Understanding Reactance in RLC Series Circuits

In a series RLC circuit, components like resistors (R), inductors (L), and capacitors (C) are connected in series. When an AC voltage is applied, each component offers opposition to the current flow. The opposition offered by inductors and capacitors is called reactance.

Reactances and Frequency

The reactance values depend on the frequency ($\small f$) of the applied AC voltage:

• Inductive Reactance ($\small X_L$): This is the opposition offered by an inductor. It is directly proportional to the frequency and inductance ($\small L$). The formula is $\small X_L = 2\pi f L$. As frequency increases, $\small X_L$ increases. $\small X_L$ is always positive.
• Capacitive Reactance ($\small X_C$): This is the opposition offered by a capacitor. It is inversely proportional to the frequency and capacitance ($\small C$). The formula is $\small X_C = \frac{1}{2\pi f C}$. As frequency increases, $\small X_C$ decreases. $\small X_C$ is always positive.

Net Reactance ($X_{net}$)

In a series RLC circuit, the total or net reactance is the difference between the inductive reactance and the capacitive reactance.

\small X_{net} = X_L - X_C

Substituting the formulas for $\small X_L$ and $\small X_C$:

\small X_{net} = 2\pi f L - \frac{1}{2\pi f C}

How Net Reactance Changes with Variable Frequency

Let's analyze how $\small X_{net}$ behaves as the frequency ($\small f$) varies:

• At very low frequencies (f > 0): $\small X_L$ ($\small 2\pi f L$) is very small (approaching zero) because $\small f$ is small. $\small X_C$ ($\small \frac{1}{2\pi f C}$) is very large (approaching infinity) because $\small f$ is in the denominator. Therefore, $\small X_{net} = X_L - X_C$ will be dominated by the large negative value of $\small -X_C$. So, at very low frequencies, $\small X_{net}$ is largely negative.
• As frequency increases: $\small X_L$ increases linearly, while $\small X_C$ decreases inversely with frequency. There will be a specific frequency where $\small X_L$ becomes equal to $\small X_C$. This frequency is called the resonance frequency ($\small f_r$).
• At resonance frequency ($\small f_r$): $\small X_L = X_C$. At this point, $\small X_{net} = X_L - X_C = 0$.
• At very high frequencies: $\small X_L$ ($\small 2\pi f L$) is very large (approaching infinity) because $\small f$ is large. $\small X_C$ ($\small \frac{1}{2\pi f C}$) is very small (approaching zero) because $\small f$ is in the denominator. Therefore, $\small X_{net} = X_L - X_C$ will be dominated by the large positive value of $\small X_L$. So, at very high frequencies, $\small X_{net}$ is largely positive.

This shows that as the frequency in a variable frequency RLC series circuit increases from a very low value, the net reactance starts negative, becomes zero at resonance, and then becomes positive at higher frequencies.

Examining Other Options

• Capacitive reactance ($\small X_C$): As discussed, $\small X_C$ starts very large at low frequency and decreases as frequency increases. It is always positive. It does not become negative or zero (unless frequency approaches infinity).
• Inductive reactance ($\small X_L$): As discussed, $\small X_L$ starts at zero (for f=0) and increases with frequency. It is always non-negative. It does not become negative.
• Impedance ($\small Z$): The impedance of an RLC series circuit is given by $\small Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + X_{net}^2}$. Since $\small X_{net}^2$ is always non-negative, and $\small R^2$ is always non-negative, $\small Z$ is always non-negative. It is smallest at resonance ($\small Z=R$) and increases as frequency moves away from resonance. It does not become negative.

Based on the analysis, the quantity that is initially negative, becomes zero, and then positive is the Net reactance.

Quantity	Behavior with Increasing Frequency	Fits Description?
Inductive Reactance ($\small X_L$)	Starts at 0, increases. Always $\small \ge 0$.	No
Capacitive Reactance ($\small X_C$)	Starts very large, decreases. Always $\small > 0$.	No
Net Reactance ($\small X_{net}$)	Starts negative, becomes 0 (at resonance), becomes positive.	Yes
Impedance ($\small Z$)	Starts large, decreases to minimum (R) at resonance, increases. Always $\small \ge 0$.	No

Therefore, in an RLC series circuit under variable frequency, the Net reactance is initially negative, becomes zero at resonance, and then becomes positive.