Resistors in a circuit obey Ohm’s circuit law and Kirchoff ’s laws. Ohm’s circuit law is written in its three forms as:
V = RI, or R = V/I or I = V/R
Where V is voltage across two points, ‘I’ is the current flowing between the points and ‘R’ is the (constant) resistance between the points. The units of these quantities are as shown in Table. These equations can be applied even to materials that do not obey Ohm’s law if the value of R for some stated set of conditions can be found.
Materials that do not obey Ohm’s law do not have a constant value of resistance, but the relationships shown above (which simply state a definition of resistance) still hold. The equations are most useful when the resistance values are constant, hence the use of the name Ohm’s law to describe the relationships.
Table Ohm’s Law and Units
Kirchoff’s laws relate to the conservation of energy, which states that energy cannot be created or destroyed, only changed into different forms. This can be expanded to laws of conservation of voltage and current. In any circuit, the voltage across each series component (carrying the same current) can be added to find the total voltage. Similarly, the total current entering a junction in a circuit must equal the sum of current leaving the junction. These laws are illustrated in Figure.
In Figure are shown the rules for finding the total resistance of resistors in series or in parallel. When a combination of series and parallel connections is used, the total resistance of each series or parallel group must be found first before finding the grand total.
The Superposition Theorem
The superposition theorem is very useful for finding the voltages and currents in a circuit with two or more sources of supply, and is usually easier to use than Kirchoff ’s law equations. Figure shows an example of the theorem in use. One supply is selected and the circuit is redrawn to show the other supply (or supplies) short-circuited (leaving only the internal resistance of each supply). The voltage and current caused by the first supply can then be calculated, using V = RI methods together with the rules for combining series and parallel resistors. Each supply is treated in turn in the same way, and finally the voltages and currents caused by each supply are added.
In this network, there are two generators
and three resistors. The generators might
be batteries, oscillators, or other signal
To find the voltage caused by the 6 V
generator, replace the 4 V generator by
its internal resistance of 0.5 kW. Using
Ohm’s law, and the potential divider
equation: V = 1.736 V.
To find the voltage caused by the 4 V
generator, the 6 V generator is replaced
by its 1 kW internal resistance. In this
case: V = 2.315 V.
Now the total voltage in the original
circuit across the 2.2 kW resistor is
simply the sum of these: 4.051 V.
The superposition principle: this states that in any linear network, the voltage at any point is the sum of the voltages caused by each generator in the circuit. To find the voltage caused by a generator replaces all other generators in the circuit by their internal resistances, and use Ohm’s law. A linear network means an arrangement of resistors and generators with the resistors obeying Ohm’s law, and the generators having a constant voltage output and constant internal resistances.
Thevenin’s (pronounced Tay-venin) theorem is, after Ohm’s circuit law, one of the most useful electrical circuit laws. The theorem states that any network of linear components, such as resistors and batteries, can be replaced in its effect by an equivalent circuit consisting only of a voltage source and a resistance in series. The size of the equivalent voltage is found by taking the open-circuit voltage between two points in the network, and the series resistance is found by calculating the resistance between the same two points assuming that the voltage source is short-circuited.
- There is a corresponding theorem, Norton’s theorem, which states that any network of linear components can also be considered to consist only of a constant current source and a resistor in parallel.
Figure shows two important networks, the potential divider and the bridge. When no current is taken from the potential divider, its output voltage V is given by:
as shown, but when current is being drawn, as is the case when a transistor is being biased by this circuit, the equivalent circuit, using Thevenin’s theorem as shown in Figure, is more useful. The bridge circuit, when no current is drawn, is said to be balanced when there is no voltage across R (which is usually a galvanometer or microammeter). In this condition:
If the bridge is not balanced, the equivalent circuit derived from using Thevenin’s theorem is, once again, more useful.