As you may recall from your introductory physics course, impedance is a quantity used to describe how different electrical elements oppose alternating current (AC). It is a two-dimensional quantity expressed as a complex number, with a real part (\(\Re(Z)\) or \(Z'\)) called the Resistance \(R\) and an imaginary part (\(\Im(Z)\) or \(Z''\)) called the Reactance \(X\):

\[ Z = R + jX \tag{1} \]

where \(j\) denotes the imaginary unit. (In what follows, we will use \(Z\) for complex impedance, written without the commonly used hat symbol.)

Besides the raw impedance, we often work with its magnitude (or modulus):

\[ |Z| = \sqrt{R^{2} + X^{2}} \tag{2} \]

which corresponds to the Euclidean distance from the origin in the complex plane, and its phase angle:

\[ \varphi = \arctan\!\left(\frac{X}{R}\right) \tag{3} \]

In impedance spectroscopy, the impedance is measured as a function of frequency, \(Z(f)\), forming a quantity that can be represented in three-dimensional space, as illustrated in Figure 1. In practice, however, this frequency sweep is usually displayed in two different two-dimensional representations. The first is the Bode plot, which shows the magnitude \(|Z(f)|\) and phase \(\varphi(f)\) as functions of frequency. The second is the Nyquist plot, which shows the relationship between the real and imaginary parts of the impedance, that is resistance \(R\) versus reactance \(X\).

For a circuit containing only an ideal resistor, the impedance is purely real:

\[ Z = R \tag{4} \]

Consequently, the magnitude and phase are frequency-independent:

\[ |Z| = R \tag{5} \]

\[ \varphi = 0 \tag{6} \]

In the Nyquist representation this appears as a single point at \((R,\,0)\) (or a vertical stack of identical points if multiple frequencies are sampled). In the Bode representation, the magnitude is a horizontal line at \(R\) and the phase is a horizontal line at \(0^\circ\), both independent of frequency, as illustrated in Figure 2.

In a circuit with a pure inductance, the situation is a bit more complex. In this case, the impedance is purely imaginary and depends linearly on frequency:

\[ Z = j \omega L \tag{7} \]

where \(\omega = 2 \pi f\) is the angular frequency and \(L\) is the inductance. From this relation, the magnitude of the impedance is

\[ |Z| = \omega L \tag{8} \]

and the phase angle is again constant:

\[ \varphi = +90^{\circ} \tag{9} \]

Thus, in the Nyquist representation the impedance appears as a vertical line along the imaginary axis, extending upward with increasing frequency. In the Bode representation, the magnitude increases linearly with frequency, while the phase remains constant at \(+90^{\circ}\). These behaviours are illustrated in Figure 3.

In a circuit with a pure capacitance, the impedance is again purely imaginary, but in contrast to the inductive case it is inversely proportional to frequency:

\[ Z = \frac{1}{j \omega C} = -\,\frac{j}{\omega C} \tag{10} \]

where \(\omega = 2 \pi f\) is the angular frequency and \(C\) is the capacitance. From this relation, the magnitude of the impedance is

\[ |Z| = \frac{1}{\omega C} \tag{11} \]

and the phase angle is constant:

\[ \varphi = -90^{\circ} \tag{12} \]

Thus, in the Nyquist representation the impedance appears as a vertical line along the negative imaginary axis, decreasing with increasing frequency. In the Bode representation, the magnitude is inversely proportional to frequency, while the phase remains constant at \(-90^{\circ}\). These behaviours are illustrated in Figure 4.

Now we move to a bit more complex situation — the series RL circuit. In real-life applications, each inductor is not ideal and, besides its inductance, includes parasitics, especially the series resistance arising from the resistance of the wire that forms the coil. Moreover, such series RL behaviour (i.e. parasitic inductance) can be observed in impedance spectra of low-resistance resistors, especially at higher frequency, where the inductance formed by the loop of contact leads begins to influence the otherwise purely resistive response.

In this case the impedance has both real and imaginary parts:

\[ Z = R + j\,\omega L \tag{13} \]

which yields the frequency-dependent magnitude (modulus)

\[ |Z| = \sqrt{R^{2} + (\omega L)^{2}} \tag{14} \]

and phase

\[ \varphi = \arctan\!\left(\frac{\omega L}{R}\right). \tag{15} \]

In the Nyquist representation, the locus appears as a vertical line at \(\Re(Z)=R\), with the imaginary part \(X=\omega L\) increasing with frequency. In the Bode representation, the magnitude rises from \(|Z|\approx R\) at low frequency toward \(|Z|\approx \omega L\) at high frequency, while the phase smoothly increases from \(0^{\circ}\) toward \(+90^{\circ}\). These behaviours are illustrated in Figure 5.

Practical note: the transition is characterised by \(\omega_{\mathrm{c}} \approx R/L\) (equivalently the time constant \(\tau=L/R\)), where \(\varphi=45^{\circ}\) and \(|Z|=\sqrt{2}\,R\).

Another common situation is the parallel RC circuit that involves the parallel connection of a resistor and a capacitor. This behaviour is commonly observed for resistors with a high nominal resistance, where parasitic capacitance occurs at higher frequencies and arises from the capacitor formed by the resistor’s contacts. Moreover, a parallel RC is a simple equivalent circuit used for the characterisation of many real-life objects (e.g. thin layers of semiconductor materials), where \(R\) describes the charge-transport (leakage/conductive) pathway and \(C\) represents the dielectric response of the layer.

To obtain the impedance of the parallel connection we add the reciprocals of the branch impedances:

\[ \frac{1}{Z} = \frac{1}{R} + j\,\omega C \tag{16} \]

For parallel circuits it is usually more efficient to work with the reciprocal of impedance, the admittance, \(Y \equiv 1/Z\). In this case:

\[ Y = \frac{1}{R} + j\,\omega C \tag{17} \]

Define the conductance and susceptance as \(G \equiv 1/R\) and \(B \equiv \omega C\), so that

\[ Y = G + jB. \tag{18} \]

The impedance is the reciprocal of the admittance:

\[ Z = \frac{1}{Y} = \frac{G - jB}{G^{2}+B^{2}}. \tag{19} \]

Substituting \(G=1/R\) and \(B=\omega C\) and simplifying gives

\[ Z = \frac{\tfrac{1}{R} - j\,\omega C}{\left(\tfrac{1}{R}\right)^{2}+(\omega C)^{2}} = \frac{R}{1+(\omega R C)^{2}} - j\,\frac{\omega R^{2} C}{1+(\omega R C)^{2}}. \tag{20} \]

Hence, the magnitude and phase are

\[ |Z| = \frac{R}{\sqrt{1+(\omega R C)^{2}}} \tag{21} \]

\[ \varphi = -\arctan(\omega R C). \tag{22} \]

Thus, in the Nyquist representation the impedance forms a semicircle of diameter \(R\), centred at \((R/2,\,0)\), starting at \(R\) (low frequency) and ending at the origin (high frequency). In the Bode representation, the magnitude is inversely proportional to frequency (a straight line of slope \(-1\) on a log–log scale) while the phase shifts from \(0^{\circ}\) to \(-90^{\circ}\). These behaviours are seen in the simulation in Figure 6.

How should we read the Nyquist diagram? If we substitute \(f \to 0\) into equation (20), only the real part remains. Thus, in the Nyquist plot the frequency sweep starts at the farthest point on the real (resistance) axis. As the frequency increases towards infinity (\(f \to \infty\)), the impedance approaches the origin of the complex plane. Changing the capacitance value does not alter the semicircular shape, but it shifts the locus of measured impedance points — effectively stretching or compressing the arc along the frequency axis.

The arc reaches its extremal value — in this representation a minimum, since we plot the loop-sided imaginary axis (\(-jX\)) — at the characteristic angular frequency

\[ \omega_{\mathrm{c}} = \frac{1}{R C}. \tag{23} \]

At this point, located at the middle of the semicircle (\(\Re(Z) = R/2\)), the impedance characteristics are:

\[ |Z| = \frac{R}{\sqrt{2}}, \quad \varphi = -45^{\circ}, \quad \Re(Z) = \frac{R}{2}, \quad (-jX)_{\min} = \frac{R}{2}. \tag{24} \]

We now move to one of the most important cases in circuit analysis — the series RLC circuit. This circuit consists of a resistor \(R\), inductor \(L\), and capacitor \(C\) connected in series. Such behaviour is commonly encountered in real-life systems: for example, in resonant circuits of radios and filters, or in impedance spectra of electrochemical and solid-state devices where resistive, inductive, and capacitive contributions appear together.

The impedance of the series RLC circuit is the sum of the three elements:

\[ Z = R + j\omega L + \frac{1}{j\omega C}. \tag{25} \]

Collecting the real and imaginary parts, we obtain:

\[ Z = R + j\left(\omega L - \frac{1}{\omega C}\right). \tag{26} \]

The magnitude of the impedance is therefore

\[ |Z| = \sqrt{R^{2} + \left(\omega L - \tfrac{1}{\omega C}\right)^{2}}, \tag{27} \]

and the phase angle is

\[ \varphi = \arctan\!\left(\frac{\omega L - \tfrac{1}{\omega C}}{R}\right). \tag{28} \]

A key feature of the RLC series circuit is the resonance frequency, at which the inductive and capacitive reactances cancel out:

\[ \omega_{0} L = \frac{1}{\omega_{0} C} \quad \Rightarrow \quad \omega_{0} = \frac{1}{\sqrt{LC}}. \tag{29} \]

At this resonance condition:

• The impedance is purely real: \(Z = R\).
• The magnitude reaches its minimum, equal to \(R\).
• The phase angle crosses zero: \(\varphi = 0^{\circ}\).

In the Nyquist representation, the series RLC spectrum appears as a straight line along the real–imaginary plane. This should be contrasted with the circle observed in the admittance representation, where the parallel contributions of \(L\) and \(C\) dominate the shape. In the Bode magnitude plot, a distinct minimum appears at the resonance frequency, while the phase shifts from capacitive (\(-90^{\circ}\)) at low frequencies, through zero at resonance, to inductive \((+90^{\circ})\) at high frequencies. These behaviours are illustrated in the simulation in Figure 7.

Finally, let us examine the parallel RLC circuit, where a resistor \(R\), inductor \(L\), and capacitor \(C\) are connected in parallel. This configuration is very common in practice, for example in resonant filters and oscillators, or as an equivalent model of dielectric and semiconductor layers where resistive, inductive, and capacitive effects coexist.

The total admittance is the sum of branch contributions:

\[ Y = \frac{1}{Z} = \frac{1}{R} + j\omega C - \frac{j}{\omega L}. \tag{30} \]

Writing \(Y = G + jB\) with \(G = 1/R\) and \(B = \omega C - \tfrac{1}{\omega L}\), the impedance is

\[ Z = \frac{1}{Y} = \frac{G - jB}{G^{2}+B^{2}}. \tag{31} \]

Hence the real and imaginary parts are

\[ \Re(Z) = \frac{1/R}{(1/R)^{2} + \left(\omega C - \tfrac{1}{\omega L}\right)^{2}}, \qquad \Im(Z) = -\frac{\omega C - \tfrac{1}{\omega L}}{(1/R)^{2} + \left(\omega C - \tfrac{1}{\omega L}\right)^{2}}. \tag{32} \]

Resonance occurs when the imaginary part vanishes, \(B(\omega_0)=0\), giving

\[ \omega_{0} = \frac{1}{\sqrt{LC}}. \tag{33} \]

At this frequency the impedance is purely real, \(Z(\omega_0)=R\). In the Nyquist diagram the locus is a circle of diameter \(R\), centred at \((R/2,0)\), with the rightmost point \((R,0)\) corresponding to resonance. In the Bode plot, the resonance shows as a maximum in \(|Z|\) and the zero crossing of the phase. These behaviours are illustrated in the simulation in Figure 8.