Total Impedance Versus Frequency

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The impedance of capacitors and inductors in a circuit depend on the frequency of the electric signal. The impedance of an inductor is directly proportional to frequency, while the impedance of a capacitor is inversely proportional to frequency.
The total impedance of the series $RLC$ circuit of Fig. 1 at any frequency is determined by
Fig. 1: Series resonant circuit.
$$Z_T = R + j X_L - j X_C = R - j (X_L - X_C)$$
The magnitude of the impedance $Z_T$ versus frequency is determined by
$$ Z_T = \sqrt{R^2 + (X_L - X_C)^2}$$
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:
$$ \bbox[10px,border:1px solid grey]{Z_T(f) = \sqrt{[R(f)]^2 + [X_L(f) - X_C(f)]^2}} \tag{1}$$
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.
Resistance versus frequency
Fig. 2: Resistance versus frequency
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
$$ y = mx + b$$
Thus, for the coil,
$$X_L = 2 \pi fL + 0 = (2\pi L)(f) + 0$$
(where $2 \pi L$ is the slope), producing the results shown in Fig. 3.
Inductive reactance versus frequency.
Fig. 3: Inductive reactance versus frequency.
For the capacitor,
$$ X_C = {1 \over 2 \pi fC}$$
or
$$ X_C f = {1 \over 2 \pi C}$$
which becomes $yx = k$, the equation for a hyperbola, where
$$y \text{(variable)}= X_C$$
$$x \text{(variable)}= f$$
$$k \text{(constant)}= { 1 \over 2 \pi C}$$
The hyperbolic curve for $X_C(f)$ is plotted in Fig. 4.
Capacitive reactance versus frequency.
Fig. 4: Capacitive reactance versus frequency.
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.
Fig. 5: Placing the frequency response of the inductive and capacitive reactance of a series RLC circuit on the same set of axes.
Fig. 5: ZT versus frequency for the series resonant circuit.
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
$$ Z_T(f) = \sqrt{[R(f)]^2 + [X_L(f) - X_C(f)]^2}\\ = \sqrt{[R(f)]^2 + [X_(f)]^2}$$
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
$$ \bbox[10px,border:1px solid grey]{\theta = \tan ^{-1} { X_L - X_C \over R}} \tag{2}$$
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
$$ \tan^{-1} (-x) = - \tan^{-1} (x) $$
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 $90^\circ$.
Fig. 7: Phase plot for the series resonant circuit.
In general, therefore, for a series resonant circuit:
$$ f < f_s : \, \text{network capacitive; I leads E}$$
$$ f > f_s : \, \text{network inductive; E leads I}$$
$$ f = f_s : \, \text{network resistive; I and E are in phase}$$

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1 comment
realnfo .
29 days
Very good information.
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