Network Functions and Fourier Analysis

Contents

**Recall Terminal Pairs or Ports** 1

**Recall Poles and Zeros in a Network Function** 3

**List out restrictions on location of Poles and Zeros in Driving Point Functions** 4

**Recall the necessary conditions for Transfer Functions** 4

**Describe Time-Domain behaviour of a function from the Pole-Zero Plot** 5

**Describe Fourier Series and its different form** 6

**Explain evaluation of Fourier-Coefficients** 7

**Describe Symmetries related to Fourier Coefficients** 8

**Describe Dirichlet Conditions to write the Fourier Series for a Periodic Function** 8

**Recall Terminal Pairs or Ports**

Terminal pairs, also known as ports, are specific points or connections in a network or circuit that serve as input and output interfaces for signals. They are used to analyze and describe the behavior of networks and circuits in terms of their input-output relationships.

In the context of network functions, terminal pairs or ports are associated with specific variables or signals. The behavior of a network can be represented using network functions, which describe how the output variables or signals relate to the input variables or signals at the network’s ports.

Here are a few examples of network functions and their associated terminal pairs:

**Transfer Function:**The transfer function describes the relationship between the input and output signals in a linear time-invariant (LTI) system. It is typically represented as H(s), where s is the complex frequency variable. The terminal pairs for a transfer function are the input port and the output port of the system.**Scattering Parameters (S-parameters):**S-parameters describe the behavior of linear electrical networks, such as amplifiers or filters, in terms of how they transmit or reflect signals at various frequencies. S-parameters are typically represented as S11, S12, S21, S22, and so on. The terminal pairs for S-parameters are specific input and output ports of the network.**Impedance Parameters (Z-parameters):**Z-parameters characterize the relationship between the voltage and current at different ports of a network. They are represented as Z11, Z12, Z21, Z22, and so on. The terminal pairs for Z-parameters are the specific voltage and current pairs at the input and output ports.**Hybrid Parameters (H-parameters):**H-parameters describe the behavior of a linear bilateral network in terms of voltage and current relationships at its ports. They are represented as h11, h12, h21, h22, and so on. The terminal pairs for H-parameters are the specific voltage and current pairs at the input and output ports.

These are just a few examples of network functions and their associated terminal pairs. The specific terminal pairs and network functions used to describe a circuit or system depend on its characteristics, topology, and the desired analysis or description of its behavior.

**Describe different Network Functions for One-Port and Two-Port Networks a) Transform Immittance Function b) Transfer Immittance Function c) Voltage Transfer Function d) Current Transfer Function**

In circuit analysis, network functions are mathematical models that describe the relationship between input and output signals in a circuit. The four main types of network functions for one-port and two-port networks are:

a) Transform immittance function: This network function relates the input and output immittances (i.e., admittance or impedance) of a one-port network. It is denoted by T and is defined as the ratio of the output immittance to the input immittance. The transform immittance function can be used to describe the behavior of passive components, such as resistors, capacitors, and inductors.

b) Transfer immittance function: This network function relates the input and output immittances of a two-port network. It is denoted by T and is defined as the ratio of the output immittance to the input immittance. The transfer immittance function can be used to describe the behavior of passive networks, such as filters and transmission lines.

c) Voltage transfer function: This network function relates the output voltage of a two-port network to its input voltage. It is denoted by V and is defined as the ratio of the output voltage to the input voltage. The voltage transfer function can be used to describe the behavior of active components, such as amplifiers and oscillators.

d) Current transfer function: This network function relates the output current of a two-port network to its input current. It is denoted by I and is defined as the ratio of the output current to the input current. The current transfer function can be used to describe the behavior of active components, such as transistors and operational amplifiers.

Overall, these network functions provide a powerful tool for analyzing and designing complex circuits, as they allow us to predict the behavior of a circuit based on its input and output signals.

**Recall Poles and Zeros in a Network Function**

In the context of network analysis, poles and zeros are important characteristics of the transfer function of a network. The transfer function is a mathematical function that describes the relationship between the input and output signals of the network.

A pole is a value of s (the Laplace transform variable) for which the transfer function becomes infinite. In other words, a pole is a value of s that makes the denominator of the transfer function equal to zero. The poles of the transfer function determine the stability and transient response of the network. A stable network has all its poles in the left-half of the complex plane.

A zero is a value of s for which the transfer function becomes zero. In other words, a zero is a value of s that makes the numerator of the transfer function equal to zero. The zeros of the transfer function determine the steady-state response of the network.

The pole-zero plot is a graphical representation of the transfer function of a system. It is a plot of the locations of the poles and zeros of the transfer function in the complex plane. The pole-zero plot is useful for understanding the behavior of the system in both the time and frequency domains.

Overall, the poles and zeros of a network function are important characteristics that provide insight into the stability, transient response, and steady-state response of the network. They can be determined from the transfer function of the network and their location in the complex plane can be analyzed using the pole-zero plot.

**List out restrictions on location of Poles and Zeros in Driving Point Functions**

Driving point functions are transfer functions that describe the relationship between the input and output signals at a single point in a network. The poles and zeros of the driving point function are important characteristics that determine the behavior of the network.

The following are the restrictions on the location of poles and zeros in driving point functions:

- Poles must be located in the left-half of the complex plane for the network to be stable.
- The number of poles must be less than or equal to the number of network variables.
- Zeros can be located anywhere in the complex plane, but it is desirable to have them in the left-half of the complex plane for stability reasons.
- If a zero is located on the imaginary axis, it must be a simple zero.
- The number of zeros must be less than or equal to the number of poles.
- The driving point function must be a proper rational function, i.e., the degree of the numerator must be less than or equal to the degree of the denominator.

Overall, the restrictions on the location of poles and zeros in driving point functions ensure the stability and proper functioning of the network. By analyzing the poles and zeros of the driving point function, we can gain insight into the behavior of the network and make improvements if necessary.

**Recall the necessary conditions for Transfer Functions**

The transfer function of a network is a mathematical function that describes the relationship between the input and output signals of the network. The transfer function is a fundamental concept in circuit analysis, and it can be used to design and analyze complex circuits.

The necessary conditions for the transfer function of a network are as follows:

**Linearity:**The network must be linear, which means that the response of the network to a sum of inputs is the sum of the responses to each input.**Time-invariance:**The network must be time-invariant, which means that the response of the network to a given input does not depend on the time at which the input is applied.**Lumpedness:**The network must be lumped, which means that it must consist of discrete circuit elements that are physically separable from each other. This condition ensures that the network can be analyzed using circuit theory.**Causality:**The network must be causal, which means that the output of the network can only depend on its past and present inputs. This implies that the transfer function must be a rational function with all its poles in the left-half of the complex plane.**Stability:**The network must be stable, which means that its output must be bounded for all bounded inputs. This implies that the transfer function must be a rational function with all its poles in the left-half of the complex plane.

Overall, these conditions ensure that the transfer function is a well-defined mathematical function that accurately describes the behavior of the network. By satisfying these conditions, we can use the transfer function to design and analyze complex circuits, and to predict the behavior of the network under different input conditions.

**Describe Time-Domain behaviour of a function from the Pole-Zero Plot**

The pole-zero plot is a graphical representation of the transfer function of a system. It is a plot of the locations of the poles and zeros of the transfer function in the complex plane. The poles and zeros of the transfer function are important characteristics of the system, and they provide valuable insights into the behavior of the system in both the frequency domain and the time domain.

In the time domain, the pole-zero plot can provide valuable information about the stability, transient response, and steady-state response of the system. The location of the poles and zeros in the complex plane can determine the nature and frequency of the response of the system.

**Stability:**The stability of the system can be determined from the location of the poles in the complex plane. If all the poles are in the left-half of the complex plane (i.e., the real parts of the poles are negative), the system is stable. If any pole is in the right-half of the complex plane (i.e., the real part of the pole is positive), the system is unstable.**Transient response:**The transient response of the system can be determined from the location of the poles in the complex plane. The closer the poles are to the imaginary axis, the slower the decay of the transient response. The further the poles are from the imaginary axis, the faster the decay of the transient response.**Steady-state response:**The steady-state response of the system can be determined from the location of the zeros in the complex plane. If the zero is close to the origin, the steady-state response of the system will be small. If the zero is far from the origin, the steady-state response of the system will be large.

Overall, the pole-zero plot provides valuable information about the behavior of the system in both the frequency domain and the time domain. By analyzing the location of the poles and zeros in the complex plane, we can predict the stability, transient response, and steady-state response of the system.

**Describe Fourier Series and its different form**

Fourier series is a mathematical technique used to represent periodic functions as an infinite sum of sine and cosine functions. The Fourier series of a periodic function f(x) with period T is given by:

f(x) = a_{0} + ∑ [a_{n} cos(nωx) + b_{n} sin(nωx)]

where ω = 2π/T, and the coefficients an and bn are given by:

a_{n} = (2/T) ∫T/2 -T/2 f(x) cos(nωx) dx, b_{n} = (2/T) ∫T/2 -T/2 f(x) sin(nωx) dx

The constant term a_{0} represents the average value of the function f(x) over one period, and the coefficients an and bn represent the amplitudes of the cosine and sine components, respectively, at each harmonic frequency nω.

The Fourier series can be expressed in different forms, depending on the type of function being analyzed and the properties of the coefficients:

**Trigonometric form:**This is the standard form of the Fourier series, as shown above, which represents the periodic function as a sum of cosine and sine terms.**Exponential form:**The Fourier series can also be expressed in terms of complex exponential functions.**Symmetry form:**If the periodic function is even or odd, then the Fourier series coefficients can be expressed in a simplified form.

**Explain evaluation of Fourier-Coefficients**

The Fourier series represents a periodic function as an infinite sum of sinusoidal functions, and the Fourier coefficients are the amplitudes of the sine and cosine functions that make up the series. The evaluation of the Fourier coefficients involves calculating the values of these coefficients using integrals.

For a periodic function f(x) with period T, the Fourier coefficients are given by:

a_{n} = (2/T) ∫T/2 -T/2 f(x) cos(nωx) dx

b_{n} = (2/T) ∫T/2 -T/2 f(x) sin(nωx) dx

where n is an integer, ω = 2π/T, and T/2 and -T/2 are the limits of integration over one period of the function.

To calculate the Fourier coefficients, we first need to determine the integrals of the product of the function f(x) with the cosine and sine functions. This can be done analytically for simple functions, or numerically using numerical integration techniques for more complex functions.

**Describe Symmetries related to Fourier Coefficients**

Symmetry properties of a periodic function can provide useful information about its Fourier series coefficients. There are several types of symmetry that can be present in a periodic function, which can be used to simplify the calculation of the Fourier coefficients or to infer certain properties of the function.

**Even symmetry:**A periodic function is said to be even if it satisfies f(-x) = f(x) for all x. If a function is even, then all of its Fourier series coefficients bn are zero because the sine function is odd and integrates to zero over the range of the function.**Odd symmetry:**A periodic function is said to be odd if it satisfies f(-x) = -f(x) for all x. If a function is odd, then all of its Fourier series coefficients an are zero because the cosine function is even and integrates to zero over the range of the function.**Half-wave symmetry:**A periodic function is said to have half-wave symmetry if it satisfies f(x+T/2) = -f(x) for all x, where T is the period of the function. If a function has half-wave symmetry, then all of its Fourier series coefficients an are zero because the cosine function is even and integrates to zero over half of the period of the function. The Fourier series coefficients bn will only be non-zero for odd values of n.**Quarter-wave symmetry:**A periodic function is said to have quarter-wave symmetry if it satisfies f(x+T/4) = -f(x) for all x. If a function has quarter-wave symmetry, then its Fourier series coefficients will have a specific pattern: the even coefficients an will be zero, and the odd coefficients b_{n}will alternate in sign and decrease in magnitude as n increases.

**Describe Dirichlet Conditions to write the Fourier Series for a Periodic Function**

The Dirichlet conditions are a set of mathematical conditions that must be satisfied by a periodic function in order for it to be expressed as a Fourier series. The conditions are as follows:

- The function must be piecewise continuous over its period.
- The function must have a finite number of maxima and minima over its period.
- The function must have a finite number of discontinuities (jumps) over its period.

These conditions ensure that the Fourier series converges to the original function and that there are no abrupt changes or singularities in the series representation.

The first condition ensures that the function is well-behaved and can be integrated and differentiated over its period. The second condition ensures that the function has a finite number of turning points, which is necessary for the Fourier series to converge uniformly. The third condition ensures that the function has no abrupt changes or jumps, which could lead to large oscillations in the Fourier series.