When Unequal Resistors Are Connected In Parallel In A Circuit

When unequal resistors are connected in parallel in a circuit, a fascinating interplay of current and resistance emerges, significantly impacting the circuit's overall behavior. Understanding this dynamic is crucial for anyone delving into electrical engineering or electronics, as it forms the bedrock for designing and analyzing complex circuits.

Parallel Resistors: A Primer

Before diving into the intricacies of unequal resistors in parallel, let's establish a fundamental understanding of parallel circuits and resistors.

Parallel Circuit: A parallel circuit is characterized by having multiple paths for current to flow from the voltage source. Components in a parallel circuit, such as resistors, are connected across each other, meaning they share the same two connection points.
Resistor: A resistor is a passive electrical component that impedes the flow of current. Its primary characteristic is its resistance, measured in ohms (Ω), which dictates the amount of voltage required to drive a certain current through it.

When resistors are connected in parallel, the voltage across each resistor is the same, equal to the voltage of the source. However, the current flowing through each resistor is inversely proportional to its resistance. This principle is key to understanding the behavior of unequal resistors in parallel.

The Dance of Unequal Resistors

Now, let's consider the scenario where we have two or more resistors with different resistance values connected in parallel. What happens to the current flow, the overall resistance of the circuit, and the power dissipated by each resistor?

Current Distribution

The most significant effect of connecting unequal resistors in parallel is the unequal distribution of current. The resistor with the lowest resistance will allow the highest current to flow through it, while the resistor with the highest resistance will allow the lowest current to flow through it. This is a direct consequence of Ohm's Law (V = IR), which states that the current (I) is directly proportional to the voltage (V) and inversely proportional to the resistance (R). Since the voltage is the same across all parallel resistors, the current will naturally favor the path of least resistance.

Imagine two water pipes connected in parallel, one wide and one narrow. The wide pipe (low resistance) will naturally carry more water than the narrow pipe (high resistance). Similarly, in an electrical circuit, the resistor with lower resistance acts as a wider path for electrons to flow.

Equivalent Resistance

The equivalent resistance of parallel resistors is always less than the smallest individual resistance. This is because the parallel connection provides more paths for current to flow, effectively reducing the overall opposition to current.

The formula for calculating the equivalent resistance (Req) of two resistors (R1 and R2) in parallel is:

1/Req = 1/R1 + 1/R2

Which can be simplified to:

Req = (R1 * R2) / (R1 + R2)

For more than two resistors in parallel, the general formula is:

1/Req = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn

Where R1, R2, R3... Rn are the individual resistances.

Example:

Let's say we have two resistors, R1 = 10 ohms and R2 = 20 ohms, connected in parallel. The equivalent resistance is:

Req = (10 * 20) / (10 + 20) = 200 / 30 = 6.67 ohms

Notice that the equivalent resistance (6.67 ohms) is less than both 10 ohms and 20 ohms.

Power Dissipation

The power dissipated by each resistor in a parallel circuit is also unequal, and directly related to the current flowing through it. Power (P) is calculated using the formula:

P = I^2 * R

Since the current (I) is different for each resistor, the power dissipated by each will also be different. The resistor with the lower resistance will dissipate more power because it carries a larger current, even though its resistance is lower. Conversely, the resistor with the higher resistance will dissipate less power.

Example (Continuing from above):

Assume the voltage source is 12V.

Current through R1 (10 ohms): I1 = V/R1 = 12V / 10 ohms = 1.2 A
Current through R2 (20 ohms): I2 = V/R2 = 12V / 20 ohms = 0.6 A

Power dissipated by R1: P1 = I1^2 * R1 = (1.2 A)^2 * 10 ohms = 14.4 W

Power dissipated by R2: P2 = I2^2 * R2 = (0.6 A)^2 * 20 ohms = 7.2 W

As you can see, the 10-ohm resistor dissipates twice as much power as the 20-ohm resistor because it carries twice the current.

Practical Implications and Applications

Understanding the behavior of unequal resistors in parallel has numerous practical implications in circuit design and analysis.

Current Limiting: Parallel resistors can be used to limit the current flowing through a specific part of a circuit. By placing a small resistor in parallel with a larger resistor, you can divert a significant portion of the current through the smaller resistor, protecting the larger resistor from excessive current and potential damage.
Voltage Division (Indirectly): While parallel resistors don't directly perform voltage division like series resistors, they influence the overall current and resistance of the circuit. This, in turn, affects the voltage distribution in other parts of the circuit, especially when combined with series resistors.
Load Balancing: In power distribution systems, parallel connections of unequal loads (which can be modeled as resistors) are common. Understanding how current distributes among these loads is crucial for ensuring efficient and safe operation of the system.
Shunt Resistors: Shunt resistors, which are low-value resistors placed in parallel with a meter, are used to extend the current measurement range of the meter. The shunt resistor diverts most of the current, allowing the meter to measure a small, proportional fraction of the total current.
LED Circuits: Parallel resistors are sometimes used in LED circuits to control the current flowing through each LED. However, it's generally recommended to use a series resistor for each LED to ensure more consistent brightness and prevent thermal runaway.
Audio Amplifiers: In audio amplifiers, parallel resistors are used in various stages for biasing transistors, setting gain, and controlling impedance. The specific values of these resistors and their arrangement are critical for achieving the desired amplifier performance.

Mathematical Analysis and Problem Solving

To effectively analyze circuits with unequal resistors in parallel, you need to be comfortable with applying Ohm's Law, Kirchhoff's Current Law (KCL), and the formulas for calculating equivalent resistance and power. Here are some common types of problems you might encounter:

Finding Equivalent Resistance: Given a set of parallel resistors, calculate the overall equivalent resistance of the combination.
Calculating Current Distribution: Given the values of the resistors and the voltage source, determine the current flowing through each resistor.
Determining Power Dissipation: Calculate the power dissipated by each resistor, given the current and resistance values.
Designing a Circuit for Specific Current or Resistance: Choose appropriate resistor values to achieve a desired equivalent resistance or current distribution in a parallel circuit.

Example Problem:

Two resistors, R1 = 50 ohms and R2 = 75 ohms, are connected in parallel to a 24V power supply.

Find the equivalent resistance (Req):

Req = (R1 * R2) / (R1 + R2) = (50 * 75) / (50 + 75) = 3750 / 125 = 30 ohms

Find the total current (Itotal) flowing from the power supply:

Itotal = V / Req = 24V / 30 ohms = 0.8 A

Find the current through each resistor (I1 and I2):

I1 = V / R1 = 24V / 50 ohms = 0.48 A
I2 = V / R2 = 24V / 75 ohms = 0.32 A

(Notice that I1 + I2 = 0.48 A + 0.32 A = 0.8 A = Itotal, which confirms Kirchhoff's Current Law)

Find the power dissipated by each resistor (P1 and P2):

P1 = I1^2 * R1 = (0.48 A)^2 * 50 ohms = 11.52 W
P2 = I2^2 * R2 = (0.32 A)^2 * 75 ohms = 7.68 W

Common Mistakes and Pitfalls

When working with parallel resistors, it's easy to make common mistakes. Here are some pitfalls to avoid:

Incorrectly Calculating Equivalent Resistance: Remember to use the reciprocal formula (1/Req = 1/R1 + 1/R2 + ...) for more than two resistors. Many people forget to take the reciprocal of the sum at the end.
Confusing Parallel and Series Circuits: It's crucial to distinguish between parallel and series circuits. In parallel circuits, the voltage is the same across all components, while in series circuits, the current is the same.
Assuming Equal Current Distribution: Don't assume that the current is equally divided among parallel resistors unless they have the same resistance.
Ignoring Power Ratings: Always check the power ratings of the resistors to ensure that they can handle the power they will dissipate in the circuit. Exceeding the power rating can lead to resistor failure.
Forgetting Kirchhoff's Current Law: KCL is a fundamental law that states that the total current entering a junction (node) must equal the total current leaving the junction. This is essential for verifying your calculations and ensuring that your circuit analysis is correct.
Using Series Formulas for Parallel Circuits (or vice versa): This is a very common mistake. Make sure you are using the correct formulas for the type of circuit you are analyzing. The equivalent resistance calculation is different in series vs parallel.
Not Drawing a Clear Circuit Diagram: Before attempting to solve a circuit problem, draw a clear and labeled circuit diagram. This will help you visualize the circuit and avoid mistakes.

Beyond Ideal Resistors: Real-World Considerations

The above discussion assumes ideal resistors, meaning their resistance remains constant regardless of temperature, voltage, or current. In reality, resistors are not ideal. Their resistance can vary with temperature (temperature coefficient), voltage (voltage coefficient), and frequency.

Tolerance: Resistors have a tolerance rating, which indicates the allowable deviation from the nominal resistance value. A 5% tolerance resistor, for example, might have a resistance value that is 5% higher or lower than its stated value. This tolerance needs to be considered in critical applications.
Temperature Coefficient: The temperature coefficient indicates how much the resistance changes with temperature. Resistors with low temperature coefficients are preferred in applications where stability is important.
Power Rating: As mentioned earlier, the power rating of a resistor is the maximum power it can dissipate without being damaged. Always choose a resistor with a power rating that is significantly higher than the expected power dissipation.
Frequency Effects: At high frequencies, resistors can exhibit inductive and capacitive effects, which can alter their impedance. These effects are typically negligible at low frequencies.

These real-world considerations are especially important in precision circuits, high-power applications, and high-frequency circuits.

Conclusion

Understanding the behavior of unequal resistors connected in parallel is fundamental to circuit analysis and design. The unequal distribution of current, the reduction in equivalent resistance, and the varying power dissipation all play crucial roles in determining the overall behavior of the circuit. By mastering the concepts and techniques discussed in this article, you'll be well-equipped to tackle a wide range of electrical and electronic challenges. Remember to pay attention to real-world considerations such as resistor tolerance and power rating to ensure reliable and safe circuit operation. Practice applying the formulas and principles discussed in this article through various example problems to solidify your understanding. With consistent effort and a keen eye for detail, you'll become proficient in analyzing and designing circuits with parallel resistors.