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## Frequency-Dependent Transmission Line

This example shows a custom frequency-dependent transmission line model. The characteristic admittance and propagation function are first derived from the frequency-dependent resistance, reactance, and susceptance. The derived values are fitted using `RF Toolbox™`. The Universal Line Model (ULM) [1] is then implemented in `Simscape™` based on the fitted parameters. The results from the frequency-dependent transmission line model and the classic pi-section transmission line model are compared.

### Specification of Parameters

Import the frequency-dependent parameters of the transmission line. These parameters are computed for an overhead linet that is 20m above the ground [2]. The ground resistivity and the skin effect of the conductor are considered nonnegligible. These parameters are pre-computed for the simulation:

• Frequency-dependent series resistance per unit length,

• Frequency-dependent series reactance per unit length,

• Frequency-dependent shunt susceptance per unit length,

• Corresponding frequency,

• Length of the transmission line,

``` Name Size Bytes Class Attributes B 1000x1 8000 double R 1000x1 8000 double X 1000x1 8000 double freq 1000x1 8000 double len 1x1 8 double ```

The frequency-dependent R, L and C are shown in these figures:

### Characteristic Admittance and Propagation Function

The characteristic admittance is expressed as , where and are the frequency-dependent series impedance and shunt admittance per unit length.

The propagation velocity is expressed as: , where is the propagation constant, and is the corresponding angular velocity.

The propagation function, , is then expressed as .

### Rational Fitting for Characteristic Admittance

To convert the characteristic admittance to the rational form, use the rational fit function `rationalfit` from `RF Toolbox™`.

where:

• is the number of poles (order of the fit).

• is the pole of .

• is the residue of .

In this case, an eight order fit is carried.

These figures show the comparison between the characteristic admittance before and after rational fitting.

### Rational Fitting for Propagation Function

The time delay of the propagation function is first removed to help reduce the order of rational fitting, where is the propagation time delay and is the propagation function without time delay. The time delay is represented by a delay unit in the model.

To convert the propagation function without time delay to the rational form, use the `rationalfit` function from `RF Toolbox™`.

where:

• is the number of poles (order of the fit).

• is the pole of .

• is the residue of .

In this case, an eight order fit is carried.

These figures show that the propagation function H (with time delay) before and after rational fitting agree.

### Universal Line Model Implemented in Simscape

In this example, only a single conductor and the ground return is considered. The equivalent circuit of the line in Laplace domain can be deduced from the Universal Line Model (ULM) [1]. Key variables are introduced as follows:

• is the voltage at terminal .

• is the current at terminal .

• is the shunt current at terminal .

• is the reflecting current from terminal .

• is the auxiliary current from terminal .

• is the propagation function.

From this equivalent circuit, the system of equations can be written as:

where:

Considering the rational form of the characteristic admittance, the shunt current on terminal one is:

To transform these equations from Laplace domain to time domain, inverse Laplace transform is then performed. This leads to:

where , and are the time domain representation of , and .

Similarly, considering the rational form of the propagation function, the auxiliary current on terminal one is:

To transform these equations from Laplace domain to time domain, inverse Laplace transform is then performed. This leads to:

Currents on the terminal two can be deduced using the same procedure. Time domain equations are finally implemented in `Simscape™` using Simscape language.

### Simulation Results from Simscape Logging

```For the first simulation case, the voltage source is generating a 60 Hz sine wave. The Pi-Section Transmission Line is using the RLC at 60 Hz, which matches the frequency of the voltage source. The plot below shows the input and output terminal voltages of the transmission line. The two models show good agreement at the steady state.```

For the second simulation case, the voltage source is generating a 60Hz sine wave with a 10 kHz modulation. The Pi-Section Transmission Line is still using the RLC at 60 Hz. It is clear that the custom frequency-dependent transmission line model is suitable for a broader band signals whereas the pi-section model is only applicable for extremely narrow band signals.

### References

[1] Morched, Atef, Bjorn Gustavsen, and Manoocher Tartibi. "A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables." IEEE Transactions on Power Delivery 14.3 (1999): 1032-1038.

[2] Dommel, Herman W. "Overhead line parameters from handbook formulas and computer programs." IEEE Transactions on Power Apparatus and Systems 2 (1985): 366-372.