Question

The response of circuits containing resistors, inductors, and capacitors depends upon the relative values of the resistors and the way they are connected. An important intermediate quantity used in describing the response of such circuits is s. Depending on the values of R, L, and C, the values of s will be either both real values, a pair of complex values, or a duplicated value. The equation that identifies the response of a series circuit shown above is s = -R/L plusminus squareroot (R/2L)^2 - 1/LC (a) Determine the values of s for a resistance of 800 Ohm. (b) Create a vector of values of R between 100 to 1000 Ohm with a step size of 5. Evaluate s at all values of R. (c) Plot value of real and imaginary parts of s vs. R. (d) What is the minimum value of R that yields pure real value of s. (e) The resonant frequency of a series RLC circuit is given by: f_r = 1/2 pi squareroot LC Compute the resonant frequency of the above circuit.

Answer 1

An RLC series circuit comprises a resistor, inductor, and capacitor, with the resonant frequency occurring when inductive and capacitive reactances cancel each other out, minimizing impedance and maximizing current flow. The resonant frequency formula is vital for applications like radio tuners.

Step-by-step explanation:

An RLC series circuit includes a resistor (R), inductor (L), and capacitor (C) connected in series with one another. The response of such a circuit to an alternating current (AC) source is complex, as the resistors, inductors, and capacitors behave differently. Resistors impede current by converting electrical energy into heat, while inductors and capacitors store energy in magnetic fields and electric fields, respectively.

In an AC circuit, the impedance caused by inductors (XL) and capacitors (XC) is frequency-dependent and may either bolster or diminish each other's effects depending on the circuit's operating frequency.

The resonant frequency (fr) is significant as at this frequency the inductive and capacitive reactances are equal in magnitude but opposite in phase, which causes them to cancel out. This results in the total impedance of the circuit being equal to just the resistive component, allowing for maximum current to flow through the circuit.

This phenomenon has practical applications, such as in tuning radio frequencies where the circuit is set to resonate at the desired frequency. The resonant frequency of a series RLC circuit is given by fr = 1 / (2π√LC).