2.1. Determination of the network harmonic impedance

The simplest method for harmonic impedance measurement supposes the changing of network status. The disturbance can be created by switching a network component in the bus where the network harmonic impedance must be measured. The main steps of this method are as follows [16]:


$$\underline{Z}\_{k} = \frac{\underline{\underline{L}I\_{k-\text{pre}}} - \underline{\underline{L}I\_{k-\text{post}}}}{\underline{I}\_{k-\text{pre}} - \underline{I}\_{k-\text{post}}} \tag{1}$$

where Uk�pre and Ik�pre are the pre-disturbance harmonic voltage and current, and Uk�post and Ik�post are the post-disturbance harmonic voltage and current.

The impedance Zk does not include the switched load.

Because the harmonic impedance determined using the expression (1) is parallel impedance, to obtain the phase of the impedance, we have to transform the parallel schema into series schema.

The series harmonic impedance can be determined using the following expression:

$$\underline{Z}\_{\rm kS} = R\_S + \mathbf{j} \cdot \mathbf{X}\_{\rm kS} \tag{2}$$

The parallel harmonic impedance can be determined using the following expression:

$$\frac{1}{\underline{Z}\_{kp}} = \frac{1}{\mathcal{R}\_p} + \frac{1}{jX\_{kp}}\tag{3}$$

$$\underline{\mathbf{Z}}\_{kp} = \frac{1}{\frac{j\mathbf{X}\_{kp} + \mathbf{R}\_p}{j\mathbf{X}\_{kp} + \mathbf{R}\_p}} = \frac{j\mathbf{X}\_{kp} \cdot \mathbf{R}\_p}{j\mathbf{X}\_{kp} + \mathbf{R}\_p} \cdot \frac{\mathbf{R}\_p - j\mathbf{X}\_{kp}}{\mathbf{R}\_p - j\mathbf{X}\_{kp}} = j \cdot \frac{\mathbf{R}\_p^2 \cdot \mathbf{X}\_{kp}}{\mathbf{R}\_p^2 + \mathbf{X}\_{kp}^2} + \frac{\mathbf{R}\_p \cdot \mathbf{X}\_{kp}^2}{\mathbf{R}\_p^2 + \mathbf{X}\_{kp}^2} \tag{4}$$

Knowing that:

$$
\underline{Z}\_{kp} = \underline{Z}\_{kS} = R\_S + j \cdot X\_{kS} \tag{5}
$$

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Results:

$$\begin{aligned} R\_S &= \frac{R\_p \cdot X\_{kp}^2}{R\_p^2 + X\_{kp}^2} \\ X\_{kS} &= \frac{R\_p^2 \cdot X\_{kp}}{R\_p^2 + X\_{kp}^2} \end{aligned} \tag{6}$$

We write now these parameters of the series impedance depending on the absolute value of the harmonic impedance, which can be practically determined using (1).

$$Z\_{kp}^2 = Z\_{kS}^2 = R\_S^2 + X\_{kS}^2 = \frac{R\_p^2 \cdot X\_{kp}^2}{R\_p^2 + X\_{kp}^2} \tag{7}$$

Using expressions (6) and (7), we can write the series resistance as follows:

$$R\_S = \frac{R\_p \cdot X\_{kp}^2}{R\_p^2 + X\_{kp}^2} \cdot \frac{R\_p}{R\_p} = \frac{Z\_{kp}^2}{R\_p} \tag{8}$$

The value of the parallel resistance, Rp, from the expression (8) can also be practically determined; it is actually the value of the harmonic impedance corresponding to the resonance frequency. Using the expression (7), the value of the series reactance can now be determined:

$$X\_{\rm kS} = \sqrt{Z\_{\rm kS}^2 - R\_{\rm S}^2} \tag{9}$$

With the values of the series parameters of the harmonic impedance, we can now determine the phase of the complex harmonic impedance:

$$\varphi\_k = \operatorname{arctg}\left(\frac{\mathbf{X}\_{kS}}{R\_S}\right) \tag{10}$$
