Resistor Circuits !
Ohm's Law:
If you wish to determine mathematically what is happening in electrical terms in simple or even complex circuits, you have to know that the current I is dependent on two factors, the voltage U and the resistance R. This dependency was described in Ohm's law named after the German physicist Georg Simon Ohm. To elaborate on this, first consider the simple circuit shown below.
If, for example, you double the voltage in this circuit, you will find that the current also increases two fold. If, whilst keeping the voltage constant, you double the resistance, the current is reduced by half. Both observations taken together are summarised in Ohm's law:
The current I increases with increasing voltage U and decreases with increasing resistance R. The current changes in direct proportion to the voltage and in inverse proportion to the resistance. |
As a mathematical expression Ohm's law is stated thus:
If you resolve the equation for voltage U, the following equation is obtained
which describes the voltage drop across a resistor through which a current I flows. If you rearrange the equation so that the resistance R appears on the left-hand side, you obtain the following relationship:
which permits the computation of the resistance from the current and voltage.
(Note: here we use the symbol U to represent voltage, as is conventional in some European countries. Elsewhere it is common to see voltage represented by the letter V, so that Ohm's law is written V = I · R, for example)
Note: resistances for which Ohm' law applies (i.e. proportionality between current and voltage) are also referred to as ohmic resistances. Metal conductors usually present an ohmic resistance, whereas the resistance of conductive fluids does not fulfil the criteria of Ohm's law.
Resistors connected in series:
If several resistors (e.g. three) are connected in a series connection with a power source, we obtain the following circuit diagram.
If measurements of current I are carried out in this circuit at different locations using an ammeter, it is discovered that the same value is always obtained regardless of the location of the measurement. In the above example this means that I_{1} = I_{2}. In general the following holds true:
In a series connection of resistors the current I through each of the resistors is the same. |
Now if you measure the voltage drops U_{1}, U_{2} and U_{3} across the individual resistors and you compare this with the total voltage U (supplied by the voltage source), you arrive at the following law:
In a series connection the total voltage is equal to the sum of the partial voltages, i.e. it holds that |
The following circuit diagram illustrates this relationship.
This law is a simplified version of Kirchhoff's second law, which states that
The algebraic sum of all voltages in a closed circuit is equal to the sum of all voltage drops across the circuit's resistors. |
This law is frequently referred to as Kirchhoff's voltage law.
The voltage drops across the individual resistors can be computed with the aid of Ohm's law from the resistor values and the current I, whereby
If the total resistance of a series connection is termed R_{tot} then, of course, Ohm's law also applies to the entire circuit, i.e. it holds true that
R_{ges} = R_{tot}
If you now apply Kirchhoff's law and Ohm's law for the individual resistors, we obtain
For the total resistance of the series circuit we thus obtain the following expression
i.e.
In a series circuit the total resistance (also called equivalent resistance) is equal to the sum of the partial resistances. |
Since the same current flows through all of the resistors, the partial voltages dropping across the resistors on account of Ohm's law and the resistors themselves respond as follows:
In a series circuit the ratio of the partial voltages is equal to the ratio of corresponding partial resistances. |
Example: A voltage of U = 10 V is applied to a threeresistors R_{1} = 10 W, R_{2} = 40 W and R_{3} = 50 W connected in series. This results in a total resistance R_{tot} = 100 W and thus a current I = U/R_{tot} = 0.1 A. For the voltage drops across the partial resistors, the following values result U_{1} = I·R_{1} = 1 V, U_{2} = I·R_{2} = 4 V and U_{3} = I·R_{3} = 5 V.
The following interactive animation shows a circuit with two resistors connected in series. When you modify the resistor values and the voltage supply of the circuit, observe how the current and the voltage drops across the resistors change
Connecting Resistors in Parallel:
Whereas in simple circuits and series connections the current only finds one path through the circuit, in parallel circuit configurations (shunted connection) there is a branching of the circuit (see the following figure). At the current's branching point (node) A the total current I is divided into partial currents I_{1}, I_{2} and I_{3}, at the current branching point B the partial currents merge again into the total current I.
A fundamental attribute of a parallel circuit is immediately obvious from the structure of the circuit itself: All of the resistors connected in parallel are located between the same points (namely A and B) and are thus all connected to the same voltage U:
When resistors are connected in parallel one and the same voltage is applied across all of the resistors. |
If you now measure the partial currents flowing in the individual branches and compare them to the total current I, you arrive at a principle which constitutes a simplified form of Kirchhoff's first law:
In a parallel circuit the total current is equal to the sum of its partial currents, i.e. it is true that |
In its basic form the law states that
The sum of the currents flowing to a node is equal to the sum of currents flowing out of the node |
and for this reason is sometimes referred to in German as the "node" rule.
Since Ohm's law naturally applies for the parallel circuit and equally for each of the resistors as well as their parallel connection, it is possible to obtain the following equation for the connection of resistors in parallel based on this simplified version of Kirchhoff's first law
R_{ges} = R_{tot}
whereby R_{tot} is the total resistance (equivalent resistance) of the parallel connection. If both sides of the equation are divided by U, you obtain the following relationship for the resistance of the parallel circuit
If in this expression you substitute the resistances by their reciprocal conductances (i.e. the inverse of the resistances), you obtain the total conductance of the parallel connection
Hence it holds true that:
In a parallel connection of resistors the reciprocal of the total resistance is equal to the sum of the reciprocal of the individual resistors and the total conductance is equal to the sum of the conductances of the individual resistors. |
In particular this means that in a parallel circuit the total resistance is always lower than the lowest resistance of any individual resistor.
Since all of the resistors are connected to the same voltage, due to Ohm's law the partial currents must respond in the completely opposite fashion to the partial resistances, i.e. it is true that
In a parallel connection of resistors the partial currents respond in inverse üroportion to the partial resistors. |
Example: A parallel circuit comprising resistors R_{1} = 20 W, R_{2} = 30 W and R_{3} = 60 W is connected to a voltage of U = 10 V. It is then true for the total resistance that
Then according to Ohm's law the following is also true for the total current I = 10 V/10 W = 1 A and for the partial currents
Combined Series and Parallel Resistor Circuits:
You frequently find both series and parallel connections in a single circuit. Circuits of this kind are sometimes referred to as combined or mixed circuits. It is possible, for example, to connect three resistors both in series or parallel connection or also in combined circuits as shown in the following graphic.
For the calculation of the total resistance R of the circuit a) you first determine the total resistance R_{1||2} of the two resistors R_{1} and R2 connected in parallel. Then the total resistance R of the series connection of R_{1||2} and R_{3} is determined. On the basis of Ohm's law and provided you know the supply voltage U you can then determine the current I_{3} and thus the voltage U_{3}. The voltage U_{12} then emerges as the difference between U and U_{3}; the currents I_{1} and I_{2} follow accordingly from Ohm's law.
Example: assuming that U = 10 V, R_{1} = 10 W, R_{2} = 40 W and R_{3} = 12 W, then the result for the total resistance is
For the currents and the voltages the following is true:
For circuit b) you first determine the total resistance R_{2-3} of the resistors connected in series R_{2} and R_{3} and then the total resistance R from the parallel connection of R_{1} and R_{2-3}.
Example: assuming that U = 10 V, R_{1} = 50 W, R_{2} = 10 W and R_{3} = 40 W, then we obtain a total resistance:
For the currents and voltages the following is then obtained