# Magnitudes of Voltages across RLC versus Frequency

Plotting the magnitude (effective value) of the voltages $V_R$, $V_L$, and $V_C$ and the current I versus frequency for the series resonant circuit on the same set of axes, we obtain the curves shown in Fig. 1. Note that the VR curve has the same shape as the I curve and a peak value equal to the magnitude of the input voltage E. The $V_C$ curve builds up slowly at first from a value equal to the input voltage since the reactance of the capacitor is infinite (open circuit) at zero frequency and the reactance of the inductor is zero (short circuit) at this frequency. As the frequency increases, $1/wC$ of the equation $$V_C = IX_C = (I)({1 \over wC})$$
Fig. 1: $V_R$, $V_L$, $V_C$, and I versus frequency for a series resonant circuit.
becomes smaller, but I increases at a rate faster than that at which $1/wC$ drops. Therefore, $V_C$ rises and will continue to rise due to the quickly rising current, until the frequency nears resonance. As it approaches the resonant condition, the rate of change of I decreases. When this occurs, the factor $1/wC$, which decreased as the frequency rose, will overcome the rate of change of I, and $V_C$ will start to drop. The peak value will occur at a frequency just before resonance. After resonance, both $V_C$ and I drop in magnitude, and $V_C$ approaches zero.
The higher the $Q_s$ of the circuit, the closer $f_{Cmax}$ will be to $f_s$, and the closer $V_{Cmax}$ will be to $Q_{s}E$. For circuits with $Q_s \ge 10$, $f_{Cmax} = f_s$, and $V_{Cmax}= Q_{s}E$. The curve for $V_L$ increases steadily from zero to the resonant frequency since both quantities $wL$ and I of the equation $$V_L = IX_L = (I)(wL)$$ increase over this frequency range. At resonance, I has reached its maximum value, but $wL$ is still rising. Therefore, $V_L$ will reach its maximum value after resonance. After reaching its peak value, the voltage $V_L$ will drop toward E since the drop in I will overcome the rise in $wL$. It approaches E because $X_L$ will eventually be infinite, and $X_C$ will be zero.
As Qs of the circuit increases, the frequency $f_{Lmax}$ drops toward $f_s$, and $V_{Lmax}$ approaches $Q_sE$. For circuits with $Q_s \geq 10$, $f_{Lmax} \cong f_s$, and $V_{Lmax} \cong Q_sE$.
The $V_L$ curve has a greater magnitude than the $V_C$ curve for any frequency above resonance, and the $V_C$ curve has a greater magnitude than the $V_L$ curve for any frequency below resonance. This again verifies that the series R-L-C circuit is predominantly capacitive from zero to the resonant frequency and predominantly inductive for any frequency above resonance. For the condition $Q_s \geq 10$, the curves of Fig. 1 will appear as shown in Fig. 2. Note that they each peak (on an approximate basis) at the resonant frequency and have a similar shape.
Fig. 2: $V_R$, $V_L$, $V_C$, and I for a series resonant circuit where $Qs \geq 10$
In review,
• $V_C$ and $V_L$ are at their maximum values at or near resonance (depending on $Q_s$).
• At very low frequencies, $V_C$ is very close to the source voltage and $V_L$ is very close to zero volts, whereas at very high frequencies, $V_L$ approaches the source voltage and $V_C$ approaches zero volts.
• Both $V_R$ and I peak at the resonant frequency and have the same shape.