Necessary condition for resonance in the circuit is, there should be two energy storing elements in the circuit. An electrical circuit is said to undergo resonance when the net current is in phase with the applied voltage.
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A circuit at resonance exhibits certain characteristics, such as maximum current or minimum current. The formula to calculate the resonant frequency is` f0 = 1/(2pisqrt(LC))`resonance practice problemsProblem: - A RLC tank circuit is composed of components having values as R = 0.2 ohm; L = 100mH and C="50" μF. Determine the resonance frequency and the corresponding input current at 24 V.
Solution:- Resonant frequency = f0 = 1/(2πsqrt(LC))= 1/(2π (100*10-3*50*10-6 )^1/2)f0 = 71.21 Hz (Answer)At resonance I = V/R= 24/0.2 = 120 Ampere. (Answer)Problem: - A coil has a resistance of 20 ohms and inductance of 80 mH and is connected in series with a 100pF capacitor. Determine at resonance, the circuit impedance and also find the resonant frequency. If the supply to the circuit is a 50 V source having an internal impedance of 2 ohms, find also the circuit current and the voltage across the capacitor.
Solution:- Resonant frequency = f0 = 1/(2π`sqrt(LC)`)= 1/(2π(80*10-3*100*10-12)^1/2)= 56.298 kHzZ = R = 20Ω [at resonance, Z = R ]I0 = V/R = 50/20 = 2.5 A [I0 = current at resonance ]Vc = IXc = 2.5 * 1/2πf0C [Vc = voltage across the capacitor ]= 2.5/( 2π* 56.298*103*100*10-12)= 70.71 KV (Answer)Problem: - A coil having an inductance and resistance of 50 mH and 100 ohms is connected in series with a capacitor and a 100 V, 1 KHz source. Obtain the value of capacitance that will cause a resonance on the circuit.
Find the circuit current at resonance frequency.Solution:- f0 = 1/(2π`sqrt(LC)`)1000 = 1/ (2π (50*10-3*C )^1/2C = 0.5 μF (Answer)I0 = V/Z = 100/100 = 1 ohm (Answer)[Z is the circuit impedance at resonance and is equal to value of resistance.]Some more resonance practice problemsProblem: - A series connected RLC circuit has R="15," L = 40mH, and C="40µF." Determine the resonant frequency and under resonant condition, calculate the current and power in the circuit. (Answer:- 125.82Hz, I="5A," power="375" Watts)Problem: - A series LCR circuit has inductance of 10 mH and resistance of 2 ohms.
What is the value of capacitance that will produce resonance? Also find the current at resonance frequency and the maximum instantaneous energy stored in the inductance at resonance. Assume the supply as 230 V, 10000 Hz. (Answer:- 0.025μF, 115 A, 132.25 Joules)Problem: - What is the resonance frequency of a series RLC circuit where R="10" ohms, L="25" mH and C="100uF." Evaluate the Q-factor also. (Answer:- 100.71 Hz, Q = 1.58)Problem: - A 5µF capacitor is connected is series with a coil having inductance of 50mH. Determine the frequency of resonance, the resistance of the coil if a 50 V source operating at resonance frequency causes a circuit current of 10 mA.
What is the Q factor of the coil? (Answer:- 318.47 Hz, Q = 0.02)Problem: - A 50 µF capacitor, when connected in series with a coil having 40 ohms resistance, resonates at 1000 Hz. Find inductance of the coil. Also obtain the circuit current if the applied voltage is 100 V. Also calculate the voltage across the capacitor and coil at resonance. (Answer:- L = 0.5 mH, I = 2.5A, Vc = 7.96V, voltage across coil = 100.31 V)