In a series resonant L_C_R circuit, if L is increased by 25% and C is decreased by 20%, then the resonant frequency willa)Increase by 10%b)Decrease by 10%c)Remain unchangedd)Increase by 2.5%Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Answer

XL1=XC1. ~ resonant frequency 1
X(L1+0.25L1)=X(C1-0.2C1) . ~ resonant frequency 2
f1=1/2π√(L1C1)
f2=1/2π√(1.25L1×0.8C1)=1/2π√((1)L1C1)
therefore, f1=f2
remain unchanged

Answer

Resonance is a phenomenon that occurs in LCR circuits when the inductive reactance (XL) and capacitive reactance (XC) are equal. At resonance, the circuit exhibits maximum impedance and maximum current. The resonant frequency (fr) is the frequency at which this occurs.

Let's analyze the effect of increasing the inductance (L) by 25% and decreasing the capacitance (C) by 20% on the resonant frequency.

Effect of increasing the inductance (L) by 25%:
- Inductive reactance (XL) is directly proportional to the inductance (L). Therefore, if L is increased by 25%, XL will also increase by 25%.
- As XL increases, the impedance of the circuit also increases.
- Since resonance occurs when XL = XC, increasing XL will shift the resonant frequency to a higher value.

Effect of decreasing the capacitance (C) by 20%:
- Capacitive reactance (XC) is inversely proportional to the capacitance (C). Therefore, if C is decreased by 20%, XC will also decrease by 20%.
- As XC decreases, the impedance of the circuit also decreases.
- Since resonance occurs when XL = XC, decreasing XC will shift the resonant frequency to a lower value.

Combining the effects of increasing L and decreasing C:
- The increase in XL and the decrease in XC will have opposite effects on the resonant frequency.
- However, the magnitudes of the changes are different (25% increase vs. 20% decrease).
- The magnitude of the change in L is greater than the magnitude of the change in C.
- Therefore, the net effect of increasing L and decreasing C is to shift the resonant frequency to a lower value.

Therefore, the resonant frequency will remain unchanged when L is increased by 25% and C is decreased by 20%. Option 'C' is the correct answer.

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