# 7

# FREQUENCY-DOMAIN ANALYSIS OF MULTICONDUCTOR LINES

In this chapter, we consider the *frequency-domain* solution of the MTL (multi-conductor transmission lines) equations for an (*n* + 1)-conductor line where the excitation sources are sinusoids that have been applied for a sufficient length of time so that the line voltages and currents are in *steady state*. We will find that, using matrix notation, the solutions to these phasor MTL equations bear a striking similarity to the solutions for two-conductor lines obtained in the previous chapter. The primary method we will use to solve these phasor MTL equations is to decouple them with a similarity transformation [A.4, 1–4]. This frequency-domain solution of the MTL equations has a long history [5–10].

## 7.1 THE MTL TRANSMISSION-LINE EQUATIONS IN THE FREQUENCY DOMAIN

We assume that the time variation of the line excitation sources is sinusoidal and the line is in steady state. Therefore, the line voltages and currents are also sinusoidal having a magnitude and a phase angle and the same frequency as the sources. Thus, the *n* line voltages and *n* line currents in the time domain are obtained from their *phasor* form in the usual fashion as

where Re{·} denotes the *real part* of the enclosed complex quuantity, and the *phasor* voltages and currents ...