Equivalent resistance what the term is and its use in engineering practice is one of the fundamental issues in electrical circuit analysis. It refers to a conversion resistance that allows any number of connected resistors to be replaced by a single equivalent component that does not change the electrical characteristics of a section of the circuit. This simplification is essential when designing, optimising and diagnosing systems composed of multiple passive components.
What is the equivalent resistance?
Understanding how resistors work in larger electrical circuits requires a simplification of their structure. When several resistors work together in a circuit section, they can be replaced by a single element with the same function in terms of resistance to the flowing current. Such a procedure greatly simplifies circuit analysis and design.
The equivalent resistance is just that, a single value of resistance that replaces any number of resistors connected together. With it, you can quickly calculate the current flowing in a circuit or the voltage drop using basic laws such as Ohm’s law. The unit of resistance is the ohm, written as Ohm.
Resistor connection types
Series connection
Resistors arranged one behind the other, in a single line. The current flows through each one in turn.
Design:
R_{\text{zast}} = R_1 + R_2 + R_3 + \ldotsExample:
R_1 = 5,\Omega,\ R_2 = 10,\Omega,\ R_3 = 15,\OmegaR_{\text{zast}} = 5 + 10 + 15 = 30,\OmegaParallel connection
The resistors are connected to the same two points. Current can flow through several paths.
Design:
\frac{1}{R_{\text{zast}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldotsAbbreviation for 2 resistors:
R_{\text{zast}} = \frac{R_1 \cdot R_2}{R_1 + R_2}Example:
R_1 = 6,\Omega,\ R_2 = 3,\OmegaR_{\text{zast}} = \frac{6 \cdot 3}{6 + 3} = \frac{18}{9} = 2,\OmegaMixed connections
Most real-world circuits do not consist solely of resistors connected in a simple way – in series or in parallel. In practice, it is much more common to see so-called mixed connections, which are a combination of both types. Such a circuit may contain sections where resistors are arranged one behind the other (in series), and right next to others that share common terminals (i.e. they are connected in parallel). The analysis of such circuits may seem difficult, but a proper step-by-step approach allows the equivalent resistance of the entire circuit to be calculated without difficulty.
The key to solving a mixed circuit is to be able to identify which resistors are connected in series and which are connected in parallel. We usually start with the most obvious parts, which can be simplified straight away. First, identify the simplest groups of connections – series or parallel – and calculate the equivalent resistance for them. We then replace such a fragment with a single resistor of the calculated resistance and analyse the simplified circuit again.
We repeat the whole process until we are left with only one resistor representing the entire circuit. This step-by-step reduction makes it easy to determine R_{text{zast}} even for very complex configurations. It can be helpful to draw simplified diagrams after each step and to systematically record the results. This not only avoids mistakes, but also teaches you how to approach problems in electronics in a logical way.
How do you calculate the equivalent resistance? Step by step
The process of calculating the equivalent resistance can be reduced to a few repeatable steps. It is important to keep things in order and analyse the circuit in stages – this way you will avoid mistakes and arrive at the correct result quickly.
Step 1: Determine the type of resistor connection.
To begin with, analyse the circuit and decide whether the resistors are connected in series, parallel or mixed. If you have more than three elements, start with the simplest parts – for example, two resistors connected directly. Note whether the current flows through them one by one (in series) or whether it splits into several paths (in parallel).
Step 2: Apply the appropriate formula.
For a serial connection, use simple addition:
R_{\text{zast}} = R_1 + R_2 + R_3 + \ldotsFor a parallel connection, apply the inverse formula:
\frac{1}{R_{\text{zast}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldotsIn the case of two resistors in parallel, you can use a convenient shortcut:
R_{\text{zast}} = \frac{R_1 \cdot R_2}{R_1 + R_2}Step 3: Calculate the resistance value and record it with the unit.
After substituting the data, perform the operations and write the result in ohms: \Omega. Remember that for parallel connections the result should be less than the smallest of the resistors.
Step 4: Simplify the layout in stages.
If you are dealing with a mixed system, carry out the reduction step by step. First simplify the parts that can be calculated and then swap them for one resistor. Repeat this process until you have one resistor left.
Tip: Draw diagrams. Mark which elements have already been recalculated. As you simplify, you can redraw the circuit – simplified and legible. This greatly reduces the risk of getting it wrong and, at the same time, develops your ability to think logically and read electrical circuits.
Examples of calculations
To get a good understanding of how equivalent resistance works in practice, it is useful to go through concrete examples. Below you will find simple tasks showing step-by-step how to calculate the equivalent resistance for different types of resistor connections: series, parallel and mixed.
Example 1: In series
R_1 = 2,\Omega,\ R_2 = 4,\Omega,\ R_3 = 6,\OmegaR_{\text{zast}} = 2 + 4 + 6 = 12,\OmegaExample 2: Parallel
R_1 = 8,\Omega,\ R_2 = 4,\OmegaR_{\text{zast}} = \frac{8 \cdot 4}{8 + 4} = \frac{32}{12} \approx 2{,}67,\OmegaExample 3: Mixed connection
R_1 = 6,} (in series with the others), [latex]R_2 = 12,} (in parallel), [latex]R_3 = 12,} (in parallel).<br>[latex display="true"]R_{\text{równ.}} = _{frac{12 } 12}{12 + 12} = _{frac{144}{24} = 6,\Omega.
R_{\text{zast}} = 6 + 6 = 12,\OmegaThe most common errors - equivalent resistance
When learning to calculate equivalent resistance, students often make the same mistakes. One of the most common is confusing series connections with parallel connections. This is a very important distinction: in a series connection, the current flows through each element in turn, while in a parallel connection it splits into several paths. An equally common mistake is using the wrong formula - for example, adding the resistance in parallel instead of calculating the inverse. Added to this are calculation problems, especially with fractions and rounding, and the omission of units, leading to incomplete or incorrect results.
In complex circuits it is also easy to get lost in the circuit structure, especially if it is not drawn clearly. Therefore, the key to avoiding mistakes is to work in stages, following the principle "from simple to complex". It is a good idea to draw out the diagrams each time and to mark the simplified parts. After each step, it is a good idea to ask yourself "does the result make sense?" - e.g. is the resistance in the parallel circuit really smaller than the smallest of the resistors. This approach not only avoids mistakes, but also allows you to better understand how circuits work.
Summary
Equivalent resistance is one of the basic tools in electrical circuit analysis. It allows us to simplify a complex circuit to a single resistance, which makes further calculations and understanding of the operation of the whole system much easier. This approach is useful in both school assignments and practical electronic projects.
The most important thing is to correctly identify the type of connection - serial, parallel or mixed - and apply the correct formula. If you work step by step, record the results with units and check the correctness of each step, calculations will quickly become intuitive and error-free.
Sources
- https://forbot.pl/blog/kurs-elektroniki-poziom-i-rezystory-id4440
- https://botland.com.pl/blog/jak-obliczyc-rezystancje-zastepcza/
- https://www.elportal.pl/pdf/k04/76_18.pdf
- https://www.tinkercad.com/circuits
- https://pl.khanacademy.org/science/physics/circuits-topic/resistance-in-circuits/a/what-is-resistance
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