Tuned Amplifier Analysis: Bandwidth and 3 dB Frequencies

A tuned amplifier is designed to amplify a specific range of frequencies around a central frequency, known as its resonant or peak output frequency. The performance of a tuned amplifier is often characterized by its quality factor (Q) and its bandwidth (BW), which dictates the range of frequencies it effectively amplifies. The 3 dB frequencies (also known as half-power frequencies) mark the points where the amplifier's output power drops to half of its maximum value (or voltage drops to \(1/\sqrt{2}\) of its maximum).

Understanding Key Concepts

• Peak Output Frequency (\(f_r\)): This is the center frequency where the amplifier provides maximum gain. In this problem, it is given as 6 MHz.
• Quality Factor (Q): A dimensionless parameter that describes how underdamped an oscillator or resonator is. For a tuned circuit, it relates the resonant frequency to the bandwidth. A higher Q indicates a narrower bandwidth and better selectivity.
• Bandwidth (BW): The range of frequencies over which the amplifier's gain is within 3 dB of its maximum gain. It is the difference between the upper and lower 3 dB frequencies.
• 3 dB Frequencies (\(f_1\) and \(f_2\)): These are the lower (\(f_1\)) and upper (\(f_2\)) frequencies at which the amplifier's output power is half of its peak power.

Formulas for Tuned Amplifier Calculations

To calculate the bandwidth and 3 dB frequencies, we use the following standard formulas for tuned amplifiers:

• Bandwidth (BW): \[ BW = \frac{f_r}{Q} \] where \(f_r\) is the peak output frequency and Q is the quality factor.
• Lower 3 dB Frequency (\(f_1\)): \[ f_1 = f_r - \frac{BW}{2} \]
• Upper 3 dB Frequency (\(f_2\)): \[ f_2 = f_r + \frac{BW}{2} \]

Step-by-Step Calculation

Given the following parameters for the tuned amplifier:

• Peak output frequency, \(f_r\) = 6 MHz
• Quality factor, Q = 60

1. Bandwidth Calculation

First, we calculate the bandwidth (BW) using the given peak output frequency and quality factor:

\[ BW = \frac{f_r}{Q} = \frac{6 \text{ MHz}}{60} \]

\[ BW = 0.1 \text{ MHz} \]

Since 1 MHz = 1000 kHz, 0.1 MHz can be converted to kHz:

\[ BW = 0.1 \times 1000 \text{ kHz} = 100 \text{ kHz} \]

So, the bandwidth of the tuned amplifier is 100 kHz.

2. 3 dB Frequencies Calculation

Next, we calculate the lower (\(f_1\)) and upper (\(f_2\)) 3 dB frequencies. We first need half of the bandwidth:

\[ \frac{BW}{2} = \frac{100 \text{ kHz}}{2} = 50 \text{ kHz} \]

Convert 50 kHz to MHz for consistency with \(f_r\):

\[ 50 \text{ kHz} = 0.05 \text{ MHz} \]

Now, calculate \(f_1\) and \(f_2\):

• Lower 3 dB Frequency (\(f_1\)):

  \[ f_1 = f_r - \frac{BW}{2} = 6 \text{ MHz} - 0.05 \text{ MHz} \]

  \[ f_1 = 5.95 \text{ MHz} \]

• Upper 3 dB Frequency (\(f_2\)):

  \[ f_2 = f_r + \frac{BW}{2} = 6 \text{ MHz} + 0.05 \text{ MHz} \]

  \[ f_2 = 6.05 \text{ MHz} \]

Therefore, the 3 dB frequencies are 5.95 MHz and 6.05 MHz.

Summary of Results

The calculated values are:

• Bandwidth (BW): 100 kHz
• Lower 3 dB frequency (\(f_1\)): 5.95 MHz
• Upper 3 dB frequency (\(f_2\)): 6.05 MHz

These results match one of the given options.