Ohm's law for instantaneous values. Ohm's law in simple terms
Ohm's law is one of the basic laws of electrical engineering. It is quite simple and is used in the calculation of almost any electrical circuits. But this law has some features of operation in AC and DC circuits in the presence of reactive elements in the circuit. These features must always be remembered.
The classic diagram of Ohm's law looks like this:
And it sounds even simpler - the current flowing in a section of the circuit will be equal to the ratio of the circuit voltage to its resistance, which is expressed by the formula:
But we know that in addition to active resistance R, there is also reactance inductance X L and capacitance X C. But you must admit that electrical circuits with purely active resistance are extremely rare. Let's look at a circuit in which an inductor L, a capacitor C, and a resistor R are connected in series:
In addition to purely active resistance R, inductance L and capacitance C also have reactance X L and X C, which are expressed by the formulas:
Where ω is the cyclic frequency of the network, equal to ω = 2πf. f – network frequency in Hz.
For direct current, the frequency is zero (f = 0), accordingly, the inductance reactance will become zero (formula (1)), and the capacitance will become infinity (2), which will lead to a break in the electrical circuit. From this we can conclude that there is no reactance of elements in DC circuits.
If we consider a classical electrical circuit using alternating current, then it will be practically no different from direct current, only a voltage source (instead of constant - alternating):
Accordingly, the formula for such a contour will remain the same:
But if we complicate the circuit and add reactive elements to it:
The situation will change dramatically. Now f is not equal to zero, which indicates that in addition to active resistance, reactance is also introduced into the circuit, which can also affect the amount of current flowing in the circuit and . Now the total resistance of the circuit (denoted as Z) and it is not equal to the active Z ≠ R. The formula will take the following form:
Accordingly, the formula for Ohm's law will change slightly:
Why is this important?
Knowing these nuances will allow you to avoid serious problems that can arise from the wrong approach to solving certain electrical problems. For example, an inductor with the following parameters is connected to an alternating voltage circuit: f nom = 50 Hz, U nom = 220 V, R = 0.01 Ohm, L = 0.03 H. The current flowing through this coil will be equal.
Ohm's law for alternating current generally has the same form as for direct current. That is, as the voltage in the circuit increases, the current in it will also increase. The difference is that in an alternating current circuit resistance is provided by elements such as an inductor and capacitance. Taking this fact into account, let us write down Ohm's law for alternating current.
Formula 1 - Ohm's law for alternating current
where z is the total resistance of the circuit.
Formula 2 - circuit impedance
In general, the impedance of an AC circuit will consist of active capacitive and inductive reactance. Simply put, the current in an alternating current circuit depends not only on the active ohmic resistance, but also on the value of capacitance and inductance.
Figure 1 - circuit containing ohmic inductive and capacitive reactance
If, for example, a capacitor is connected to a DC circuit, then there will be no current in the circuit, since a DC capacitor is an open circuit. If inductance appears in the DC circuit, the current will not change. Strictly speaking, it will change, since the coil will have ohmic resistance. But the change will be negligible.
If the capacitor and coil are connected in an alternating current circuit, then they will resist the current in proportion to the value of the capacitance and inductance, respectively. In addition, a phase shift between voltage and current will be observed in the circuit. In general, the current in a capacitor leads the voltage by 90 degrees. In inductance it lags by 90 degrees.
Capacitance depends on the size of the capacitance and the frequency of the alternating current. This dependence is inversely proportional, that is, with increasing frequency and capacitance, the resistance will decrease.
DC circuits
Before studying Ohm's law, let's consider two circuits for connecting resistances. Figure 1 shows a circuit for connecting resistances in series, and Figure 2 shows a circuit for connecting resistances in parallel.
Let's figure out how to calculate the total resistance of a circuit (such a resistance can be replaced by a circuit of resistances - let's call it Re).
For resistances connected in series, the total resistance is equal to the sum of the circuit resistances. 
This is obvious since the input current I flows in series through all resistances (current IR) and does not change. For any number of resistances connected in series, they are added together to obtain the total resistance.
When resistances are connected in parallel, the input current I branches into separate currents I1, I2,..., In-1, In which depend on the value of the resistances through which they flow. 
If the resistances have different values, then the currents passing through them will also be different. The total resistance of two parallel-connected resistances is calculated by the formula: 1/Re = 1/R1 + 1/R2 or Re = (R1*R2) / (R1+R2). If there are more than two resistances, then the total resistance is calculated using the formula shown in the figure.
Ohm's law for DC circuits
The simplest electrical circuit is a power source (GB1 in the figure) and a load resistance R. This electrical circuit is characterized by three main parameters: the current I passing through the load resistance, the power source voltage U and the load resistance R.
According to Ohm's law, any of these three parameters can be calculated using the formulas shown in the figure.
Now, having certain knowledge, let’s calculate the power at the resistances R1, R2 and R3 turned on as shown in the figure. R1 = 100 ohms, R2 = 200 ohms and R3 = 200 ohms. 
All three parameters have a linear relationship with each other. For example, an increase in load resistance leads to a proportional decrease in the current in the circuit.
We know that the current passing through the load resistance heats the load, for example, an electric soldering iron, an iron, etc. Therefore, when calculating electrical circuits, one should take into account such an important parameter as the electrical power of the load. It is important to know power, firstly in order to select the right power source and secondly, in order to calculate the load of the corresponding power. If the load power is insufficient, the load resistance will heat up and eventually burn out. Power is designated by the letter P and is measured in units - WATT (denoted by W or W). Power is calculated using the formulas shown in the figure.
First, let's find the total resistance Re of parallel-connected resistances R2 and R3. Re = R2 * R3 / (R2+R3). Re = 200 * 200 / (200+200) = 40000 / 400 = 100 Ohm. Let's find the current I1. I1 = U / (R1+Re) = 12 / (100+100) = 12 / 200 = 0.06A. By definition, the current I1 is equal to the current Ie, therefore we can find the voltage Ure across the resistance Re. Ure = Ie * Re = 0.06 * 100 = 6 V.
Let's find the power Pr1 on resistor R1.
Pr1 = I1 2 * R1 = 0.06 2 * 100 = 0.0036 * 100 = 0.36W.
Let's find the power Pr2 on resistor R2.
Pr2 = Ure 2 / R2 = 6 2 / 200 = 36 / 200 = 0.18W.
Since R2=R3, the power on R3 is equal to the power on R2 and is equal to 0.18W (Watt).
DC measurements
In practice, it is often necessary to measure some parameters of a section of an electrical circuit. Typically this is current, voltage or resistance. For each type of measurement there is its own measuring device. To measure voltage, a device called a VOLTMETER is used, to measure current, an AMPERMETER device, and to measure resistance, an OHMETER.
To repair, adjust and configure radio and electronic equipment, universal measuring instruments called TESTER or MULTIMETER are usually used. These devices can, depending on the position of the mode switch, measure current, voltage or resistance. 
Based on the display of measurement results, multimeters are divided into digital and analog. The figure shows both types of devices. Both devices have common measurement limit switches. The measurement limits are grouped by functional purpose (in the figure the groups are indicated by V, A and Ohm).
In the figure, the measurement limit switches for both devices are set to measure voltage with a limit of 5 Volts. The + and - signs indicate terminals for connecting conductors to the circuit being measured.
The electric current in the circuit is measured with an AMMETER. The ammeter is connected in series with the load.
When measuring an unknown current, it is better to set the current measurement limit to the maximum value so as not to damage the device. In the figure, the measurement limit is set to measure the maximum current value of 1 Ampere. Electrical VOLTAGE at a resistance or section of a circuit is measured with a VOLTMETER. The voltmeter is connected in parallel with the resistance or section of the circuit.
When measuring an unknown voltage, the measurement limit on the voltmeter should be set to the maximum value. If the instrument readings are too small, the measurement limit should be gradually reduced.
Electrical RESISTANCE is measured with an OHMETER. An ohmmeter is connected in parallel to the resistance being measured.
A prerequisite for measuring resistance must be that the electrical circuit must be de-energized. It is also necessary to keep in mind that the resistance being measured should not be connected in parallel to another resistance.
Ohm's law- a physical law that defines the relationship between electrical quantities - voltage, resistance and current for conductors.
It was first discovered and described in 1826 by the German physicist Georg Ohm, who showed (using a galvanometer) the quantitative relationship between electromotive force, electric current and the properties of the conductor as a proportional relationship.
Subsequently, the properties of a conductor capable of resisting electric current based on this dependence began to be called electrical resistance (Resistance), denoted in calculations and diagrams by the letter R and measured in Ohms in honor of the discoverer.
The source of electrical energy itself also has internal resistance, which is usually denoted by the letter r.
Ohm's law for a circuit section
From the school physics course, everyone is well aware of the classical interpretation of Ohm’s Law:
The current strength in a conductor is directly proportional to the voltage at the ends of the conductor and inversely proportional to its resistance.
This means if there is resistance to the ends of the conductor R= 1 ohm voltage applied U= 1 Volt, then the magnitude of the current I in the conductor will be equal to 1/1 = 1 Ampere.
This leads to two more useful relationships:
If a current of 1 Ampere flows in a conductor with a resistance of 1 Ohm, then at the ends of the conductor there is a voltage of 1 Volt (voltage drop).
If there is a voltage of 1 Volt at the ends of the conductor and a current of 1 Ampere flows through it, then the resistance of the conductor is 1 Ohm.
The above formulas in this form can be applied to alternating current only if the circuit consists only of active resistance R.
In addition, it should be remembered that Ohm's Law is valid only for linear circuit elements.
A simple online calculator is provided for practical calculations.
Ohm's law. Calculation of voltage, resistance, current, power.
After resetting, enter any two known parameters.
Ohm's law for a closed circuit
If you connect an external circuit with a resistance to the power source R, current will flow in the circuit taking into account the internal resistance of the source:
I- Current strength in the circuit.
- Electromotive force (EMF) - the magnitude of the power source voltage independent of the external circuit (without load). Characterized by the potential energy of the source.
r- Internal resistance of the power supply.
For electromotive force, external resistance R and internal r are connected in series, which means the amount of current in the circuit is determined by the value of the emf and the sum of the resistances: I = /(R+r) .
The voltage at the terminals of the external circuit will be determined based on the current and resistance R relationship, which has already been discussed above: U = IR.
Voltage U, when connecting the load R, will always be less than the EMF by the value of the product I*r, which is called the voltage drop across the internal resistance of the power supply.
We encounter this phenomenon quite often when we see partially discharged batteries or accumulators in operation.
As the discharge progresses, their internal resistance increases, therefore, the voltage drop inside the source increases, which means the external voltage decreases U = - I*r.
The lower the current and internal resistance of the source, the closer in value its EMF and voltage at its terminals U.
If the current in the circuit is zero, therefore = U. The circuit is open, the emf of the source is equal to the voltage at its terminals.
In cases where the internal resistance of the source can be neglected ( r≈ 0), the voltage at the source terminals will be equal to the EMF ( ≈ U) regardless of the external circuit resistance R.
This power source is called voltage source.
Ohm's law for alternating current
If there is inductance or capacitance in an AC circuit, its reactance must be taken into account.
In this case, the entry for Ohm's Law will look like:
Here Z- total (complex) resistance of the circuit - impedance. It includes active R and reactive X components.
Reactance depends on the ratings of the reactive elements, on the frequency and shape of the current in the circuit.
You can learn more about complex resistance on the impedance page.
Taking into account the phase shift φ
created by reactive elements, Ohm's Law is usually written for sinusoidal alternating current in a complex form:
Complex current amplitude. = I amp e jφ
- complex voltage amplitude. = U amp e jφ
- complex resistance. Impedance.
φ
- phase shift angle between current and voltage.
e- constant, the base of the natural logarithm.
j- imaginary unit.
I amp, U amp- amplitude values of sinusoidal current and voltage.
Nonlinear elements and circuits
Ohm's law is not a fundamental law of nature and can be applied in limited cases, for example, for most conductors.
It cannot be used to calculate voltage and current in semiconductor or vacuum devices, where this dependence is not proportional and can only be determined using the current-voltage characteristic (volt-ampere characteristic). This category of elements includes all semiconductor devices (diodes, transistors, zener diodes, thyristors, varicaps, etc.) and vacuum tubes.
Such elements and the circuits in which they are used are called nonlinear.
Georg Simon Ohm began his research inspired by the famous work of Jean Baptiste Fourier, “The Analytical Theory of Heat.” In this work, Fourier represented the heat flow between two points as a temperature difference, and associated the change in heat flow with its passage through an irregularly shaped obstacle made of heat-insulating material. Similarly, Ohm caused the occurrence of electric current by a potential difference.
Based on this, Ohm began to experiment with different conductor materials. In order to determine their conductivity, he connected them in series and adjusted their length so that the current strength was the same in all cases.
It was important for such measurements to select conductors of the same diameter. Ohm, measuring the conductivity of silver and gold, obtained results that, according to modern data, are not accurate. Thus, Ohm's silver conductor conducted less electric current than gold. Om himself explained this by saying that his silver conductor was coated with oil and because of this, apparently, the experiment did not give accurate results.
However, this was not the only problem that physicists, who at that time were engaged in similar experiments with electricity, had problems with. Great difficulties in obtaining pure materials without impurities for experiments and difficulties in calibrating the diameter of the conductor distorted the test results. An even bigger snag was that the current strength was constantly changing during the tests, since the source of the current was alternating chemical elements. Under such conditions, Ohm derived a logarithmic dependence of the current on the resistance of the wire.
A little later, the German physicist Poggendorff, who specialized in electrochemistry, suggested that Ohm replace the chemical elements with a thermocouple made of bismuth and copper. Om began his experiments again. This time he used a thermoelectric device powered by the Seebeck effect as a battery. To it he connected in series 8 copper conductors of the same diameter, but of different lengths. To measure the current, Ohm suspended a magnetic needle over the conductors using a metal thread. The current running parallel to this arrow shifted it to the side. When this happened, the physicist twisted the thread until the arrow returned to its original position. Based on the angle at which the thread was twisted, one could judge the value of the current.
As a result of a new experiment, Ohm came to the formula:
X = a / b + l
Here X– intensity of the magnetic field of the wire, l– wire length, a– constant source voltage, b– resistance constant of the remaining elements of the circuit.
If we turn to modern terms to describe this formula, we get that X– current strength, A– EMF of the source, b + l– total circuit resistance.
Ohm's law for a circuit section
Ohm's law for a separate section of a circuit states: the current strength in a section of a circuit increases as the voltage increases and decreases as the resistance of this section increases.
I=U/R
Based on this formula, we can decide that the resistance of the conductor depends on the potential difference. From a mathematical point of view, this is correct, but from a physics point of view, it is false. This formula is applicable only for calculating the resistance on a separate section of the circuit.
Thus, the formula for calculating the conductor resistance will take the form:
R = p ⋅ l / s
Ohm's law for a complete circuit
The difference between Ohm's law for a complete circuit and Ohm's law for a section of a circuit is that now we must take into account two types of resistance. This is “R” the resistance of all components of the system and “r” the internal resistance of the source of electromotive force. The formula thus takes the form:
I = U / R + r
Ohm's law for alternating current
Alternating current differs from direct current in that it changes over certain time periods. Specifically, it changes its meaning and direction. To apply Ohm's law here, you need to take into account that the resistance in a circuit with direct current may differ from the resistance in a circuit with alternating current. And it differs if components with reactance are used in the circuit. Reactance can be inductive (coils, transformers, chokes) or capacitive (capacitor).
Let's try to figure out what the real difference is between reactive and active resistance in a circuit with alternating current. You should already understand that the value of voltage and current in such a circuit changes over time and, roughly speaking, have a wave form.
If we diagram how these two values change over time, we get a sine wave. Both voltage and current rise from zero to a maximum value, then, falling, pass through zero and reach a maximum negative value. After this, they rise again through zero to the maximum value and so on. When it is said that current or voltage is negative, it means that it moves in the opposite direction.
The whole process occurs with a certain frequency. The point where the voltage or current value from the minimum value rising to the maximum value passes through zero is called phase.
In fact, this is just a preface. Let's return to reactive and active resistance. The difference is that in a circuit with active resistance, the current phase coincides with the voltage phase. That is, both the current value and the voltage value reach a maximum in one direction at the same time. In this case, our formula for calculating voltage, resistance or current does not change.
If the circuit contains reactance, the phases of the current and voltage shift from each other by ¼ of a period. This means that when the current reaches its maximum value, the voltage will be zero and vice versa. When inductive reactance is applied, the voltage phase "overtakes" the current phase. When capacitance is applied, the current phase "overtakes" the voltage phase.
Formula for calculating the voltage drop across inductive reactance:
U = I ⋅ ωL
Where L is the inductance of the reactance, and ω – angular frequency (time derivative of the oscillation phase).
Formula for calculating the voltage drop across capacitance:
U = I / ω ⋅ C
WITH– reactance capacitance.
These two formulas are special cases of Ohm's law for variable circuits.
The complete one will look like this:
I=U/Z
Here Z– The total resistance of a variable circuit is known as impedance.
Scope of application
Ohm's law is not a basic law in physics, it is just a convenient dependence of some values on others, which is suitable in almost any practical situation. Therefore, it will be easier to list situations when the law may not work:
- If there is inertia of charge carriers, for example in some high-frequency electric fields;
- In superconductors;
- If the wire heats up to such an extent that the current-voltage characteristic ceases to be linear. For example, in incandescent lamps;
- In vacuum and gas radio tubes;
- In diodes and transistors.