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# Transmission Line Loss: Models and Equations

### Key Takeaways

• Transmission lines utilize admittance models instead of the more familiar impedance models.

• Models vary significantly for transmission line models under specific lengths.

• Constant calculations capture the error between long transmission lines and shorter models. Transmission line loss is one of the most critical considerations in power distribution.

It’s an inescapable problem that merits serious consideration for efficiency: in the real world, long-distance transmission can experience significant losses due to electrical and material factors. Not only does this inefficiency present a loss of value, but it also harms the long-term integrity of the cable due to heat-related aging. To improve costs by eliminating waste and minimizing repair/replacement schedules, engineers utilize different line models to simulate transmission line loss, requiring considerably more complexity for improved accuracy.

Comparing Loss and Lossless Transmission Line Models

 Loss Lossless Practical; provides a better indication of how circuits operate in the real-world. Uses an attenuation constant and phase constant to relate the impedance/conductance and reactance/susceptance of lumped and discrete line models. Theoretical; offers a quick ballpark calculation to serve as a framework (though not necessarily a guide) for analysis. Can only relate the phase constant of the reactance/susceptance between lumped and discrete element models as attenuation constant is zero by definition for lossless lines.

## Differentiating Transmission Lines from Standard Circuit Models

Numerically and symbolically, transmission lines can use admittance, or the reciprocal of impedance, to represent circuit parameters. Like impedance, admittance is a complex value containing a real and imaginary component: conductance and susceptance. Mathematically, it’s important to note that the imaginary-valued nature of reactance and susceptance is a negative reciprocal relationship. That is, capacitive susceptance is positive, and inductive susceptance is negative.

The admittance components are responsible for different mechanisms of current flow, with the conductance responsible for the moving charge carriers. In contrast, susceptance gives way to displacement current due to time-varying electric fields. As a result, admittance is a frequency-dependent value, which nullifies static-frequency capacitor equations like c = qv and necessitates a more apt relationship.

Returning to transmission: these lines often use lumped element circuit models with paths of least resistance (or, equivalently, paths of greatest conductance) to minimize losses during operation. These circuits are reducible to a parallel configuration of the total conductance and magnetic current loss for simplification. Concurrently, benefits from a hybrid impedance-admittance analysis allow for multiple evaluation angles:

• From an impedance design perspective, series voltage drops and power loss (arising from the finite conductance of materials) result in real losses along the line; self-inductance does not contribute toward any loss.

• From an admittance design perspective, the shunt conductance also results in real losses in the circuit and leakage current at the insulators. Depending on the complexity of the admittance circuit and the model's needs, both the shunt conductance and the shunt capacitance may be negligible.

Generally, the decision to ignore admittance characteristics falls under the distance or size of the transmission circuit:

• Short - A short-line model (for a distance up to and including 80 km/50 miles) uses lumped elements, accounts for series impedance, and neglects shunt capacitance.

• Medium - A medium-line model (distances greater than 80 km/50 miles and up to 250 km/155 miles) must account for shunt capacitance but still uses a lumped element model for analysis ease.

• Long - At distances greater than 250 km/155 miles, a lumped element becomes noticeably inaccurate as it fails to capture more of the intricacies of the circuit’s performance. Mathematically, this requires using a differential length term to improve network analysis calculations.

## Calculating for Transmission Line Loss

Encapsulating the scope of longer transmission line loss requires the treatment of transmission lines as two-port networks, allowing a determination of the lines’ chain parameters. Consider a two-port network (i.e., an open circuit at the input and the load with some circuit elements between them); this format will contain an input current, an output current, and two voltage drops across the input and network of the load. The relationship between current and voltage is: with constants A, B, C, and D are complex numbers. Constants A and D are dimensionless, but B and D represent impedance and admittance, with units of ohms and siemens, respectively. When combined with the appropriate line model for the transmission difference and appropriate application of Kirchoff’s Laws, it’s possible to further define these chain parameters with additional constraints like the propagation constant: that contains the attenuation constant alpha and phase constant beta. The attenuation constant alpha represents line loss due to series resistance/shunt conductance, while the phase constant tracks the change in phase due to series reactance/shunt susceptance.

With these constants, it’s possible to directly compare a distributed transmission line (i.e., a long transmission line) and a lumped medium transmission line. After applying the appropriate network analysis equations, both the impedance and admittance of the long-line transmission model are equivalent to that of the medium-line transmission model after folding in a correction factor for adjustments.

The simplest analysis for transmission lines ignores loss entirely to eliminate many of these extra performance factors of the circuit. While this is less accurate than the approaches above, a ballpark calculation of network parameters can be valuable in certain situations before graduating to more complex models. The attenuation constant alpha goes to zero for the propagation constant, and the remaining equation is:

𝛾 =jβ

In terms of parameters, the surge impedance of the line, calculated as the phase constant of the impedance and admittance, is which produces the familiar equation for the characteristic impedance of the transmission line.