how it is transmitted through a circuit, and how to use this quantity in calculations. But before moving on to the main part, let us turn to the history of the discovery of electric current and its sources, as well as the definition of what electromotive force is.

Story

Electricity as a source of energy has been known since ancient times, because nature itself generates it in huge volumes. A striking example is lightning or an electric ramp. Despite such closeness to humans, it was possible to curb this energy only in the middle of the seventeenth century: Otto von Guericke, burgomaster from Magdeburg, created a machine that allows generating an electrostatic charge. In the mid-eighteenth century, Peter von Muschenbroek, a scientist from Holland, created the world's first electric capacitor, named the Leyden jar in honor of the university where he worked.

Perhaps, the era of real discoveries dedicated to electricity begins with the work of Luigi Galvani and Alessandro Volta, who studied, respectively, electrical currents in muscles and the emergence of current in so-called galvanic cells. Further research opened our eyes to the connection between electricity and magnetism, as well as to several very useful phenomena (such as electromagnetic induction), without which it is impossible to imagine our lives today.

But we will not delve into magnetic phenomena and will focus only on electrical ones. So, let's look at how electricity arises in galvanic cells and what it is all about.

What is a galvanic cell?

We can say that it produces electricity due to chemical reactions occurring between its components. The simplest galvanic cell was invented by Alessandro Volta and named after him as a voltaic column. It consists of several layers alternating with each other: a copper plate, a conductive gasket (in home version The design uses cotton wool soaked in salt water) and a zinc plate.

What reactions take place in it?

Let's take a closer look at the processes that allow us to generate electricity using a galvanic cell. There are only two such transformations: oxidation and reduction. When one element, the reducing agent, is oxidized, it gives up electrons to another element, the oxidizing agent. The oxidizing agent, in turn, is reduced by accepting electrons. In this way, charged particles move from one plate to another, and this, as is known, is called electric current.

And now let’s move smoothly to the main topic of this article - EMF source current. And first, let's look at what this electromotive force (EMF) is.

What is EMF?

This quantity can be represented as the work of forces (namely “work”) performed when a charge moves along a closed electrical circuit. Very often they also make clarifications that the charge must necessarily be positive and unit. And this is an essential addition, since only under these conditions can the electromotive force be considered an accurate measurable quantity. By the way, it is measured in the same units as voltage: volts (V).

EMF of current source

As you know, each battery or battery has its own resistance value that it can produce. This value, the emf of the current source, shows how much work is done by external forces to move charge along the circuit in which the battery or accumulator is connected.

It is also worth clarifying what type of current the source produces: constant, alternating or pulsed. Galvanic cells, including accumulators and batteries, always produce only direct electric current. The EMF of the current source in this case will be equal in magnitude to the output voltage at the contacts of the source.

Now it’s time to figure out why such a quantity as EMF is needed in general, and how to use it when calculating other quantities of an electrical circuit.

EMF formula

We have already found out that the EMF of the current source is equal to the work of external forces to move the charge. For greater clarity, we decided to write down the formula for this quantity: E = A external forces / q, where A is work, and q is the charge on which work was done. Please note that the total charge is taken, not the unit charge. This is done because we consider the work of forces to move all charges in a conductor. And this work to charge ratio will always be constant for this source, since no matter how many charged particles you take, the specific amount of work for each of them will be the same.

As you can see, the formula for electromotive force is not so complicated and consists of only two quantities. It's time to move on to one of the main questions arising from this article.

Why is EMF needed?

It has already been said that EMF and voltage are actually the same quantities. If we know the values ​​of the EMF and the internal resistance of the current source, then it will not be difficult to substitute them into Ohm’s law for a complete circuit, which looks like this: I=e/(R+r), where I is the current strength, e is the EMF, R is circuit resistance, r - internal resistance of the current source. From here we can find two characteristics of the chain: I and R. It should be noted that all these arguments and formulas are valid only for the chain direct current. In the case of a variable, the formulas will be completely different, since it obeys its own oscillatory laws.

But it still remains unclear what application the EMF of a current source has. In a circuit, as a rule, there are a lot of elements that perform their function. In any phone there is a board, which is also nothing more than electrical circuit. And each such circuit requires a current source to operate. And it is very important that its EMF matches the parameters for all elements of the circuit. Otherwise, the circuit will either stop working or burn out due to high voltage inside her.

Conclusion

We think this article was useful for many. After all, in modern world It is very important to know as much as possible about what surrounds us. Including essential knowledge about the nature of electric current and its behavior inside circuits. And if you think that such a thing as an electrical circuit is used only in laboratories and you are far from it, then you are very mistaken: all devices that consume electricity actually consist of circuits. And each of them has its own current source, which creates an EMF.

8.5. Thermal effect of current

8.5.1. Current source power

Total power of the current source:

P total = P useful + P losses,

where P is useful - useful power, P useful = I 2 R ; P losses - power losses, P losses = I 2 r; I - current strength in the circuit; R - load resistance ( external circuit); r is the internal resistance of the current source.

Apparent power can be calculated using one of three formulas:

P full = I 2 (R + r), P full = ℰ 2 R + r, P full = I ℰ,

where ℰ is the electromotive force (EMF) of the current source.

Net power- this is the power that is released in the external circuit, i.e. on a load (resistor), and can be used for some purposes.

Net power can be calculated using one of three formulas:

P useful = I 2 R, P useful = U 2 R, P useful = IU,

where I is the current strength in the circuit; U is the voltage at the terminals (clamps) of the current source; R - load resistance (external circuit).

Power loss is the power that is released in the current source, i.e. in the internal circuit, and is spent on processes taking place in the source itself; The power loss cannot be used for any other purposes.

Power loss is usually calculated using the formula

P losses = I 2 r,

where I is the current strength in the circuit; r is the internal resistance of the current source.

During a short circuit, the useful power goes to zero

P useful = 0,

since there is no load resistance in the event of a short circuit: R = 0.

The total power during a short circuit of the source coincides with the loss power and is calculated by the formula

P full = ℰ 2 r,

where ℰ is the electromotive force (EMF) of the current source; r is the internal resistance of the current source.

Useful power has maximum value in the case when the load resistance R is equal to the internal resistance r of the current source:

R = r.

Maximum useful power:

P useful max = 0.5 P full,

where Ptot is the total power of the current source; P full = ℰ 2 / 2 r.

Explicit formula for calculation maximum useful power as follows:

P useful max = ℰ 2 4 r .

To simplify the calculations, it is useful to remember two points:

• if with two load resistances R 1 and R 2 the same useful power is released in the circuit, then internal resistance current source r is related to the indicated resistances by the formula

r = R 1 R 2 ;

• if the maximum useful power is released in the circuit, then the current strength I * in the circuit is half the strength of the short circuit current i:

I * = i 2 .

Example 15. When shorted to a resistance of 5.0 Ohms, a battery of cells produces a current of 2.0 A. The short circuit current of the battery is 12 A. Calculate the maximum useful power of the battery.

Solution . Let us analyze the condition of the problem.

1. When a battery is connected to a resistance R 1 = 5.0 Ohm, a current of strength I 1 = 2.0 A flows in the circuit, as shown in Fig. a , determined by Ohm's law for complete chain:

I 1 = ℰ R 1 + r,

where ℰ - EMF of the current source; r is the internal resistance of the current source.

2. When the battery is short-circuited, a short-circuit current flows in the circuit, as shown in Fig. b. The short circuit current is determined by the formula

where i is the short circuit current, i = 12 A.

3. When a battery is connected to a resistance R 2 = r, a current of force I 2 flows in the circuit, as shown in Fig. in , determined by Ohm's law for the complete circuit:

I 2 = ℰ R 2 + r = ℰ 2 r;

in this case, the maximum useful power is released in the circuit:

P useful max = I 2 2 R 2 = I 2 2 r.

Thus, to calculate the maximum useful power, it is necessary to determine the internal resistance of the current source r and the current strength I 2.

In order to find the current strength I 2, we write the system of equations:

i = ℰ r , I 2 = ℰ 2 r )

and divide the equations:

i I 2 = 2 .

This implies:

I 2 = i 2 = 12 2 = 6.0 A.

In order to find the internal resistance of the source r, we write the system of equations:

I 1 = ℰ R 1 + r, i = ℰ r)

and divide the equations:

I 1 i = r R 1 + r .

This implies:

r = I 1 R 1 i − I 1 = 2.0 ⋅ 5.0 12 − 2.0 = 1.0 Ohm.

Let's calculate the maximum useful power:

P useful max = I 2 2 r = 6.0 2 ⋅ 1.0 = 36 W.

Thus, the maximum usable power of the battery is 36 W.

Ohm's law for a complete circuit, the definition of which concerns the value of electric current in real circuits, depends on the current source and the load resistance. This law also has another name - Ohm's law for closed circuits. The operating principle of this law is as follows.

As the most simple example, electric lamp, which is a consumer of electric current, together with the current source is nothing more than a closed one. This electrical circuit is clearly shown in the figure.

An electric current passing through a light bulb also passes through the current source itself. Thus, while passing through the circuit, the current will experience the resistance of not only the conductor, but also the resistance, directly, of the current source itself. In the source, resistance is created by the electrolyte located between the plates and the boundary layers of the plates and electrolyte. It follows that in a closed circuit, its total resistance will consist of the sum of the resistances of the light bulb and the current source.

External and internal resistance

Load resistance, in in this case of a light bulb connected to a current source is called external resistance. The direct resistance of the current source is called internal resistance. For a more visual representation of the process, all values ​​must be designated conventionally. I - , R - external resistance, r - internal resistance. When current flows through an electrical circuit, in order to maintain it, there must be a potential difference between the ends of the external circuit, which has the value IxR. However, current flow is also observed in the internal circuit. This means that in order to maintain electric current in the internal circuit, a potential difference at the ends of the resistance r is also necessary. The value of this potential difference is equal to Iхr.

Battery electromotive force

The battery must have the following value of electromotive force capable of maintaining the required current in the circuit: E=IxR+Ixr. From the formula it can be seen that the electromotive force of the battery is the sum of external and internal. The current value must be taken out of brackets: E=I(r+R). Otherwise you can imagine: I=E/(r+R) . The last two formulas express Ohm's law for a complete circuit, the definition of which is as follows: in a closed circuit, the current strength is directly proportional to the electromotive force and inversely proportional to the sum of the resistances of this circuit.

Goal of the work: study the method of measuring EMF and internal resistance of a current source using an ammeter and voltmeter.

Equipment: metal tablet, current source, ammeter, voltmeter, resistor, key, clamps, connecting wires.

To measure the EMF and internal resistance of the current source, an electrical circuit is assembled, the diagram of which is shown in Figure 1.

An ammeter, a resistance and a switch connected in series are connected to the current source. In addition, a voltmeter is also connected directly to the output jacks of the source.

EMF is measured by reading a voltmeter with the switch open. This method of determining EMF is based on a corollary from Ohm’s law for a complete circuit, according to which, with an infinitely large resistance of the external circuit, the voltage at the source terminals is equal to its EMF. (See the paragraph "Ohm's Law for a Complete Circuit" in the Physics 10 textbook).

To determine the internal resistance of the source, key K is closed. In this case, two sections can be roughly distinguished in the circuit: external (the one that is connected to the source) and internal (the one that is located inside the current source). Since the source EMF is equal to the sum of the voltage drops in the internal and external sections of the circuit:

ε = Ur+UR, ThatUr = ε -UR (1)

According to Ohm's law for a section of the chain U r = I · r(2). Substituting equality (2) into (1) we get:

I· r = ε - Ur , whence r = (ε - UR)/ J

Therefore, in order to find out the internal resistance of a current source, it is necessary to first determine its EMF, then close the switch and measure the voltage drop across the external resistance, as well as the current strength in it.

Progress

1. Prepare a table to record the results of measurements and calculations:

Draw a diagram in your notebook to measure the EMF and internal resistance of the source.

After checking the circuit, assemble the electrical circuit. Unlock the key.

Measure the magnitude of the source emf.

Close the key and determine the readings of the ammeter and voltmeter.

Calculate the internal resistance of the source.

1. Determination of emf and internal resistance of a current source by graphical method

Goal of the work: study the measurements of emf, internal resistance and short circuit current of the current source, based on the analysis of the graph of the dependence of the voltage at the output of the source on the current in the circuit.

Equipment: galvanic cell, ammeter, voltmeter, resistor R 1 , variable resistor, key, clamps, metal tablet, connecting wires.

From Ohm’s law for a complete circuit it follows that the voltage at the output of the current source depends in direct proportion to the current in the circuit:

since I =E/(R+r), then IR + Ir = E, but IR = U, whence U + Ir = E or U = E – Ir (1).

If you plot the dependence of U on I, then from its points of intersection with the coordinate axes you can determine E, I K.Z. - the strength of the short circuit current (the current that will flow in the source circuit when the external resistance R becomes zero).

EMF is determined by the point of intersection of the graph with the voltage axis. This point on the graph corresponds to the state of the circuit in which there is no current in it and, therefore, U = E.

The strength of the short circuit current is determined by the point of intersection of the graph with the current axis. In this case, the external resistance R = 0 and, therefore, the voltage at the source output U = 0.

The internal resistance of the source is found by the tangent of the angle of inclination of the graph relative to the current axis. (Compare formula (1) with a mathematical function of the form Y = AX + B and remember the meaning of the coefficient for X).

Progress

To record the measurement results, prepare a table:

1. After the teacher checks the circuit, assemble the electrical circuit. Set the variable resistor slider to the position at which the resistance of the circuit connected to the current source is maximum.

Determine the current in the circuit and the voltage at the source terminals at the maximum resistance value of the variable resistor. Enter the measurement data into the table.

Repeat current and voltage measurements several times, decreasing the value each time variable resistance so that the voltage at the source terminals decreases by 0.1V. Stop measurements when the current in the circuit reaches 1A.

Plot the points obtained in the experiment on a graph. Plot voltage along the vertical axis, and current along the horizontal axis. Draw a straight line through the points.

Continue the graph until it intersects with the coordinate axes and determine the values ​​of E and I K.Z.

Measure the EMF of the source by connecting a voltmeter to its terminals with the external circuit open. Compare the EMF values ​​obtained by the two methods and indicate the reason for the possible discrepancy in the results.

Determine the internal resistance of the current source. To do this, calculate the tangent of the angle of inclination of the constructed graph to the current axis. Since the tangent of an angle in a right triangle is equal to the ratio opposite side to the adjacent side, then practically this can be done by finding the ratio E / I K.Z