Resistors In Series and Parallel
Wherever there is a flow of charged particles, we say that it is an electric current. The flow of electricity is always stable and constant. You will find that it flows from higher potential to lower potential in a circuit. A regular circuit contains conductors (usually copper wires), resistors, switch to turn on and off the circuit, and a power source. A resistor is an electrical component that provides resistance or limits the flow of current in the circuit. For example, we can consider a tube light used in our household as a resistor.
In general use, we have a combination of resistors used in all circuits. We can either have resistors in series or resistors in parallel. In some cases, we can have both series and parallel combinations of resistors in a single circuit.
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The above figure shows a simple circuit consisting of a conductor, a resistor, and a battery.
Series Combination of Resistors
When we have resistors in series, the current flows through them one after the other. Each resistor will have the same current flowing through them. Across a series of resistors, there will always be a voltage drop.
To calculate the equivalent resistance, we need to derive the series resistance formula. To obtain the equation, we use Ohm's law. According to the law, the potential drop 'V' is given as V=IR, where 'I' is the current, and 'R' is the resistance of the circuit.
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The image shows the circuit of resistors placed in series.
According to Kirchhoff’s loop law we have,
\[\sum_{i=1}^{N}\] \[V_{i}\] = 0
V − V1 − V2 − V3 = 0,
V = V1 + V2 + V3
V = IR1 + IR2 + IR3
I = VR1 + R2 + R3 = VRS
Therefore, we get the series resistance formula as,
RS = R1 + R2 + R3 + . . . + RN−1 + RN
\[R_{s}\] = \[\sum_{i=1}^{N}\] \[R_{i}\]
Parallel Combination of Resistors
When we have resistors in parallel, the electric current divides itself to travel through the different branches. The voltage drops across each resistor will be equal, unlike the resistors in series. Since there are a lot of resistors connected in parallel, we must find the total resistance. To do that we need to derive the resistors in the parallel formula. To acquire the equation, we use Ohm's law. According to the law the equation for the electric current 'I' is given as I = V/R, where 'V' is the potential drop, and 'R' is the resistance of the circuit.
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The image shows the circuit of resistors placed in series.
According to Kirchhoff’s junction rule we have,
\[\sum\] \[I_{in}\] = \[\sum\] \[I_{out}\]
I = I1 +I2
I = \[\frac{V_{1}}{R_{1}}\] + \[\frac{V_{2}}{R_{2}}\] = \[\frac{V}{R_{1}}\] + \[\frac{V}{R_{2}}\]
I = V \[\left ( \frac{1}{R_{1}} + \frac{1}{R_{2}} \right )\]
\[R_{p}\] = \[\left ( \frac{1}{R_{1}} + \frac{1}{R_{2}} \right )^{-1}\]
Therefore, we get the resistors in the parallel formula as,
\[R_{p}\] = \[\left ( \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \cdots + \frac{1}{R_{N-1} + \frac{1}{R_{N}}} \right )^{-1}\]
\[R_{p}\] = \[\left ( \sum_{i=1}^{N} \frac{1}{R_{i}} \right )^{-1}\]
Combination of Resistors In Series And Parallel
In practice, you will never find simple electrical circuits, where the resistors are only placed in series or parallel. Instead, you will find complex connections with multiple resistors connected in series and parallel at the same time. Now, just because the circuit looks complicated, it doesn't mean that it is difficult to calculate the resistance of the circuit. All you need to do is break the connections into small parts so that you can calculate the equivalent resistance easily. Your main goal is to keep decreasing the number of resistors by using the formula of resistance in series and parallel. We will now try to solve some questions on resistors in series and parallel.
Solved Problems
Question 1) Consider a circuit with a voltage of 9V, and consisting of five resistors. Calculate the equivalent resistance, and the current ‘I’ through the resistors.
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Answer 1) Looking at the figure, we can see that the resistors are in series. Therefore, we will use the series resistance formula to calculate the equivalent resistance.
Given: V = 9V
R1 = R2 = R3 = R4 = 20Ω
R5 = 10Ω
The equivalent resistance is given as,
RS = R1 + R2 + R3 + R4 + R5 = 20Ω + 20Ω + 20Ω + 20Ω + 10Ω = 90Ω
The total resistance with the correct number of significant digits is Req = 90Ω.
Using Ohm’s law, we can calculate the current in the circuit.
I = V/RS = 9V/90Ω = 0.1A
Question 2) Three resistors R1 = 1.00Ω, R2 = 2.00Ω, and R3 = 2.00Ω, are connected in parallel. The battery has a voltage of 3V. Calculate the equivalent resistance, and current ‘I’ through the circuit.
Answer 2) Since the resistors are connected in parallel, we will use the resistors in parallel formula to calculate the equivalent resistance.
Given: V = 3V
R1 = 1.00Ω
R2 = 2.00Ω
R3 = 2.00Ω
The equivalent resistance is given as,
\[R_{p}\] = \[\left ( \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} \right )^{-1}\]
\[R_{p}\] = \[\left ( \frac{1}{1} + \frac{1}{2} + \frac{1}{2} \right )^{-1}\]
\[R_{p}\] = 0.50Ω
Therefore, we get the equivalent resistance as \[R_{eq}\] = 0.50Ω.
Using Ohm’s law, we can calculate the current in the circuit.
I = V/Rp = 3V/0.5Ω = 6A
FAQs on Resistors in Series and Parallel Configuration
1. What is the main difference between connecting resistors in series and in parallel?
The main difference lies in how current and voltage behave in the circuit.
- In a series configuration, resistors are connected end-to-end, providing only a single path for the current. Therefore, the current is the same through each resistor, but the total voltage is divided among them.
- In a parallel configuration, resistors are connected across the same two points, creating multiple paths for the current. Here, the voltage across each resistor is the same, but the total current from the source is divided among the branches.
2. What are the formulas for calculating equivalent resistance in series and parallel circuits?
The formulas for calculating the equivalent resistance (R_eq) differ for each configuration:
- For resistors in series: The equivalent resistance is the sum of the individual resistances. The formula is:
R_series = R1 + R2 + R3 + ... + Rn - For resistors in parallel: The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances. The formula is:
1/R_parallel = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn
3. Why does the voltage divide across resistors in a series circuit but remain the same in a parallel one?
This happens due to the nature of electric potential and current paths. In a series circuit, the current flows through each resistor sequentially. Each resistor causes a 'potential drop' (V=IR), and since they are on the same path, these drops add up to the total source voltage. In a parallel circuit, all resistors are connected between the same two common points in the circuit. As voltage is simply the potential difference between two points, all components connected between these same two points must have the same voltage across them.
4. In which real-world applications are series and parallel resistor combinations most commonly used?
Both configurations have distinct, practical uses:
- Series circuits are often used in applications where current regulation or a specific voltage drop is needed. Examples include fuses, which are placed in series to protect a circuit, and old decorative fairy lights, where if one bulb burns out, the entire string goes off.
- Parallel circuits are fundamental to applications where independent operation is crucial. The most common example is household wiring, where all appliances receive the same voltage (e.g., 220V) and can be turned on or off without affecting the others.
5. How should one approach solving a complex circuit with a mix of series and parallel resistors?
The most effective strategy is to simplify the circuit in stages. First, identify the simplest combinations within the larger circuit. Look for small groups of resistors that are clearly in series or parallel. Calculate the equivalent resistance for that small group. Then, redraw the circuit, replacing that group with its single equivalent resistor. Repeat this process, simplifying group by group, until the entire circuit is reduced to a single equivalent resistance.
6. What happens to the total resistance if you add another resistor to a series circuit versus a parallel circuit?
The effect is opposite for the two configurations.
- When you add a resistor to a series circuit, you are increasing the total length of the path the current must travel. This increases the total equivalent resistance.
- When you add a resistor to a parallel circuit, you are providing an additional path for the current to flow. This makes it easier for the overall current to pass through the circuit, thereby decreasing the total equivalent resistance.
7. How can you visually identify if resistors are in series or parallel in a circuit diagram?
You can identify the configuration by tracing the path of the current:
- Resistors are in series if they are connected one after another in a single, unbranched line. The current has only one path to follow through them.
- Resistors are in parallel if their terminals are connected to the same two common points (or junctions). This creates multiple branches, forcing the current to split and flow through each branch simultaneously.
