How to determine internal resistance. Electromotive force. Internal resistance of the current source
At the ends of the conductor, and therefore the current, the presence of external forces of a non-electrical nature is necessary, with the help of which the separation of electrical charges occurs.
By outside forces are any forces acting on electrically charged particles in a circuit, with the exception of electrostatic (i.e., Coulomb).
Third-party forces set in motion charged particles inside all current sources: in generators, power plants, galvanic cells, batteries, etc.
When a circuit is closed, an electric field is created in all conductors of the circuit. Inside the current source, charges move under the influence of external forces against Coulomb forces (electrons move from a positively charged electrode to a negative one), and throughout the rest of the circuit they are driven by an electric field (see figure above).
In current sources, in the process of separating charged particles, different types of energy are converted into electrical energy. Based on the type of converted energy, the following types of electromotive force are distinguished:
- electrostatic- in an electrophore machine, in which mechanical energy is converted into electrical energy by friction;
- thermoelectric- in a thermoelement - the internal energy of the heated junction of two wires made of different metals is converted into electrical energy;
- photovoltaic- in a photocell. Here the conversion of light energy into electrical energy occurs: when certain substances are illuminated, for example, selenium, copper (I) oxide, silicon, a loss of negative electrical charge is observed;
- chemical- in galvanic cells, batteries and other sources in which chemical energy is converted into electrical energy.
Electromotive force (EMF)- characteristics of current sources. The concept of EMF was introduced by G. Ohm in 1827 for direct current circuits. In 1857, Kirchhoff defined EMF as the work of external forces during the transfer of a unit electric charge along a closed circuit:
ɛ = A st /q,
Where ɛ — EMF of the current source, A st- work of outside forces, q- amount of transferred charge.
Electromotive force is expressed in volts.
We can talk about electromotive force at any part of the circuit. This is the specific work of external forces (work to move a single charge) not throughout the entire circuit, but only in a given area.
Internal resistance of the current source.
Let there be a simple closed circuit consisting of a current source (for example, a galvanic cell, battery or generator) and a resistor with a resistance R. The current in a closed circuit is not interrupted anywhere, therefore, it also exists inside the current source. Any source represents some resistance to current. It's called internal resistance of the current source and is designated by the letter r.
In the generator r- this is the winding resistance, in a galvanic cell - the resistance of the electrolyte solution and electrodes.
Thus, the current source is characterized by the values of EMF and internal resistance, which determine its quality. For example, electrostatic machines have a very high EMF (up to tens of thousands of volts), but at the same time their internal resistance is enormous (up to hundreds of megohms). Therefore, they are unsuitable for generating high currents. Galvanic cells have an EMF of only approximately 1 V, but the internal resistance is also low (approximately 1 Ohm or less). This allows them to obtain currents measured in amperes.
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. In a passenger car, we will power the 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, but consideration of its declared electrical characteristics - voltage and charge (capacity) - 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 between real sources and ideal generators, 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 an electrical circuit inside, 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 cannot be removed 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 to limit the electrical power that can be obtained in the load supplied 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, to determine two unknowns, it is necessary to make two measurements: measure the voltage at the output of a 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 is the output voltage at current I1, Uout2 is the output voltage at current I2. 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 (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, that is, with 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, one should be wary of overloading the two-terminal network with excessive current, that is, the current should not exceed the maximum permissible value 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 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.
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 load voltage 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 that is currently being developed
Let's try to solve this problem using a specific example. The electromotive force of the power source is 4.5 V. A load was connected to it, and a current equal to 0.26 A flowed through it. The voltage then became equal to 3.7 V. First of all, imagine that a serial circuit of an ideal voltage source of 4.5 V, the internal resistance of which is zero, as well as a resistor, the value of which needs to be found. It is clear that in reality this is not the case, but for calculations the analogy is quite suitable.
Step 2
Remember that the letter U only denotes voltage under load. To designate the electromotive force, another letter is reserved - E. It is impossible to measure it absolutely accurately, because you will need a voltmeter with infinite input resistance. Even with an electrostatic voltmeter (electrometer), it is huge, but not infinite. But it’s one thing to be absolutely accurate, and another to have an accuracy acceptable in practice. The second is quite feasible: it is only necessary that the internal resistance of the source be negligible compared to the internal resistance of the voltmeter. In the meantime, let's calculate the difference between the EMF of the source and its voltage under a load consuming a current of 260 mA. E-U = 4.5-3.7 = 0.8. This will be the voltage drop across that “virtual resistor”.
Step 3
Well, then everything is simple, because the classical Ohm’s law comes into play. We remember that the current through the load and the “virtual resistor” is the same, because they are connected in series. The voltage drop across the latter (0.8 V) is divided by the current (0.26 A) and we get 3.08 Ohms. Here is the answer! You can also calculate how much power is dissipated at the load and how much is useless at the source. Dissipation at load: 3.7*0.26=0.962 W. At the source: 0.8*0.26=0.208 W. Calculate the percentage ratio between them yourself. But this is not the only type of problem to find the internal resistance of a source. There are also those in which the load resistance is indicated instead of the current strength, and the rest of the initial data is the same. Then you need to do one more calculation first. The voltage under load (not EMF!) given in the condition is divided by the load resistance. And you get the current strength in the circuit. After which, as physicists say, “the problem is reduced to the previous one”! Try to create such a problem and solve it.
