Impedance and Admittance Formulas

To analyze a transient in a distributed-parameter line, a traveling-wave theory is explained for both single- and multiconductor systems. A method to introduce a velocity difference and attenuation in the multiconductor system is described together with a field test results. Impedance and admittance formulas of not ordinary conductors, such as a finite-length conductor and a vertical one, are also explained. 

Application examples of the theory described in this chapter are given so as to understand the necessity of the theory. 

Finally, the Electromagnetic Transients Program (EMTP), which has been widely used all over the world, is briefly explained. 

It should be noted that all the theories and formulas in this chapter are based on transverse electromagnetic (TEM) wave propagation. 

In general, the impedance and admittance of a conductor are composed of the conductor internal impedance Z and the outer-media impedance Zq. The same is applied to the admittance [1]  

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First, this book will illustrate a transient on a single-phase line from a physical viewpoint, and how it can be solved analytically by an electric circuit theory. The impedance and admittance formulas of an overhead line will also be described. Approximate formulas that can be computed using a pocket calculator will be explained to show that a transient can be analytically evaluated via hand calculation. Since a real power line contains three phases, a theory to deal with a multiphase line will be developed. Finally, the book describes how to tackle a real transient in a power system. A computer simulation tool is necessary for this— specifically the well-known simulation tool Electro Magnetic Transients Program (EMTP), originally developed by the U.S. Department of Energy, Bonneville Power Administration— which is briefly explained in Chapter 1. 

In this chapter, a theory of distributed-parameter circuits is explained starting from the approximate impedance and admittance formulas of an overhead conductor. The derivation of the approximate formulas is described from the viewpoint of the physical behavior of current and voltage on a conductor. 

This section explains impedance and admittance formulas of nonuniform lines, such as finite-length horizontal and vertical conductors based on a plane wave assumption. The formulas are applied to analyze a transient on a nonuniform line by an existing circuit theory-based simulation tool such as the EMTP [9,11]. The impedance formula is derived based on Neumann s inductance formula by applying the idea of complex penetration depth explained earlier. The admittance is obtained from the impedance assuming the wave propagation velocity is the same as the light velocity in free space in the same manner as an existing admittance formula, which is almost always used in steady-state and transient analyses on an overhead line. 

The penetration depth is physically defined as the depth for an electromagnetic wave penetrating into a conductor when the wave hits the conductor surface. The physical concept of the penetration depth is very useful to explain the behavior of a current and a voltage on a conductor and also to derive impedance and admittance formulas of various conductor shapes and geometrical configuration. However, it should be reminded that the concept is based on TEM wave propagation and thus is not applicable to non-TEM propagation. Also, remind that it is just an approximation. 

The significance of the proximity effect on conductor impedance is well-known. There are a number of papers that derive a theoretical formula of impedance and admittance [23, 24, 25, 26, 27, 28-29] and discuss impedance variation due to proximity based on numerical simulations [30, 31, 32-33]. The proximity effect may be very important in a steady-state power system s performance from a power loss viewpoint some quantitative results at a frequency of 50 or 60 Hz have been published [34, 35, 36-37]. 

The applicable range of Equation 6.7 needs to be discussed. Figure 6.32 illustrates a typical gas pipeline used in Japan [59,60]. Its impedance and admittance are easily evaluated by hand based on approximate formulas [45,59,63], as given in Appendix 6.A.4, and are given here as... 

Theory of Distributed-Parameter Circuits and Impedance/Admittance Formulas... 

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