Question:

A circuit contains a resistor, an inductor, and a capacitor connected in series. The resistance of the resistor is R ohms, the inductance of the inductor is L henries, and the capacitance of the capacitor is C farads. The circuit is connected to a sinusoidal voltage source of magnitude V0 volts and frequency f hertz.

a) Calculate the total impedance (Z) of the circuit in terms of R, L, C, f, and V0.

b) Consider a scenario where R = 10 Ω, L = 0.5 H, C = 0.01 F, f = 100 Hz, and V0 = 50 V. Calculate the total impedance (Z) in this scenario.

c) Based on the calculated value of Z, describe the behavior of the circuit. Is it primarily resistive, inductive, or capacitive? Explain your reasoning.

Answer:

a) The total impedance (Z) of the circuit is given by the formula:

Z = √((R^2) + ((ωL -1/ωC)^2))

where ω = 2πf (angular frequency in radians per second).

b) Given: R = 10 Ω, L = 0.5 H, C = 0.01 F, f = 100 Hz, and V0 = 50 V.

First, calculate ω: ω = 2πf = 2π(100 Hz) = 200π radians/second

Now, substitute the values into the impedance formula:

Z = √((10^2) + (((200π)(0.5) -1/(200π)(0.01))^2))

Simplifying further:

Z = √(100 + (100π - 500π)^2)

Z = √(100 + (-400π)^2)

Z = √(100 + 160,000π^2)

Using a calculator, we get:

Z ≈ √(100 + 500,796.3274)

Z ≈ √500,896.3274

Z ≈ 707.07 Ω (rounded to two decimal places)

c) The calculated value of Z is approximately 707.07 Ω.

The behavior of the circuit can be determined based on the relative magnitudes of the resistive, inductive, and capacitive components of the impedance.

In this case, the resistance R is much smaller than the inductive reactance ωL and the capacitive reactance 1/ωC. Therefore, the circuit is primarily inductive and capacitive.

This is because the inductive reactance increases with increasing frequency, while the capacitive reactance decreases with increasing frequency. The combined effect results in an impedance that is dominated by the inductive and capacitive components rather than the resistive component.

Hence, the circuit exhibits behavior characteristic of an L-C circuit, where energy is continuously exchanged between the inductor and the capacitor.