If you measure the rms voltages across the resistor, capacitor, and inductor in a series \(R L C\) circuit, will they add to the rms generator voltage?

RMS voltage, or root mean square voltage, is a critical concept in the study of alternating current (AC) circuits, including series RLC circuits. RMS value represents the equivalent value of DC voltage that would deliver the same power to a load. In simpler terms, it is a measure of the effective or average voltage that performs work.

For example, if an AC voltage source has a peak value of 100V, the RMS voltage will be lower, around 70.7V, because the alternating nature of the AC supply means that it only reaches its peak for an instant. The formula for RMS voltage for a sinusoidal waveform is the peak voltage divided by the square root of two, or approximately 0.707. In practice, measuring the RMS voltage gives you a practical value to use when designing and analyzing circuits because it accounts for the fluctuating nature of AC.

When dealing with AC circuits, the concept of phasors is indispensable. A phasor is a complex number that represents the magnitude and phase angle of a sinusoidal function. The phasor sum, therefore, refers to the vector addition of individual phasors to get the total or resultant phasor.

For the series RLC circuit, voltage phasors across each element are not simply additive due to their phase differences. They are represented as vectors in a two-dimensional plane. The RMS voltage across the generator, or supply voltage, is a result of adding these vectors head-to-tail geometrically, not arithmetically. This method takes into account both the magnitude and direction (phase) of each voltage component, hence providing a true representation of the sum.

Reactance is an opposition to the flow of AC caused by capacitors and inductors. For capacitors, this is known as reactive capacitance, denoted as \(X_C\). It is the property of a capacitor that restricts the change of voltage in an AC circuit. A key characteristic of reactive capacitance is that it decreases with increasing frequency---the higher the frequency of the AC signal, the less opposition the capacitor provides.

The formula for reactive capacitance is \(X_C = \frac{1}{\text{2}\pi f C}\), where \(C\) is the capacitance in farads and \(f\) is the frequency in hertz. Understanding reactive capacitance is necessary for calculating the voltage drop across a capacitor in a series RLC circuit, which is crucial for analyzing the circuit's behavior in response to different frequencies.

Conversely, reactive inductance, denoted as \(X_L\), is the property of an inductor that resists changes in current in an AC circuit. Unlike capacitive reactance, inductive reactance increases with the frequency---the inductor opposes the change in current more significantly at higher frequencies.

The formula for reactive inductance is \(X_L = \text{2}\pi f L\), where \(L\) is the inductance in henrys and \(f\) is the frequency in hertz. In a series RLC circuit, this inductive reactance plays a pivotal role alongside the capacitive reactance in determining the overall impedance and behavior of the circuit at different frequencies. Balancing the effects of reactive capacitance and inductive reactance is essential for applications like tuning circuits, where precise control over circuit response is critical.