Chapter 30: Introduction to Network Analysis [cite: 375, 376]

Introduction

Network Analysis is a structured technique used to mathematically analyse a network of interconnected components [cite: 380-382]. This is applied when a circuit is too complex to simply be analysed by series and parallel rules [cite: 383]. In such cases, other means are required to find current and voltage drops [cite: 384, 388].

Network analysis is used in heavily complex systems. When these systems are communicating signals with information content, there is a primary concern with getting signals from one point to another with maximum efficiency [cite: 390-401].

Laws Determining Current and Voltage in AC Networks [cite: 404, 405]

The fundamental laws and relationships governing these networks include:

• Ohm’s Law equivalent for AC: i = v / z, where z is the complex impedance and v is the voltage drop across the impedance [cite: 406-413].
• The laws for impedance in series and parallel [cite: 414]:
  For series: Z_T = Z₁ + Z₂ + Z₃ + … + Z_n [cite: 415, 416]
  For parallel: 1 / Z_T = 1 / Z₁ + 1 / Z₂ + 1 / Z₃ + … + 1 / Z_n [cite: 417-427]

Kirchhoff’s Laws

• Kirchhoff’s Current Law: At any point in an electrical circuit, the phasor sum of the currents flowing towards that junction is equal to the phasor sum of the currents flowing away from that junction [cite: 430-434].
• Kirchhoff’s Voltage Law: In any closed loop in a network, the phasor sum of the voltage drops (i.e., the products of current and impedance) taken around the loop is equal to the phasor sum of the EMFs acting in that loop [cite: 435-440].

Kirchhoff’s laws can be used to determine the voltage and current of any point in the circuit, often by understanding mesh currents and node voltages [cite: 442].

Complex Theorems for D.C and A.C. [cite: 443]

These are more complex theorems that can be used as alternatives to the use of Kirchhoff’s laws to solve problems involving both D.C and A.C electrical networks [cite: 444-448]. They include:

• (a) The Superposition Theorem (Chapter 32) [cite: 449]
• (b) Thevenin’s Theorem (Chapter 33) [cite: 449]
• (c) Norton’s Theorem (Chapter 33) [cite: 450]
• (d) The Maximum Power Transfer Theorem (Chapter 35) [cite: 451, 452]

In addition to these theorems, star-delta (T-π) and delta-star (π-T) methods are often used to simplify circuits [cite: 456-458].

Solutions to Simultaneous Equations Using Determinants [cite: 469]

For n loops in a circuit, n simultaneous equations with n unknowns are formed using Kirchhoff’s laws. One way to solve this is by elimination and substitution. Another powerful way is by using Determinants [cite: 471-475].

Network Analysis Using Kirchhoff’s Laws [cite: 479]

If the current flowing through each branch is required, the following three-step procedure may be used [cite: 483-486]:

1. Label the branch current; the direction is arbitrary, but it is useful to assume they are leaving the positive voltage end [cite: 487-495].
2. Divide the circuit up into loops, and then apply Kirchhoff’s Voltage Law (KVL) to each loop in turn. Loop direction (clockwise or anticlockwise) is a matter of choice and does not have to be the same for every loop [cite: 496-513].
3. Solve the simultaneous equations to find the current in a particular part of the circuit [cite: 514-519].

Problems & Examples