A sine-wave generator with 15.0-V peak output is applied across cH a series R L C circuit. At th

A sine-wave generator with 15.0-V peak output is applied...

Key Concepts

Frequency Relationships in RLC Circuits

The frequency at which a series RLC circuit resonates is determined by its inductance and capacitance, typically expressed by the formula f? = 1/(2??(LC)). This frequency relationship is crucial for designing circuits with specific responses and is used to calculate unknown values when the resonant frequency is known.

Reactance (Inductive and Capacitive)

Inductive reactance (proportional to frequency and inductance) and capacitive reactance (inversely proportional to frequency and capacitance) determine how the inductor and capacitor respond to alternating current. The interplay between these two types of reactance, especially their cancellation at resonance, is essential for understanding the circuit behavior at different frequencies.

Impedance in RLC Circuits

The impedance of a series RLC circuit is a combination of resistance and the net reactance from the inductor and capacitor. Off resonance, the impedance is given by the square root of the sum of the squared resistance and the squared difference between inductive and capacitive reactances. This relationship is fundamental in determining element values when the current response changes with frequency.

Resonance in RLC Circuits

Resonance occurs when the inductive reactance and capacitive reactance in a series RLC circuit cancel each other out, resulting in the circuit’s impedance being purely resistive. At this frequency, the current reaches its maximum for a given applied voltage, and this property is used to directly measure or calculate the resistance in the circuit.

Series RLC Circuit Analysis

A series RLC circuit is a fundamental electrical network consisting of a resistor, an inductor, and a capacitor connected in series. The analysis typically involves determining the voltage, current, and impedance relationships among the circuit elements, which can be used to deduce unknown circuit parameters when certain measurements are known.

Transcript

00:01 Hi, here in this given problem peak voltage supplied by the generator with the help of generator that is given as 15 .0 volt and at resonance, at resonance as we know x l becomes equal to x c, inductive reactance is equal to capacitive reactance.

00:24 So, impedance of the circuit is given by its resistance only and resonating angular frequency it is given to be 775 hertz and we know an expression for it that is 1 by 2 pi multiplied by square root of the product of inductance with the capacitance.

00:48 So, first of all and yeah one more thing i naught peak current at resonance that is given to be 112 milliampere.

00:58 So, using ohms law r will be given by v naught by i naught means this is 15 .0 divided by 112 multiplied by 10 to the power minus 3 and the resistance of this circuit is calculated to be equal to 133 .9 ohm or rounding it off we can say this is 134 ohm and it becomes one of the answer for this given problem.

01:28 Now, using the value of this f naught 775 is equal to 1 by 2 pi means 2 times of 3 .14 multiplied by square root of l c.

01:41 So, we can say l c will be given by 1 by 2 times of 3 .14 and multiplied by 775 and it will be having a square of it.

01:57 So, this l c product of inductance with the capacitance it comes out to be equal to 4 .22 multiplied by 10 to the power minus 8.

02:11 So, we find value of this inductance in terms of capacitance it will be given by 4 .22 multiplied by 10 to the power minus 8 and divided by c.

02:22 Now, frequency is changed to 1 .22 kilo hertz means this is 1220 hertz at this frequency peak value of current i naught dash becomes 44 .8 milliampere.

02:44 So, now using ohms law again z means impedance of this circuit that will be given by v naught by i naught dash v naught is same 15 .0 i naught dash that is that is given to be 44 .8 milliampere or multiplied by 10 to the power minus 3.

03:06 So, impedance of this circuit calculated to be equal to 334 .8 ohm.

03:13 So, now using the vector sum of reactance and resistance z is given as x l minus x c square plus r square.

03:28 So, we can say x l minus x c that will be given as square root of z square minus r square plugging in the known values for z we know that is 334 .8 square minus r already we have found it to be equal to 133 .9 without rounding it off we are taking exact value...

Please give Ace some feedback