Question

What is the Q (Quality factor) of a series circuit that resonates at 10 kHz, has equal reactance of 5 kilo-ohms each, and a resistor value of 50 ohms?

Answer 1

The **Quality factor (Q)** of a series resonant circuit is a dimensionless parameter that describes how "sharp" or selective the resonance is in a system. It provides an indication of the efficiency of energy storage in the reactive components (inductors and capacitors) relative to energy dissipation in the resistive components. In a series resonant circuit, the quality factor gives an insight into how quickly the resonance occurs and how much energy is lost as heat.

### Understanding a Series Resonant Circuit

A typical series resonant circuit consists of three components:
- **Inductor (L)**: Stores energy in its magnetic field.
- **Capacitor (C)**: Stores energy in its electric field.
- **Resistor (R)**: Dissipates energy in the form of heat.

At resonance, the inductive reactance (\(X_L = 2\pi f L\)) and the capacitive reactance (\(X_C = \frac{1}{2\pi f C}\)) cancel each other out, so the impedance of the circuit is purely resistive and is equal to the resistance \(R\).

The resonance frequency \(f_r\) is given by the formula:

\[
f_r = \frac{1}{2 \pi \sqrt{LC}}
\]

At this frequency, the voltage across the inductor and capacitor is at its maximum, and the current through the circuit is also at its maximum.

### Definition of the Quality Factor (Q)

The quality factor \(Q\) quantifies the sharpness or selectivity of resonance. In a series resonant circuit, it is defined as:

\[
Q = \frac{\text{Energy stored in the circuit}}{\text{Energy dissipated per cycle}}
\]

For a series RLC circuit, the quality factor can also be expressed mathematically as:

\[
Q = \frac{\omega_r L}{R}
\]

where:
- \(\omega_r = 2\pi f_r\) is the angular resonance frequency,
- \(L\) is the inductance,
- \(R\) is the resistance of the circuit.

Alternatively, since \(Q\) relates to the sharpness of the resonance peak, it can also be defined in terms of the bandwidth of the resonance curve:

\[
Q = \frac{f_r}{\Delta f}
\]

where:
- \(f_r\) is the resonance frequency,
- \(\Delta f\) is the bandwidth, which is the frequency range over which the power falls to half its maximum value (also known as the 3 dB bandwidth).

### Interpretation of the Quality Factor

- **High Q factor**: Indicates that the circuit has a narrow bandwidth and thus a sharp resonance. This means the circuit is highly selective and stores energy efficiently compared to the energy lost in each cycle. High-Q circuits are desirable in applications like filters or tuners, where selectivity is important.
  
- **Low Q factor**: Indicates a broader resonance peak and greater energy dissipation per cycle. The circuit is less selective, meaning it allows a wider range of frequencies to pass. This happens when the resistance \(R\) in the circuit is relatively high, leading to greater energy loss as heat.

### Practical Example

Consider a series resonant circuit with the following values:
- Inductance \(L = 100 \, \mu H\),
- Capacitance \(C = 1 \, \mu F\),
- Resistance \(R = 10 \, \Omega\).

1. **Resonant frequency**:
   
\[
f_r = \frac{1}{2\pi \sqrt{LC}} = \frac{1}{2\pi \sqrt{(100 \times 10^{-6})(1 \times 10^{-6})}} = 15.92 \, kHz
\]

2. **Quality factor**:

\[
Q = \frac{\omega_r L}{R} = \frac{2\pi (15.92 \times 10^3) \times 100 \times 10^{-6}}{10} = 10
\]

So, the circuit has a \(Q\)-factor of 10. This indicates a moderate level of selectivity, meaning the resonance is not extremely sharp but not too broad either.

### Summary of Key Points:
- The **Quality factor (Q)** describes the sharpness of resonance in a series resonant circuit.
- A higher \(Q\) means a narrower resonance peak, higher selectivity, and lower energy dissipation.
- A lower \(Q\) means a broader resonance, lower selectivity, and greater energy dissipation.
- \(Q\) is directly proportional to the inductance \(L\) and inversely proportional to the resistance \(R\).
- It is an essential parameter in designing circuits like filters, oscillators, and tuners where controlling the bandwidth is important.

Answer 2

The Quality factor (Q) of a series resonant circuit can be calculated using the formula:

\[
Q = \frac{\text{Resonant frequency} (f_r)}{\text{Bandwidth}}
\]

Alternatively, for a series resonant circuit, \( Q \) can also be determined by:

\[
Q = \frac{X_L}{R}
\]

Where:
- \( X_L \) is the inductive reactance (which is equal to the capacitive reactance \( X_C \) at resonance in a series circuit),
- \( R \) is the resistance in the circuit.

Given:
- Resonant frequency, \( f_r = 10 \text{ kHz} \),
- Reactances \( X_L = X_C = 5 \text{ k}\Omega = 5000 \text{ ohms} \),
- Resistance \( R = 50 \text{ ohms} \).

Now, substitute the known values into the formula:

\[
Q = \frac{X_L}{R} = \frac{5000 \text{ ohms}}{50 \text{ ohms}} = 100
\]

Thus, the Quality factor \( Q \) of the series circuit is:

\[
Q = 100
\]