In a series of LCR circuits ,impedance Z is the same at frequencies f1 & f2. Therefore the resonant frequency of the circuit is_______________.

In a series LCR circuit, when the impedance \( Z \) is the same at two different frequencies \( f_1 \) and \( f_2 \), it indicates a specific relationship between the inductance \( L \), capacitance \( C \), and the frequencies involved. To determine the resonant frequency of the circuit, we need to delve into the characteristics of resonance in LCR circuits.

Understanding Resonance in LCR Circuits

Resonance occurs in an LCR circuit when the inductive reactance \( X_L \) equals the capacitive reactance \( X_C \). At this point, the circuit can oscillate at its natural frequency, which is known as the resonant frequency \( f_0 \). The formula for the resonant frequency is given by:

f₀ = \frac{1}{2\pi\sqrt{LC}}

Impedance and Its Implications

The impedance \( Z \) of a series LCR circuit is defined as:

Z = \sqrt{R^2 + (X_L - X_C)^2}

Where:

• R is the resistance.
• X_L is the inductive reactance, given by \( X_L = 2\pi f L \).
• X_C is the capacitive reactance, given by \( X_C = \frac{1}{2\pi f C} \).

When the impedance is the same at two frequencies \( f_1 \) and \( f_2 \), it suggests that the difference in reactance at these frequencies compensates for any resistance in the circuit. This leads to the conclusion that the resonant frequency \( f_0 \) lies between \( f_1 \) and \( f_2 \).

Finding the Resonant Frequency

To find the resonant frequency, we can use the property of the circuit where the impedance remains constant. If \( Z(f_1) = Z(f_2) \), then:

X_L(f_1) - X_C(f_1) = X_L(f_2) - X_C(f_2)

This condition implies that the resonant frequency \( f_0 \) is the frequency at which the circuit can oscillate freely without any net reactance. Since the impedance is the same at both frequencies, the resonant frequency can be expressed as:

f₀ = \sqrt{f₁ \cdot f₂}

This formula indicates that the resonant frequency is the geometric mean of the two frequencies where the impedance is equal. Thus, if you know \( f_1 \) and \( f_2 \), you can easily calculate \( f_0 \).

Example Calculation

Let’s say \( f_1 = 100 \) Hz and \( f_2 = 400 \) Hz. The resonant frequency \( f_0 \) would be:

f₀ = \sqrt{100 \cdot 400} = \sqrt{40000} = 200 \text{ Hz}

This example illustrates how the resonant frequency can be determined when the impedance is constant at two different frequencies.

In summary, when the impedance \( Z \) is the same at frequencies \( f_1 \) and \( f_2 \) in a series LCR circuit, the resonant frequency \( f_0 \) can be found using the geometric mean of those two frequencies. This relationship is fundamental in understanding how LCR circuits behave at different frequencies and is crucial for applications in electronics and signal processing.