# Quality factor and freq bandwidth in a series RLC circuit

• Engineering
In summary: In the first part of the experiment with a resistance of 470 Ω, Q is 2.88, because there is twice as much energy stored as lost in the system. When the resistance is increased to 1 kΩ, the Q factor drops to 1.58, because there is now only 1.57 times as much energy stored as lost in the system.

## Homework Statement

How and why does quality factor change in a series RLC circuit when the resistance in the circuit increases? I've made a small experiment circuit which looks like one in the picture:

The current frequency is 1 kHz and the voltmeter should show 2V throughout the whole experiment. Inductance L = 100 mH and capacitance C = 47nF. Now, in the first part of the experiment, I use R = 470 Ω. I get that the max current at the resonant frequency is Imax = 4.13 mA and the resonant frequency is fr = 2436 Hz. Then I calculate boundary frequencies ( left and right of the resonant frequency, where the current Ig = Imax/sqrt(2)) and boundary frequencies are: fg1 = 2076 Hz and fg2 = 2920 Hz, which makes the frequency bandwidth Δf = fg2 - fg1 = 844 Hz.
Then, the quality factor is: Q1 = 2.88

Now, for the second part of the experiment, I use R = 1 kΩ. I get that Imax = 1.97 mA at the resonant frequency fr = 2420 Hz (it's probably the same as the resonant frequency in the first case, but the instruments aren't as precise). Boundary frequencies are: fg1 = 1706 Hz and fg2 = 3234 Hz, which makes the frequency bandwidth Δf = 1528 Hz.
Then, the quality factor is: Q2 = 1.58

My question is, why is the quality factor dependent upon the resistance? Why does the resistance affect the frequency bandwidth and quality factor (for lower R, quality factor is higher and vice versa)?

## Homework Equations

Just theoretical explanation.

## The Attempt at a Solution

The experiment explained above.

The only mechansm for energy loss in the system is the resistance, since pure reactances are lossless. So with no resistance, the system would be 100% efficient and all energy added to it from the source would be stored and returned.

Q factor is directly proportional to the ratio: (energy stored)/( energy lost) per cycle. So if the losses approach zero, then Q approaches infinity.

## Homework Statement

How and why does quality factor change in a series RLC circuit when the resistance in the circuit increases? I've made a small experiment circuit which looks like one in the picture:

The current frequency is 1 kHz and the voltmeter should show 2V throughout the whole experiment. Inductance L = 100 mH and capacitance C = 47nF. Now, in the first part of the experiment, I use R = 470 Ω. I get that the max current at the resonant frequency is Imax = 4.13 mA and the resonant frequency is fr = 2436 Hz. Then I calculate boundary frequencies ( left and right of the resonant frequency, where the current Ig = Imax/sqrt(2)) and boundary frequencies are: fg1 = 2076 Hz and fg2 = 2920 Hz, which makes the frequency bandwidth Δf = fg2 - fg1 = 844 Hz.
Then, the quality factor is: Q1 = 2.88

Now, for the second part of the experiment, I use R = 1 kΩ. I get that Imax = 1.97 mA at the resonant frequency fr = 2420 Hz (it's probably the same as the resonant frequency in the first case, but the instruments aren't as precise). Boundary frequencies are: fg1 = 1706 Hz and fg2 = 3234 Hz, which makes the frequency bandwidth Δf = 1528 Hz.
Then, the quality factor is: Q2 = 1.58

My question is, why is the quality factor dependent upon the resistance? Why does the resistance affect the frequency bandwidth and quality factor (for lower R, quality factor is higher and vice versa)?

## Homework Equations

Just theoretical explanation.

## The Attempt at a Solution

The experiment explained above.
Q = wL/R or 1/wRC.

## 1. What is the quality factor in a series RLC circuit?

The quality factor (Q) in a series RLC circuit is a measure of the energy efficiency of the circuit. It is defined as the ratio of the reactance of the circuit to its resistance, and is represented by the formula Q = XL / R. A higher quality factor indicates a more efficient circuit, as it means that the reactance is significantly larger than the resistance.

## 2. How does the quality factor affect the bandwidth of a series RLC circuit?

The quality factor is directly related to the bandwidth of a series RLC circuit. A higher quality factor leads to a narrower bandwidth, meaning that the circuit is able to filter out a narrower range of frequencies. On the other hand, a lower quality factor results in a wider bandwidth, allowing a greater range of frequencies to pass through the circuit.

## 3. What is the significance of the resonant frequency in a series RLC circuit?

The resonant frequency is the frequency at which the capacitive and inductive reactances in a series RLC circuit cancel each other out, resulting in a purely resistive circuit. At this frequency, the circuit exhibits the highest quality factor and the lowest impedance. It is an important parameter to consider when designing a series RLC circuit for a specific application.

## 4. How does the addition of resistance affect the quality factor of a series RLC circuit?

In a series RLC circuit, the addition of resistance decreases the quality factor. This is because the resistance introduces losses in the circuit, reducing its energy efficiency. As a result, the reactance becomes smaller relative to the resistance, resulting in a lower quality factor.

## 5. Can the quality factor and bandwidth of a series RLC circuit be adjusted?

Yes, the quality factor and bandwidth of a series RLC circuit can be adjusted by changing the values of the circuit's components. For example, the quality factor can be increased by increasing the inductance or capacitance in the circuit, or by reducing the resistance. Similarly, the bandwidth can be narrowed by increasing the quality factor, or widened by decreasing it.

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