Miller's Theorem

3 Key excerpts on "Miller's Theorem"

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  eBook - ePub

  Introductory Electrical Engineering With Math Explained in Accessible Language

  • Magno Urbano(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  24 Millman’s Theorem : Circuit Analysis

  24.1 Introduction

  In this chapter, we examine Millman’s theorem, another very useful set of techniques for circuit analysis, specifically for circuits containing multiple voltage sources.

  24.2 Millman’s Theorem

  Millman’s theorem states that any circuit containing multiple voltage sources, each one in series with its own resistance, can be replaced by one voltage source (VEQ ) in series with a resistance (REQ ) (see Figure 24.1 ).

  Figure 24.1

  Millman’s theorem.

  24.2.1 The Theory

  According to Millman’s theorem, each of the voltage sources and their respective resistors produce a current, and the sum of all currents is equal to the total current produced by the circuit, as shown in Figure 24.2 .

  Figure 24.2

  Sum of individual currents.

  Mathematically, this can be expressed as

  or generically as the following postulate.

  MILLMAN’S POSTULATE

  • i represents the circuit’s total current, in Amperes.
  • ix represents each of the parallel currents, in Amperes.

  The individual element being considered by the theorem is a voltage source in series with a resistor, which we will call a voltage source block.

  According to the Millman’s theorem, these voltage source blocks are in parallel, each one contributing with its own current to the circuit’s total current, as shown in Figure 24.2

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  eBook - ePub

  Computational Electromagnetics with MATLAB, Fourth Edition

  • Matthew N.O. Sadiku(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  7   Transmission-Line-Matrix Method  

  Excuses are the most important tools of non-achievers .

  —Unknown   7.1Introduction

  The link between field and circuit theories has been exploited in developing numerical techniques to solve certain types of partial differential equations (PDEs) arising in field problems with the aid of equivalent electrical networks [1 ]. There are three ranges in the frequency spectrum for which numerical techniques for field problems in general have been developed. In terms of the wavelength λ and the approximate dimension ℓ of the apparatus, these ranges are [2 ]:

  λ > > ℓ

  λ ≈ ℓ

  λ < < ℓ

  In the first range, the special analysis techniques are known as circuit theory ; in the second, as microwave theory ; and in the third, as geometric optics (frequency independent). Hence, the fundamental laws of circuit theory can be obtained from Maxwell’s equations by applying an approximation valid when λ >> ℓ . However, it should be noted that circuit theory was not developed by approximating Maxwell’s equations, but rather was developed independently from experimentally obtained laws. The connection between circuit theory and Maxwell equations (summarizing field theory) is important; it adds to the comprehension of the fundamentals of electromagnetics. According to Silvester and Ferrari, circuits are mathematical abstractions of physically real fields; nevertheless, electrical engineers at times feel they understand circuit theory more clearly than fields [3 ].

  The idea of replacing a complicated electrical system by a simple equivalent circuit goes back to Kirchhoff and Helmholtz. As a result of Park’s [4 ], Kron’s [5 ,6 ] and Schwinger’s [7 ,8 ] works, the power and flexibility of equivalent circuits become more obvious to engineers. The recent applications of this idea to scattering problems, originally due to Johns and Beurle [9

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  eBook - ePub

  Amplifiers and Oscillators

  Optimization by Simulation

  • François de Dieuleveult, François De Dieuleveult(Authors)
  • 2018(Publication Date)
  • ISTE Press - Elsevier

    (Publisher)

  BC . The second repercussion relates to the choice of the circuit and the dimensioning of the components. Mathematical analysis shows that the gain and the high cutoff frequency are related and we must dimension the components taking into account the fundamental relations.

  Figure 2.7 Miller effect in an amplifier

  Because it is not possible, with the simple structure of the common emitter assembly, to separate the two parameters, gain and cutoff frequency, we can then look for other circuits, other configurations, which overcome these disadvantages.

  2.2 Common base amplifier

  2.2.1 Calculation of the transfer function

  A schematic diagram of the common base amplifier is shown in Figure 2.8 . As before, the elements responsible for the polarization of the transistor are assumed to be negligible for the alternative regime. Only the elements that play a major role in the frequency response of the amplifier are retained.

  Figure 2.8 Common base amplifier stage

  The transfer function of the stage is obtained by solving the system of two equations [2.39] , which describes the circuit shown in Figure 2.8 . Thanks to Millman’s theorem which allows the description of the circuit with only two equations. The system is solved by eliminating V

  BE

  between the two equations [2.39] .

  −

  V BE

  =

  V IN

  R IN

  +

  g m

  V BE

  1

  R IN

  +

  1

  R E

  +

  C BE

  s

  V OUT

  = −

  g m

  V BE

  R L

  R C

  C BC

  s + 1

     [2.39]

  The transfer function is given in [2.40] . In contrast to the previous cases, it is meaningless to look for the roots of a second-degree polynomial as this transfer function is written naturally, like the product of two terms of order 1, showing directly the two cutoff pulsations given by equations [2.41] .

  V OUT

  s

  V IN

  s

  =

  g m

  R E

  R C

  R E

  +

  R IN

  +

  g m

  R E

  R IN

  1

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