The Frequency of an LC Circuit and its Error Analysis

The frequency of an LC circuit is a critical parameter in many applications, including resonant communication and filtering circuits. The formula for the frequency, f, is given by:

f frac{1}{2pisqrt{LC}}

where L is the inductance and C is the capacitance. This equation is derived from the fundamental relationship between the inductor and the capacitor. To understand the error propagation in frequency due to errors in inductance and capacitance, let's delve into the error analysis.

Direct Error Propagation Analysis

Given the frequency formula, we can start by taking the natural logarithm on both sides:

ln f ln left(frac{1}{2pisqrt{LC}}right)

This simplifies to:

ln f ln left(frac{1}{2pi}right) - frac{1}{2} left(ln L ln Cright)

For the error propagation, we use the total differential:

frac{delta f}{f} -frac{1}{2} left(frac{delta L}{L} frac{delta C}{C}right)

If 1% L and 2% C, then the maximum permissible error becomes:

frac{delta f}{f} frac{1}{2} (1% 2%) 1.5%

This is the direct error propagation analysis assuming the worst-case conditions where both L and C could be at their extreme values.

Probabilistic Error Analysis

Without knowing the actual values, we need to consider the probabilistic error, which depends on the specifications of the components. The exact error cannot be determined without specific values, but we can estimate it by considering the worst-case scenarios.

Worst Case Conditions

Considering L and C are high by their respective percentages, the formula becomes:

frac{1}{1.01sqrt{1.02}} approx 0.97068 or about a 3% decrease in frequency.

Conversely, when L and C are low by their respective percentages, the formula becomes:

frac{1}{0.99sqrt{0.98}} approx 1.030715 or about a 3% increase in frequency.

Solving for Probabilistic Error Using RSS

A more accurate approach is to use the Root-Sum-Square (RSS) method to combine the tolerances. Here's the RSS of the relative errors:

sqrt{(1.01^2 1.02^2)/4} approx 0.2236

This value represents about a 2% error in either direction, but it’s often considered too conservative.

To provide a more realistic estimate, we can use the concept of standard error for a Gaussian distribution. The standard error is given by dividing the tolerance by sqrt{3}:

frac{0.2236}{sqrt{3}} approx 0.129

Combining this with RSS, we get:

sqrt{(0.129^2 0.129^2)/2} approx 0.0903

This represents about a 1% error, but in reality, you might want to be a bit more aggressive with this error margin. If you were building 100 of these circuits, about 62 or fewer would be expected to be within this error range.

For a more conservative approach, you could multiply the value by a constant, say 3, to get 0.27, where over 99 circuits would be expected to be within this value.

Conclusion

Understanding the error propagation in an LC circuit is crucial for accurate design and performance prediction. The worst-case analysis and the probabilistic approach provide a more comprehensive view of the error, allowing for better design and reliability. The concept of standard error, though not widely understood, can be a useful tool in error analysis.