Assessment of the Impact of Per Unit Parameters Errors on Wave and Output Parameters in a Transmission Line

Abstract

The assessment of the impact of per unit length parameter errors on the determination of wave parameters, currents, and voltages at the end of the line has been presented in the paper. The impact on the above-mentioned values has been indicated. This paper presents an assessment of the impact of per unit parameter errors on the determination of both wave parameters, as well as currents and voltages at the end of a transmission line, although it is mainly focused on indicating which of the per unit parameters have the strongest impact on the above-mentioned values. For this purpose, elements of incremental sensitivity have been used.

1. Introduction

In order to determine currents and voltages in circuits with distributed parameters—transmission lines, the knowledge of the so-called wave parameters (the wave impedance and the propagation constant) is necessary. These parameters can be determined knowing the per unit parameters of the line, i.e., resistance, inductance, capacitance, and leakage given per unit of line length (usually 1 km). The determination of these quantities is often inaccurate, e.g., in an overhead line the value of capacitance and leakage depends on the weather conditions, and is usually specified in certain ranges. The study of sensitivity of wave parameters to errors in the determination of unit parameters was presented in the paper [1]. This task was performed based on the definition of the relative sensitivity of electrical circuits [2,3]. This paper presents an assessment of the impact of per unit parameter errors on the determination of both wave parameters, as well as currents and voltages at the end of a transmission line, although it is mainly focused on indicating which of the per unit parameters have the strongest impact on the above-mentioned values. For this purpose, elements of incremental sensitivity have been used [4].

2. Linear Circuits with Distributed Parameters

A transmission line is an electrical circuit with distributed parameters, supplied from a sinusoidal voltage source. The length of the electric circuit is comparable to the voltage wavelength λ [2]. The circuit is composed of elements R₀, G₀, L₀, and C₀, which determine the losses of active power related to heat generation and leakage, and as a result of accumulation of energy in magnetic and electric fields [5]. They are called the primary parameters of a transmission line [5]. In the case of circuits with distributed parameters in relation to the length of the line, the longitudinal parameters (resistance R₀, inductance L₀, and lateral parameters: capacitance C₀, and leak conductivity G₀ [6]) are described. Their values are specified as Ω/m, S/m, H/m, and F/m [7,8,9]. It is conventionally assumed that if l is greater than 0.1 λ, or the transmission of impulse signals with a short rise time, as is the case in high voltage cables of spark ignition engines [3,4,6,10], then such a line should be treated as a circuit with distributed parameters. This paper shall examine the properties of a homogeneous line, whose primary parameters are uniformly distributed along the line [7,9]. When analyzing an equivalent circuit of a line with distributed parameters, the following are considered:

—

voltage loss on the wire resistance distributed uniformly along the line;

—

voltage loss on the wire inductance distributed uniformly along the line;

—

leakage current through the insulation distributed uniformly along the line (for cable line: cable insulation, for overhead line: air insulation);

—

line capacitance distributed uniformly, where the plates of the capacitor are represented by two wires or a single wire and the earth.

A transmission line can be represented as a cascade connection of elementary sections of Δx length, in which the resistance, capacitor and coil are lumped elements. Instantaneous values of voltage u(x,t) and current i(x,t) at each point of the line are functions of two independent variables: distance and time. If the line parameters are distributed uniformly along the line, the transmission line is homogeneous. A line is linear if its parameters are not dependent on voltage or current at a given point in the line.

Figure 1 presents a model of a homogeneous two-wire transmission line with the length l. Terminals 1–1′, called line input terminals, are connected to the e(t) voltage source with internal impedance Z₁. A receiver with impedance Z₂ is connected to output terminals 2–2′.

At any point on the line distant by x₁ from the beginning of the line, the voltage is u(x₁,t) and the current is i(x₁,t), in which case for the input terminals x = 0, and for the output terminals x = l [3,10,11].

The unit resistance of a single-wire line can be determined using the formula:

$$R_0=\frac{1000}{\gamma \text{}S}$$

(1)

where γ is the specific conductivity of the wire [m/Ω mm²], and S is the cross-section of the wire [mm²].

When a two-wire line is considered, the result needs to be multiplied by two.

The per unit inductance for a single wire with a length l = 1 m is expressed by the formula:

$$L=\frac{\mu }{4\pi }\left(0.5+ln\frac{a}{R}\right)$$

(2)

where a is the distance between the axes of the cables, R is the wire radius, and $\mu$ is the electric permittivity of the environment (for air assumed as $\mu =4\pi \cdot {10}^{-7}$ H/m).

The per unit capacitance for a two-wire line with a length l = 1 m is expressed by the formula:

$$C_0=\frac{\pi {\epsilon }_0}{\ln \frac{a}{R}}$$

(3)

where ${\epsilon }_0$ is the electric permittivity of the environment—air.

3. Transmission Line Wave and Output Parameters

The term wave parameters of a transmission line most commonly refers to: wave impedance expressed by the Equations (6) and (7):

$$Z_c=\sqrt{\frac{R_0+j\omega L}{G_0=j\omega C}}$$

(4)

where R₀ is the per unit length resistance, G₀ is the per unit length inductance, L₀ is the per unit length capacitance, and C₀ is the per unit length leakage.

The propagation constant:

$$\gamma =\sqrt{\left(R_0+j\omega L_0\right)\cdot \left(G_0+j\omega C_0\right)}$$

(5)

Knowledge of these parameters allows for the determination of the input impedance:

$$Z_{we}=Z_C\frac{Z_{2}ch\gamma \text{}l+Z_{c}sh\gamma \text{}l}{Z_{2}sh\gamma \text{}l+Z_{c}ch\gamma \text{}l}$$

(6)

where Z₂ is the line load impedance, and l is the line length.

Then, for a known voltage U₁ at the beginning of the line, the current I₁ at the beginning of the line can be determined:

$$I_1=\frac{U_1}{Z_{we}}$$

(7)

and then, voltage U₂ and current I₂ at the end of the line:

$$\left\{\begin{array}{l}{U}_2=U_{1}ch\gamma \text{}l-Z_c{I}_{1}sh\gamma \text{}l\\ I_1=-\frac{U_1}{Z_{C_1}}sh\gamma \text{}l+I_{1}ch\gamma \text{}l\end{array}\right.$$

(8)

The sensitivity of wave parameters as well as the output parameters of a transmission line can be traced by changing the per unit length parameters of this line [12,13,14].

4. Numerical Experiments

Numerical experiments were carried out for a 200 kHz frequency, for an overhead line with following per unit length parameters:

R₀ = 0.68 Ω/km; G₀ = 50 µS/km;

L₀ = 0.128 mH/km; C₀ = 0.01 µF/km;

with an impedance load Z₂ = 100 Ω

In order to indicate which of the per unit length parameters have the strongest impact on the values characterizing the transmission line, multiple calculations were carried out, each time substituting a different changed value of a per unit length parameter. These changes consisted in increasing and decreasing the value by 1%. The obtained results—percentage changes—are presented in the form of

Table 1. Changes to the wave impedance modulus.

$\left|{\mathit{Z}}_{\mathit{C}}\right|$
	+1%	−1%
R0	8.980621718802 × 10−6	−8.891264700064 × 10−6
G0	−7.95516852989 × 10−6	7.876015737812 × 10−6
L0	0.498747363629691	−0.501247263019991
C0	−0.496273219279826	0.503773449510121

,

Table 2. Changes to damping constant.

$\mathit{\alpha }=\mathbf{Re}\left\{\mathit{\gamma }\right\}$
	+1%	−1%
R0	0.515151228520686	−0.515151275908188
G0	0.484848713525002	−0.484848750704316
L0	−0.013838466672733	0.016490175659639
C0	0.016313672812472	−0.013964997756260

,

Table 3. Changes to phase constant.

$\mathit{\beta }=\mathbf{Im}\left\{\mathit{\gamma }\right\}$
	+1%	−1%
R0	2.85161812570 × 10−7	−2.4048216284705 × 10−7
G0	−2.27573141879 × 10−7	2.6715112510711 × 10−7
L0	0.49875597170012	−0.5012560025127765
C0	0.49875647683926	−0.5012565178567181

,

Table 4. Changes to the real part of the input resistance.

$\mathbf{Re}\left\{{\mathit{Z}}_{\mathit{w}\mathit{e}}\right\}$
	+1%	−1%
R0	0.000384588324788	−0.0003889650747472
G0	0.043391174085347	−0.0433956466098439
L0	−5.732231738707145	5.9768553112726378
C0	3.562167621061453	−3.2793821336200557

,

Table 5. Changes to the imaginary part of the input resistance.

$\mathbf{Im}\left\{{\mathit{Z}}_{\mathit{w}\mathit{e}}\right\}$
	+1%	−1%
R0	−0.002777179953811	0.0027783246452566
G0	0.001528422276180	−0.0015294640405447
L0	−0.041812016570611	−0.0221985063733944
C0	−1.189631026642754	1.2554145000339767

,

Table 6. Changes to the I₁ current modulus.

$\left|{\mathit{I}}_1\right|$
	+1%	−1%
R0	0.002747571139330	−0.0027486055903140
G0	−0.000971236507435	0.0009719814772497
L0	−0.031856013256254	0.0939895328519183
C0	1.233753020617313	−1.2645765148497845

,

Table 7. Changes to the argument of the current I₁.

$\mathbf{arg}\left\{{\mathit{I}}_1\right\}$
	+1%	−1%
R0	0.002372834589791	−0.0023694989848461
G0	−0.044547012164130	0.0445533464751429
L0	5.724421549430766	−5.9109167085244178
C0	−2.381912054750487	1.9818184348223934

,

Table 8. Changes to the U₂ voltage modulus.

$\left|{\mathit{U}}_2\right|$
	+1%	−1%
R0	−0.009870362379686	0.0098708010242115
G0	−0.010998843739695	0.0110003885878087
L0	−0.053850964577535	0.0838634668968119
C0	0.630798491006207	−0.6478547211964083

,

Table 9. Changes to the argument of the voltage U₂.

$\mathbf{arg}\left\{{\mathit{U}}_2\right\}$
	+1%	−1%
R0	0.000288807602249	−0.000288916464849
G0	−0.002528487115057	0.002528654755186
L0	−2.317110575876312	2.323205351184315
C0	−2.751776829843434	2.741194932584926

,

Table 10. Changes to the I₂ current modulus.

$\left|{\mathit{I}}_2\right|$
	+1%	−1%
R0	−0.009870362379715	0.0098708010242179
G0	−0.010998843739739	0.0110003885878038
L0	−0.053850964577548	0.0838634668968006
C0	0.630798491006164	−0.6478547211963962

and

Table 11. Changes to the argument of the current I₂.

$\mathbf{arg}\left\{{\mathit{I}}_2\right\}$
	+1%	−1%
R0	0.000288807602261	−0.0002889164648371
G0	−0.002528487115069	0.0025286547551618
L0	−2.317110575876356	2.3232053511843256
C0	−2.751776829843455	2.7411949325849654

.

The highest and lowest values of the modulus are highlighted by bold font in the tables. On the basis of the analysis of the above tables, it can be stated that the biggest influence on the damping factor is the change of R_0, and on the phase factor the change of C₀. C₀ also has the greatest impact on the modulus, and the argument of both the output voltage and current [15,16,17,18,19].

In order to evaluate how the values of wave and output parameters change when per unit length parameters are changed, a proprietary program has been written in the MATLAB environment. It allows for the acquisition of graphs of wave and output parameters values for a given frequency, load, input parameters, and variability of individual per unit length parameters (e.g., ±20%) at any value of supply voltage U₁ and load Z₂. The presented graphs were created for U₁ = 100 V, Z₂ = 100 Ω. Thus, with a change of per unit length resistance within ±20%, the damping factor α linearly increases from−10% to +10%, regardless of frequency (Figure 2a, Figure 3a and Figure 4a), while the phase factor β reaches a minimum around −5% and then increases, but these changes are very minor, approximately 10⁻⁵% for f = 100 kHz (Figure 2b), 10⁻⁶% for f = 500 kHz (Figure 3b), and 10⁻⁷% for f = 1 MHz (Figure 4b). As far as the wave impedance modulus and argument are concerned, they increase almost linearly, with the exception that the modulus changes for low frequencies are around 10⁻³% (Figure 2c) and for large ones, around 10⁻⁵% (Figure 3c and Figure 4c), whereas the argument changes are large, i.e., ±300% (Figure 2d, Figure 3d and Figure 4d).

Changes to the damping factor alpha when changing parameter L₀ are non−linear, and vary from 1% to 0.2%, regardless of frequency (Figure 5a, Figure 6a and Figure 7a). The factor β rises almost linearly from −10% to 10%, regardless of frequency (Figure 5b, Figure 6b and Figure 7b). The impedance modulus rises almost linearly in the range of −10% to 10% for all tested frequencies (Figure 5c, Figure 6c and Figure 7c). The argument changes linearly in the range from approximately 200% to −400%, with minor changes for individual frequencies (Figure 5d, Figure 6d and Figure 7d).

Changes to the damping factor alpha when changing parameter C₀ are non−linear, and vary from 0.25% to 1.4%, regardless of frequency (Figure 8a, Figure 9a and Figure 10a). The factor β rises almost linearly from −10% to 10%, regardless of frequency (Figure 8b, Figure 9b and Figure 10b). The impedance modulus rises almost linearly in the range of 12% to −8% for all tested frequencies (Figure 8c, Figure 9c and Figure 10c). The argument changes linearly in the range from approximately −400% to 280%, with minor changes for individual frequencies (Figure 8d, Figure 9d and Figure 10d).

Changes to the damping factor alpha when changing the G₀ parameter are non−linear, and vary from 1% to 0.2% regardless of frequency (Figure 11a, Figure 12a and Figure 13a). The factor β rises almost linearly from −10% to 10%, regardless of frequency (Figure 11b, Figure 12b and Figure 13b). The impedance modulus rises almost linearly in the range of −10% to 10% for all tested frequencies (Figure 11c, Figure 12c and Figure 13c). The argument changes linearly in the range from approximately 40% to −200% with minor changes for individual frequencies (Figure 11d, Figure 12d and Figure 13d).

Changes to the U₂ voltage modulus occur when the R₀ parameter is changed linearly, and vary from 0.4% to −0.4%, depending on the frequency. The higher the frequency, the smaller the change (Figure 14a, Figure 15a and Figure 16a). The U₂ argument increases almost linearly from −2% to 2%, depending on the frequency (Figure 14b, Figure 15b and Figure 16b). The I₂ Modulus changes linearly from 0.4% to −0.4% for lower frequencies, and 0.04% to −0.04% for higher tested frequencies (Figure 14c, Figure 15c and Figure 16c). The I₂ argument rises linearly from −2.1% to 2.1% for 500 kHz, and −0.13% to 0.13% for 1 MHz (Figure 14d, Figure 15d and Figure 16d).

Changes of U₂ voltage modulus occur when L₀ parameter is changed parabolically, and change from 7.5% to −3%, regardless of frequency. (Figure 17a, Figure 18a and Figure 19a). The U₂ argument falls almost linearly from 50% to −50%, independently of the frequency (Figure 17b, Figure 18b and Figure 19b). The I₂ modulus changes parabolically in the range of 7.5% to 4%, with slight differences depending on the frequency (Figure 17c, Figure 18c and Figure 19c). The I₂ argument falls linearly in the range from 50% to −50%, independently of the frequency (Figure 17d, Figure 18d and Figure 19d).

Changes of the U₂ voltage modulus occur when parameter C₀ is changed non−linearly, and change from −14% to 8% independently of frequency. (Figure 20a, Figure 21a and Figure 22a). The U₂ argument falls almost linearly from 50% to −50%, independently of frequency (Figure 20b, Figure 21b and Figure 22b). The I₂ modulus changes from −14% to 8%, independently of the frequency (Figure 20c, Figure 21c and Figure 22c). The I₂ argument decreases linearly in the range from 50% to −50%, independently of the frequency (Figure 20d, Figure 21d and Figure 22d).

Changes of the U₂ voltage modulus occur when the G₀ parameter is changed almost linearly, and change from 0.09% to −0.09% for 500 kHz, and 0.04% to −0.04% for 1 MHz (Figure 23a and Figure 24a). The U₂ argument decreases almost linearly from 0.02% to −0.02% at 500 kHz, and 0.01% to −0.01% at 1 MHz (Figure 23b and Figure 24b). The I₂ modulus runs almost linearly, and changes from 0.09% to −0.09% for 500 kHz, and 0.04% to −0.04% for 1 MHz (Figure 23c and Figure 24c). The I₂ argument falls almost linearly from 0.02% to −0.02% at 500 kHz, and 0.01% to −0.01% at 1 MHz (Figure 23d and Figure 24d).

5. Conclusions

The paper presents the evaluation of the impact of per unit parameter errors on the determination of wave parameters, as well as currents and voltages at the end of the line. On the basis of numerical calculations, it was found that errors in the determination of per unit inductance have the greatest influence on the value of the part of the input resistance and on I₁ current argument. However, the change of per unit resistance and leakage have the smallest effect on the value of the phase constant. It can also be seen that increasing per unit length parameters by 1% does not always result in an increase of output parameters.

A thorough analysis of the above diagrams may be used to construct optimal high voltage cables used in the ignition systems of spark−ignition engines, where the most important parameter is the U₂ voltage modulus [3,4,10,20].