The resistive circuits can be analyzed by reducing a network of resistors in series and parallel to an equivalent resistance, for which the current and voltage values can be obtained by means of Ohm's law; known these values, you can proceed backwards and calculate the currents and voltages at the ends of each resistance of the network.
This article briefly illustrates the equations necessary to carry out an analysis of this type, along with some practical examples. Additional reference sources are also indicated, although the article itself provides sufficient detail to be able to put the concepts acquired into practice without the need for further study. The "step-by-step" approach is used only in sections where there is more than one step.
The resistances are represented in the form of resistors (in the schematic, as zigzag lines), and the circuit lines are intended as ideal, and therefore with zero resistance (at least in relation to the resistances shown).
A summary of the main steps is set out below.
Steps
Step 1. If the circuit contains more than one resistor, find the equivalent resistance "R" of the entire network, as shown in the "Combination of Series and Parallel Resistors" section
Step 2. Apply Ohm's Law to this resistance value “R”, as illustrated in the section “Ohm's Law”
Step 3. If the circuit contains more than one resistor, the current and voltage values calculated in the previous step can be used, in Ohm's law, to derive the voltage and current of every other resistor in the circuit
Ohm's Law
Parameters of Ohm's law: V, I, and R.
Ohm's law can be written in 3 different forms depending on the parameter to be obtained:
(1) V = IR
(2) I = V / R
(3) R = V / I
"V" is the voltage across the resistance (the "potential difference"), "I" is the intensity of current flowing through the resistance, and "R" is the resistance value. If the resistance is a resistor (a component that has a calibrated resistance value) it is normally indicated with "R" followed by a number, such as "R1", "R105", etc.
Form (1) is easily convertible into forms (2) or (3) with simple algebraic operations. In some cases, instead of the symbol "V", "E" is used (for example, E = IR); "E" stands for EMF or "electromotive force", and is another name for voltage.
Form (1) is used when both the value of the current flowing through a resistance and the value of the resistance itself are known.
Form (2) is used when both the value of the voltage across the resistance and the value of the resistance itself are known.
Form (3) is used to determine the value of the resistance, when both the voltage value across it and the intensity of current flowing through it are known.
The units of measurement (defined by the International System) for Ohm's law parameters are:
- The voltage across the resistor "V" is expressed in Volts, symbol "V". The abbreviation "V" for "volt" is not to be confused with the voltage "V" that appears in Ohm's law.
- The intensity of current "I" is expressed in Ampere, often abbreviated to "amp" or "A".
- Resistance "R" is expressed in Ohms, often represented by the Greek capital letter (Ω). The letter "K" or "k" expresses a multiplier for "one thousand" ohms, while "M" or "MEG" for one "million" ohms. Often the symbol Ω is not indicated after the multiplier; for example, a 10,000 Ω resistor can be indicated with "10K" rather than "10 K Ω".
Ohm's law is applicable for circuits containing only resistive elements (such as resistors, or the resistances of conductive elements such as electrical wires or PC board tracks). In the case of reactive elements (such as inductors or capacitors) Ohm's law is not applicable in the form described above (which contains only "R" and does not include inductors and capacitors). Ohm's law can be used in resistive circuits if the applied voltage or current is direct (DC), if it is alternating (AC), or if it is a signal that varies randomly over time and is examined at a given instant. If the voltage or current is sinusoidal AC (as in the case of the 60 Hz household network), the current and voltage are usually expressed in volts and amps RMS.
For additional information about Ohm's law, its history and how it is derived, you can consult the related article on Wikipedia.
Example: Voltage drop across an electric wire
Let's assume we want to calculate the voltage drop across an electric wire, with resistance equal to 0.5 Ω, if it is crossed by a current of 1 ampere. Using the form (1) of Ohm's law we find that the voltage drop across the wire is:
V. = IR = (1 A) (0.5 Ω) = 0.5 V (that is, 1/2 volt)
If the current had been that of the home network at 60 Hz, suppose 1 amp AC RMS, we would have obtained the same result, (0, 5), but the unit of measurement would have been "volts AC RMS".
Resistors in Series
The total resistance for a "chain" of resistors connected in series (see figure) is simply given by the sum of all resistances. For "n" resistors named R1, R2, …, Rn:
R.total = R1 + R2 +… + Rn
Example: Series resistors
Let's consider 3 resistors connected in series:
R1 = 10 Ohm
R2 = 22 Ohm
R3 = 0.5 Ohm
Total resistance is:
R.total = R1 + R2 + R3 = 10 + 22 + 0.5 = 32.5 Ω
Parallel resistors
The total resistance for a set of resistors connected in parallel (see figure) is given by:
The common notation for expressing the parallelism of resistances is (""). For example, R1 in parallel with R2 is denoted by "R1 // R2". A system of 3 resistors in parallel R1, R2 and R3 can be indicated with "R1 // R2 // R3".
Example: Parallel resistors
In the case of two resistors in parallel, R1 = 10 Ω and R2 = 10 Ω (of identical value), we have:
It is called "less than the minor", to indicate that the value of the total resistance is always less than the smallest resistance among those that make up the parallel.
Combination of Resistors in Series and Parallel
Networks that combine resistors in series and parallel can be analyzed by reducing the "total resistance" to an "equivalent resistance".
Steps
- In general, you can reduce the resistances in parallel to an equivalent resistance using the principle described in the section “Resistors in Parallel”. Remember that if one of the branches of the parallel consists of a series of resistors, you must first reduce the latter to an equivalent resistance.
- You can derive the total resistance of a series of resistors, R.total simply by adding up the individual contributions.
- It uses Ohm's law to find, given a voltage value, the total current flowing in the network, or, given the current, the total voltage across the network.
- The total voltage, or current, calculated in the previous step is used to calculate the individual voltages and currents in the circuit.
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Apply this current or voltage in Ohm's law to derive the voltage or current across each resistor in the network. This procedure is briefly illustrated in the following example.
Note that for large networks it may be necessary to perform several iterations of the first two steps.
Example: Series / Parallel Network
SeriesParallelCircuit_313 For the network shown on the right, it is first necessary to combine the resistors in parallel R1 // R2, to then obtain the total resistance of the network (across the terminals) by:
R.total = R3 + R1 // R2
Suppose we have R3 = 2 Ω, R2 = 10 Ω, R1 = 15 Ω, and a 12 V battery applied to the ends of the network (therefore Vtotal = 12 volts). Using what is described in the previous steps we have:
SeriesParallelExampleEq_708 The voltage across R3 (indicated by VR3) can be calculated using Ohm's law, given that we know the value of the current passing through the resistance (1, 5 amperes):
V.R3 = (Itotal) (R3) = 1.5 A x 2 Ω = 3 volts
The voltage across R2 (which coincides with that across R1) can be calculated using Ohm's law, multiplying the current I = 1.5 amps by the parallel of resistors R1 // R2 = 6 Ω, thus obtaining 1.5 x 6 = 9 volts, or by subtracting the voltage across R3 (VR3, calculated earlier) from the battery voltage applied to the network 12 volts, that is, 12 volts - 3 volts = 9 volts. Known this value, it is possible to obtain the current that crosses the resistance R2 (indicated with IR2)) by means of Ohm's law (where the voltage across R2 is indicated by VR2"):
THER2 = (VR2) / R2 = (9 volts) / (10 Ω) = 0.9 amps
Similarly, the current flowing through R1 is obtained, by means of Ohm's law, by dividing the voltage across it (9 volts) by the resistance (15 Ω), obtaining 0.6 amps. Note that the current through R2 (0.9 amps), added to the current through R1 (0.6 amps), equals the total current of the network.
