Total Impedance Versus Frequency

The total impedance of the series $RLC$ circuit of Fig. 1 at any frequency is determined by The magnitude of the impedance $Z_T$ versus frequency is determined by The total impedance versus frequency curve for the series resonant circuit of Fig. 1 can be found by applying the impedance versus frequency curve for each element of the equation just derived, written in the following form: where $Z_T(f)$ "means" the total impedance as a function of frequency. For the frequency range of interest, we will assume that the resistance R does not change with frequency, resulting in the plot of Fig. 2. The curve for the inductance, as determined by the reactance equation, is a straight line intersecting the origin with a slope equal to the inductance of the coil. The mathematical expression for any straight line in a two dimensional plane is given by Thus, for the coil, (where $2 \pi L$ is the slope), producing the results shown in Fig. 3. For the capacitor, or which becomes $yx = k$, the equation for a hyperbola, where The hyperbolic curve for $X_C(f)$ is plotted in Fig. 4. In particular, note its very large magnitude at low frequencies and its rapid drop off as the frequency increases. If we place Fig. 3 and Fig. 4 on the same set of axes, we obtain the curves of Fig. 5. The condition of resonance is now clearly defined by the point of intersection, where $X_L = X_C$. For frequencies less than fs, it is also quite clear that the network is primarily capacitive ($X_C > X_L$). For frequencies above the resonant condition, $X_L > X_C$, and the network is inductive. Applying to the curves of Fig. 5, where $X_(f) = X_L(f) - X_C(f)$ we obtain the curve for $Z_T( f )$ as shown in Fig. 6. The minimum impedance occurs at the resonant frequency and is equal to the resistance R. Note that the curve is not symmetrical about the resonant frequency (especially at higher values of $Z_T$). The phase angle associated with the total impedance is For the $\tan^{-1} x$ function (resulting when $X_L > X_C$), the larger x is, the larger the angle $\theta$ (closer to $90^\circ$). However, for regions where $X_C > X_L$, one must also be aware that At low frequencies, $X_C > X_L$, and $\theta$ will approach $-90^\circ$ (capacitive), as shown in Fig. 7, whereas at high frequencies, $X_L > X_C$, and $\theta$ will approach $90^\circ$. In general, therefore, for a series resonant circuit: