Wien Bridge Oscillator Resonant Frequency Calculation

The Wien Bridge oscillator is a type of electronic oscillator that generates sine waves. It is commonly used in audio frequency generators and other applications where a low-distortion sine wave is required. The key components that determine its operating frequency are the resistors (R) and capacitors (C) in its frequency-determining network.

Wien Bridge Oscillator Frequency Formula

For a standard Wien Bridge oscillator, the resonant frequency (\(f_r\)) at which it oscillates is determined by the values of the resistance (R) and capacitance (C) in its frequency-selective feedback network. The formula for the resonant frequency is given by:

\[f_r = \frac{1}{2\pi RC}\]

Where:

• \(f_r\) is the resonant frequency in Hertz (Hz)
• \(\pi\) (pi) is approximately 3.14159
• \(R\) is the resistance in Ohms (\(\Omega\))
• \(C\) is the capacitance in Farads (F)

Given Values for Resonant Frequency Calculation

From the question, we are provided with the following values for the Wien Bridge oscillator:

• Resistance (R) = 51 k\(\Omega\)
• Capacitance (C) = 0.001 \(\mu\)F

Before plugging these values into the formula, it's crucial to convert them into their base SI units (Ohms and Farads):

• R = 51 k\(\Omega\) = \(51 \times 10^3\) \(\Omega\)
• C = 0.001 \(\mu\)F = \(0.001 \times 10^{-6}\) F = \(1 \times 10^{-9}\) F

Step-by-Step Resonant Frequency Calculation

Now, let's substitute the converted R and C values into the resonant frequency formula:

\[f_r = \frac{1}{2\pi RC}\]

\[f_r = \frac{1}{2 \times 3.14159 \times (51 \times 10^3 \, \Omega) \times (1 \times 10^{-9} \, F)}\]

First, calculate the product of R and C:

\[RC = (51 \times 10^3) \times (1 \times 10^{-9})\]

\[RC = 51 \times 10^{(3-9)}\]

\[RC = 51 \times 10^{-6} \, \text{seconds}\]

Next, substitute this value back into the frequency formula:

\[f_r = \frac{1}{2 \times 3.14159 \times (51 \times 10^{-6})}\]

\[f_r = \frac{1}{6.28318 \times 51 \times 10^{-6}}\]

\[f_r = \frac{1}{320.44218 \times 10^{-6}}\]

\[f_r = \frac{1}{0.00032044218}\]

Finally, calculate the resonant frequency:

\[f_r \approx 3120.73 \, \text{Hz}\]

Final Resonant Frequency Result

The calculated resonant frequency for the given Wien Bridge oscillator with R = 51 k\(\Omega\) and C = 0.001 \(\mu\)F is approximately 3120.7 Hz.