A source is a device that converts mechanical, chemical, thermal and some other forms of energy into electrical energy. In other words, the source is an active network element designed to generate electricity. Various types The sources available in the electrical network are voltage sources and current sources. These two concepts in electronics are different from each other.
Constant voltage source
A voltage source is a device with two poles; its voltage is constant at any time, and the current passing through it has no effect. Such a source will be ideal, having zero internal resistance. In practical conditions it cannot be obtained.
An excess of electrons accumulates at the negative pole of the voltage source, and a deficiency of electrons at the positive pole. The states of the poles are maintained by processes within the source.
Batteries
Batteries store chemical energy internally and are capable of converting it into electrical energy. The batteries cannot be recharged, which is their disadvantage.
Batteries
Rechargeable batteries are rechargeable batteries. When charging, electrical energy is stored internally as chemical energy. During unloading, the chemical process occurs in the opposite direction and electrical energy is released.
Examples:
- Lead-acid battery cell. It is made from lead electrodes and electrolytic liquid in the form of sulfuric acid diluted with distilled water. The voltage per cell is about 2 V. In car batteries, six cells are usually connected in a series circuit, and the resulting voltage at the output terminals is 12 V;
- Nickel-cadmium batteries, cell voltage – 1.2 V.
Important! For small currents, batteries and accumulators can be considered as a good approximation of ideal voltage sources.
AC voltage source
Electricity is produced at power stations using generators and, after voltage regulation, is transmitted to the consumer. AC voltage home network 220 V in power supplies for various electronic devices is easily converted to a lower value when using transformers.
Current source
By analogy, just as an ideal voltage source creates a constant voltage at the output, the task of a current source is to produce a constant current value, automatically controlling the required voltage. Examples are current transformers (secondary winding), photocells, collector currents of transistors.
Calculation of the internal resistance of the voltage source
Real voltage sources have their own electrical resistance, which is called “internal resistance”. The load connected to the source terminals is designated as “external resistance” - R.
A battery of batteries generates EMF:
ε = E/Q, where:
- E – energy (J);
- Q – charge (C).
The total emf of a battery cell is its open circuit voltage when there is no load. It can be checked with good accuracy using a digital multimeter. The potential difference measured at the output terminals of the battery when it is connected to a load resistor will be less than its voltage when the circuit is open, due to the flow of current through the external load and through the internal resistance of the source, this leads to the dissipation of energy in it as thermal radiation .
The internal resistance of a chemical battery is between a fraction of an ohm and a few ohms and is mainly due to the resistance of the electrolytic materials used in the manufacture of the battery.
If a resistor with resistance R is connected to a battery, the current in the circuit is I = ε/(R + r).
Internal resistance is not a constant value. It is affected by the type of battery (alkaline, lead-acid, etc.), and changes depending on the load value, temperature and period of use of the battery. For example, with disposable batteries, the internal resistance increases during use, and the voltage therefore drops until it reaches a state that is unsuitable for further use.
If the emf of the source is a predetermined quantity, the internal resistance of the source is determined by measuring the current flowing through the load resistance.
- Since the internal and external resistance in the approximate circuit are connected in series, you can use Ohm's and Kirchhoff's laws to apply the formula:
- From this expression r = ε/I – R.
Example. A battery with a known emf ε = 1.5 V is connected in series with a light bulb. The voltage drop across the light bulb is 1.2 V. Therefore, the internal resistance of the element creates a voltage drop: 1.5 - 1.2 = 0.3 V. The resistance of the wires in the circuit is considered negligible, the resistance of the lamp is not known. Measured current passing through the circuit: I = 0.3 A. It is necessary to determine the internal resistance of the battery.
- According to Ohm's law, the resistance of the light bulb is R = U/I = 1.2/0.3 = 4 Ohms;
- Now, according to the formula for calculating the internal resistance, r = ε/I – R = 1.5/0.3 – 4 = 1 Ohm.
When short circuit external resistance drops to almost zero. The current can only be limited by the small resistance of the source. The current generated in such a situation is so strong that the voltage source may be damaged by the thermal effects of the current and there is a risk of fire. The risk of fire is prevented by installing fuses, for example in car battery circuits.
The internal resistance of a voltage source is an important factor when deciding how to deliver the most efficient power to a connected electrical appliance.
Important! Maximum power transfer occurs when the internal resistance of the source is equal to the resistance of the load.
However, under this condition, remembering the formula P = I² x R, an identical amount of energy is transferred to the load and dissipated in the source itself, and its efficiency is only 50%.
Load requirements must be carefully considered to decide on the best use of the source. For example, a lead-acid car battery must deliver high currents at a relatively low voltage of 12 V. Its low internal resistance allows it to do this.
In some cases, high voltage power supplies must have extremely high internal resistance to limit short-circuit current.
Features of the internal resistance of the current source
An ideal current source has infinite resistance, but for genuine sources one can imagine an approximate version. The equivalent electrical circuit is a resistance connected to the source in parallel and an external resistance.
The current output from the current source is distributed as follows: part of the current flows through the highest internal resistance and through the low load resistance.
The output current will be the sum of the currents in the internal resistance and the load Io = In + Iin.
It turns out:
In = Io – Iin = Io – Un/r.
This relationship shows that as the internal resistance of the current source increases, the more the current across it decreases, and the load resistor receives most of the current. Interestingly, voltage will not affect the current value.
Real source output voltage:
Uout = I x (R x r)/(R +r) = I x R/(1 + R/r).
Current strength:
Iout = I/(1 + R/r).
Output power:
Rout = I² x R/(1 + R/r)².
Important! When analyzing circuits, we proceed from the following conditions: when the internal resistance of the source significantly exceeds the external one, it is a current source. When, on the contrary, the internal resistance is significantly less than the external one, this is a voltage source.
Current sources are used to supply electricity to measuring bridges, operational amplifiers, and these can be various sensors.
Video
The need to introduce the term can be illustrated by the following example. Let's compare two chemical DC sources with the same voltage:
- Car lead-acid battery with a voltage of 12 volts and a capacity of 55 Ah
- Eight AA batteries connected in series. The total voltage of such a battery is also 12 volts, the capacity is much smaller - approximately 1 Ah
Despite the same voltage, these sources differ significantly when operating at the same load. So, car battery is capable of delivering a large current to the load (the car engine starts from the battery, while the starter consumes a current of 250 amperes), and the starter does not rotate at all from a chain of batteries. The relatively small capacity of the batteries is not the reason: one amp-hour in the batteries would be enough to rotate the starter for 14 seconds (at a current of 250 amps).
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 the two-terminal network does not contain sources, then “ internal resistance" for such a two-terminal network means the same as Just"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:
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. Two-terminal network only externally 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 energy source without any electrical circuit (for example, a galvanic cell), then the internal resistance is almost purely active (unless we are talking about very high frequencies), it is due to 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 the effects non-electric nature. So, for example, in power it can be limited by the contact area of the substances participating in the reaction, in a hydroelectric power station 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:
The influence of internal resistance on the properties of a two-terminal network
The effect of internal resistance is an integral property of any active two-terminal network. The main result of the presence of internal resistance is to limit the electrical power that can be obtained in the load supplied from this two-terminal network.
Let there be a two-terminal network, which can be described by the above equivalent circuit. A 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 U out = φ 2 − φ 1) at two different load currents. Then the unknown parameters can be found from the system of equations:
| (Voltages) |
Where U out1 I 1, Uout2- output voltage at current I 2. By solving the system of equations, we find the unknown unknowns:
Typically, a simpler technique is used to calculate the internal resistance: the voltage in the no-load mode and the current in the short-circuit mode of the two-terminal network are found. In this case, system () is written as follows:
Where U oc- output voltage in idle mode (eng. open circuit), 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:
| (Internal Resistance) |
Measurement
Concept measurement applicable to real device(but not to the diagram). Direct measurement with an ohmmeter is impossible, since it is impossible to connect the probes of the device to the internal resistance terminals. Therefore, 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.
Sometimes the following simple measurement method is used, which does not require calculations:
- 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, one should be wary of overloading the two-terminal circuit with excessive current, that is, the current should not exceed the maximum permissible values for this two-terminal network.
Reactive internal resistance
If the equivalent circuit of a two-terminal network contains reactive elements - capacitors and/or inductors, then calculation Reactive internal resistance is carried out in the same way as active resistance, but instead of resistor resistances, 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 made by the complex amplitude method.
Measurement reactance has some special features because it is a complex-valued function rather than a scalar value:
- You can search for various parameters of a complex value: modulus, argument, only the real or imaginary part, as well as the entire complex number. Accordingly, the measurement technique will depend on what we want to obtain.
- Any of the listed parameters depends on frequency. Theoretically, to obtain by measurement full information about internal reactive resistance, it is necessary to remove addiction on frequency, that is, carry out measurements at everyone frequencies that the source of a given two-terminal network can generate.
Application
In most cases, we should not talk about application internal resistance, and about accounting its negative impact, since internal resistance is rather a negative effect. However, in some systems a nominal internal resistance is essential.
Simplification of equivalent circuits
The representation of a two-terminal network as a combination of a voltage generator and internal resistance is the simplest and most frequently used equivalent circuit of a two-terminal network.
Source-Load Matching
Matching the source and load is the choice of the ratio of the load resistance and the internal resistance of the source in order to achieve the specified properties of the resulting system (as a rule, they try to achieve the maximum value of any parameter for a given source). Most commonly used following types approvals:
Current and power matching should be used with caution as there is a risk of overloading the source.
High Voltage Reduction
Sometimes a large resistance is artificially added to the source (it is added to the internal resistance of the source) in order to significantly reduce the voltage received from it. However, adding a resistor as additional resistance (the so-called quenching resistor) leads to useless power being allocated to it. To avoid wasting energy, AC systems use reactive damping impedances, most often capacitors. This is how capacitor power supplies are built. Similarly, using a capacitive tap from a high-voltage power line, you can obtain small voltages to power any autonomous devices.
Minimizing noise
When amplifying weak signals, the task often arises of minimizing the noise introduced by the amplifier into the signal. For this purpose special low noise amplifiers, however, they are designed in such a way that the lowest noise figure is achieved only within a certain range of the output impedance of the signal source. For example, a low noise amplifier provides minimal noise only over the source output impedance range of 1 kΩ to 10 kΩ; if the signal source has a lower output impedance (for example, a microphone with an output impedance of 30 Ohms), then a step-up transformer should be used between the source and the amplifier, which will increase the output impedance (as well as the signal voltage) to the required value.
Restrictions
The concept of internal resistance is introduced through an equivalent circuit, so the same restrictions apply as for the applicability of equivalent circuits.
Examples
Internal resistance values are relative: what is considered small, for example, for a galvanic cell, is very large for powerful battery. Below are examples of two-terminal networks and the values of their internal resistance r. Trivial cases of two-terminal networks no sources are specifically stated.
Low internal resistance
High internal resistance
Negative internal resistance
There are two-terminal networks whose internal resistance has negative meaning. In normal active resistance, energy dissipation occurs, in reactive In resistance, energy is stored and then released back to the source. The peculiarity of negative resistance is that it itself is a source of energy. Therefore, negative resistance does not occur in its pure form; it can only be simulated electronic circuit, which necessarily contains a source of energy. Negative internal resistance can be achieved in circuits by using:
- elements with negative differential resistance, such as tunnel diodes
Systems with negative resistance are potentially unstable and therefore can be used to build self-oscillators.
see also
Links
Literature
- Zernov N.V., Karpov V.G. Theory radio circuits. - M. - L.: Energy, 1965. - 892 p.
- Jones M.H. Electronics - practical course. - M.: Tekhnosphere, 2006. - 512 p.
Let's say there is a simple electrical closed circuit that includes a current source, for example a generator, galvanic cell or battery, and a resistor with a resistance R. Since the current in the circuit is not interrupted anywhere, it flows inside the source.
In such a situation, we can say that any source has some internal resistance that prevents current flow. This internal resistance characterizes the current source and is designated by the letter r. For a battery, internal resistance is the resistance of the electrolyte solution and electrodes; for a generator, it is the resistance of the stator windings, etc.
Thus, the current source is characterized by both the magnitude of the EMF and the value of its own internal resistance r - both of these characteristics indicate the quality of the source.
Electrostatic high-voltage generators (like the Van de Graaff generator or the Wimshurst generator), for example, are distinguished by a huge EMF measured in millions of volts, while their internal resistance is measured in hundreds of megaohms, which is why they are unsuitable for producing large currents.
Galvanic elements (such as a battery), on the contrary, have an EMF of the order of 1 volt, although their internal resistance is of the order of fractions or, at most, tens of ohms, and therefore currents of units and tens of amperes can be obtained from galvanic elements.
This diagram shows a real source with an attached load. Its internal resistance, as well as the load resistance, are indicated here. According to, the current in this circuit will be equal to:
Since the section of the external circuit is homogeneous, the voltage across the load can be found from Ohm’s law:
Expressing the load resistance from the first equation and substituting its value into the second equation, we obtain the dependence of the load voltage on the current in a closed circuit:
In a closed loop, the EMF is equal to the sum of the voltage drops across the elements of the external circuit and the internal resistance of the source itself. The dependence of load voltage on load current is ideally linear.
The graph shows this, but experimental data on a real resistor (crosses near the graph) always differ from the ideal:
Experiments and logic show that at zero load current, the voltage on the external circuit is equal to the source emf, and at zero load voltage, the current in the circuit is equal to . This property of real circuits helps to experimentally find the emf and internal resistance of real sources.
Experimental determination of internal resistance
To experimentally determine these characteristics, plot the dependence of the voltage on the load on the current value, then extrapolate it to the intersection with the axes.
At the point of intersection of the graph with the voltage axis is the value of the source emf, and at the point of intersection with the current axis is the value of the short circuit current. As a result, the internal resistance is found by the formula:
The useful power developed by the source is released to the load. The dependence of this power on the load resistance is shown in the figure. This curve starts from the intersection of the coordinate axes at the zero point, then increases to the maximum power value, after which it drops to zero when the load resistance is equal to infinity.
To find the maximum load resistance at which the maximum power will theoretically develop at a given source, the derivative of the power formula with respect to R is taken and set equal to zero. Maximum power will develop when the external circuit resistance is equal to the internal resistance of the source:
This provision about the maximum power at R = r allows us to experimentally find the internal resistance of the source by plotting the dependence of the power released on the load on the value of the load resistance. Having found the real, and not theoretical, load resistance that provides maximum power, the real internal resistance of the power supply is determined.
The efficiency of a current source shows the ratio of the maximum power allocated to the load to the total power, which in this moment develops
An electric current in a conductor arises under the influence of an electric field, causing free charged particles to move in a direction. Generating particle current is a serious problem. To build such a device that will maintain the field potential difference for a long time in one state is a task that was only possible for humanity to solve by the end of the 18th century.
First attempts
The first attempts to “store electricity” for its further research and use were made in Holland. The German Ewald Jürgen von Kleist and the Dutchman Pieter van Musschenbroek, who conducted their research in the town of Leiden, created the world's first capacitor, later called the “Leyden jar”.
The accumulation of electric charge already took place under the influence of mechanical friction. It was possible to use a discharge through a conductor for a certain, fairly short period of time.
The victory of the human mind over such an ephemeral substance as electricity turned out to be revolutionary.
Unfortunately, the discharge (electric current created by the capacitor) lasted so short that it could not be created. In addition, the voltage supplied by the capacitor gradually decreases, which leaves no possibility of receiving long-term current.
It was necessary to look for another way.
First source
The Italian Galvani's experiments on "animal electricity" were an original attempt to find a natural source of current in nature. Hanging the legs of dissected frogs on the metal hooks of an iron grid, he drew attention to the characteristic reaction of the nerve endings.
However, Galvani's conclusions were refuted by another Italian, Alessandro Volta. Interested in the possibility of obtaining electricity from animal organisms, he conducted a series of experiments with frogs. But his conclusion turned out to be the complete opposite of previous hypotheses.
Volta noticed that a living organism is only an indicator of an electrical discharge. When current passes, the muscles of the paws contract, indicating a potential difference. The source of the electric field turned out to be the contact of dissimilar metals. The farther apart they are in the series of chemical elements, the more significant the effect.
Plates of dissimilar metals, lined with paper disks soaked in an electrolyte solution, created the necessary potential difference for a long time. And even though it was low (1.1 V), the electric current could be studied for a long time. The main thing is that the tension remained unchanged for just as long.
What's happening
Why does this effect occur in sources called “galvanic cells”?
Two metal electrodes placed in a dielectric play different roles. One supplies electrons, the other accepts them. The process of redox reaction leads to the appearance of an excess of electrons on one electrode, which is called the negative pole, and a deficiency on the second, which we will designate as the positive pole of the source.
In the simplest galvanic cells, oxidation reactions occur on one electrode, reduction reactions on the other. Electrons come to the electrodes from the outer part of the circuit. The electrolyte is a conductor of ion current inside the source. The force of resistance controls the duration of the process.
Copper-zinc element
It is interesting to consider the principle of operation of galvanic cells using the example of a copper-zinc galvanic cell, the action of which comes from the energy of zinc and copper sulfate. In this source, a copper plate is placed in a solution and a zinc electrode is immersed in a zinc sulfate solution. The solutions are separated by a porous spacer to avoid mixing, but they must come into contact.
If the circuit is closed, the surface layer of zinc is oxidized. In the process of interaction with the liquid, zinc atoms, turning into ions, appear in the solution. Electrons are released at the electrode, which can take part in the formation of current.
Once on the copper electrode, electrons take part in the reduction reaction. Copper ions come from the solution to the surface layer; during the reduction process, they turn into copper atoms, depositing on the copper plate.
Let's summarize what is happening: the process of operation of a galvanic cell is accompanied by the transition of electrons from the reducing agent to the oxidizing agent along the external part of the circuit. Reactions occur on both electrodes. An ion current flows inside the source.
Difficulty of use
In principle, any of the possible redox reactions can be used in batteries. But there are not so many substances capable of working in technically valuable elements. Moreover, many reactions require expensive substances.
Modern rechargeable batteries have a simpler structure. Two electrodes placed in one electrolyte fill the vessel - the battery body. Such design features simplify the structure and reduce the cost of batteries.
Any galvanic cell is capable of producing direct current.
The current resistance does not allow all the ions to appear on the electrodes at the same time, so the element operates for a long time. The chemical reactions of ion formation sooner or later stop, and the element is discharged.
The current source is of great importance.
A little about resistance
The use of electric current, undoubtedly, brought scientific and technological progress to a new level and gave it a gigantic impetus. But the force of resistance to the flow of current gets in the way of such development.
On the one hand, electric current has invaluable properties used in everyday life and technology, on the other hand, there is significant resistance. Physics, as a science of nature, tries to establish a balance and bring these circumstances into line.
Current resistance arises due to the interaction of electrically charged particles with the substance through which they move. It is impossible to exclude this process under normal temperature conditions.
Resistance
The current source and the resistance of the external part of the circuit have a slightly different nature, but the same in these processes is the work done to move the charge.
The work itself depends only on the properties of the source and its filling: the qualities of the electrodes and electrolyte, as well as for the external parts of the circuit, the resistance of which depends on the geometric parameters and chemical characteristics of the material. For example, the resistance of a metal wire increases with its length and decreases with increasing cross-sectional area. When solving the problem of how to reduce resistance, physics recommends using specialized materials.
Current work
In accordance with the Joule-Lenz law, an amount of heat is released in conductors proportional to the resistance. If the amount of heat is denoted by Q int. , current strength I, its flow time t, then we get:
- Q internal = I 2 r t,
where r is the internal resistance of the current source.
In the entire chain, including both its internal and external parts, the total amount of heat will be released, the formula of which is:
- Q total = I 2 r t + I 2 R t = I 2 (r +R) t,
It is known how resistance is denoted in physics: the external circuit (all elements except the source) has a resistance R.
Ohm's law for a complete circuit
Let us take into account that the main work is performed by external forces inside the current source. Its value is equal to the product of the charge transferred by the field and the electromotive force of the source:
- q · E = I 2 · (r + R) · t.
Understanding that the charge is equal to the product of the current strength and the time it flows, we have:
- E = I (r + R).
In accordance with cause-and-effect relationships, Ohm's law has the form:
- I = E: (r + R).
In a closed circuit, the EMF of the current source is directly proportional and inversely proportional to the total (impact) resistance of the circuit.
Based on this pattern, it is possible to determine the internal resistance of the current source.
Source discharge capacity
The main characteristics of sources include discharge capacity. Maximum amount electricity obtained during operation under certain conditions depends on the strength of the discharge current.
In the ideal case, when certain approximations are made, the discharge capacity can be considered constant.
For example, a standard battery with a potential difference of 1.5 V has a discharge capacity of 0.5 Ah. If the discharge current is 100 mA, it works for 5 hours.
Methods for charging batteries
Using batteries will drain them. charging of small-sized elements is carried out using a current whose strength does not exceed one tenth of the source capacity.
The following charging methods are available:
- using constant current for a given time (about 16 hours with a current of 0.1 battery capacity);
- charging with a decreasing current to a given potential difference;
- use of asymmetrical currents;
- sequential application of short pulses of charging and discharging, in which the time of the first exceeds the time of the second.
Practical work
A task is proposed: determine the internal resistance of the current source and the emf.
To perform it, you need to stock up on a current source, an ammeter, a voltmeter, a slider rheostat, a key, and a set of conductors.
Use will allow you to determine the internal resistance of the current source. To do this, you need to know its EMF and the value of the rheostat resistance.
The calculation formula for the current resistance in the external part of the circuit can be determined from Ohm's law for the circuit section:
- I=U:R,
where I is the current strength in the external part of the circuit, measured with an ammeter; U is the voltage across the external resistance.
To increase accuracy, measurements are taken at least 5 times. What is it for? The voltage, resistance, current (or rather, current strength) measured during the experiment are used further.
To determine the EMF of the current source, we take advantage of the fact that the voltage at its terminals when the switch is open is almost equal to the EMF.
Let's assemble a circuit of a battery, a rheostat, an ammeter, and a key connected in series. We connect a voltmeter to the terminals of the current source. Having opened the key, we take its readings.
The internal resistance, the formula of which is obtained from Ohm’s law for a complete circuit, is determined by mathematical calculations:
- I = E: (r + R).
- r = E: I - U: I.
Measurements show that the internal resistance is significantly less than the external one.
The practical function of accumulators and batteries is widely used. The indisputable environmental safety of electric motors is beyond doubt, but creating a capacious, ergonomic battery is a problem of modern physics. Its solution will lead to a new round of development of automotive technology.
Small-sized, lightweight, high-capacity rechargeable batteries are also extremely necessary in mobile devices. electronic devices. The amount of energy used in them is directly related to the performance of the devices.
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 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.
Total 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 the complete circuit:
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.