Network Synthesis

Contents

**Describe relationship between Pole positions and Stability** 1

**Describe and apply Routh-Hurwitz stability criterion to determine stability of a polynomial** 2

**Describe and construct Bode diagram** 4

**Describe the fundamental concepts and procedure of Network Synthesis** 5

**Determine whether the given function is Hurwitz polynomial or not** 6

**Define Positive Real Functions** 7

**Determine whether the given function is Positive Real Function or not** 8

**Synthesise LC, RC, and RL networks in Foster’s I form** 9

**Synthesise LC, RC, and RL networks in Foster’s II form** 10

**Synthesise LC, RC, and RL networks in Cauer’s I form** 11

**Synthesise LC, RC, and RL networks in Foster’s II form** 12

**Describe relationship between Pole positions and Stability**

In network theory, the pole positions of a system are closely related to its stability. The poles of a system are the values of s (complex numbers) for which the transfer function of the system becomes infinite or undefined. They are important indicators of the system’s dynamic behavior and response.

The relationship between pole positions and stability can be understood as follows:

- Stability Criterion:

In general, for a system to be stable, all the poles of its transfer function must have negative real parts. This criterion is known as the “Pole Placement Criterion” or the “Nyquist Stability Criterion.”

- Stable Poles:

If all the poles of a system have negative real parts, the system is considered stable. Stable poles indicate that the system’s response will decay over time, ensuring the stability of the system.

- Unstable Poles:

If any pole of a system has a positive real part, the system is considered unstable. Unstable poles indicate that the system’s response will grow or oscillate over time, leading to instability and undesirable behavior.

- Marginally Stable Poles:

Marginally stable poles are poles with zero real parts. In this case, the system may exhibit critical behavior, and its stability depends on the presence of other poles with negative real parts. Marginally stable systems can be unstable or stable depending on the nature of the remaining poles.

- Relationship with Transfer Function:

The pole positions directly affect the behavior of the transfer function and, consequently, the system’s stability. The location of the poles in the complex plane determines the system’s response characteristics, such as damping, overshoot, settling time, and oscillations.

By analyzing the pole positions, engineers can assess the stability of a network or system and make design decisions to ensure stable and reliable operation. Techniques like pole-zero analysis, Bode plots, and root locus plots are commonly used to determine the pole positions and assess system stability.

It’s important to note that stability is a fundamental aspect of network theory and plays a crucial role in ensuring the reliable and efficient operation of various electrical, electronic, and control systems.

**Describe and apply Routh-Hurwitz stability criterion to determine stability of a polynomial**

The Routh-Hurwitz stability criterion is a mathematical method used to determine the stability of a polynomial equation with real coefficients. It provides a systematic way to determine the number of roots of the polynomial equation that lie in the right half of the complex plane, which correspond to unstable poles in the transfer function of a control system.

To apply the Routh-Hurwitz stability criterion, follow these steps:

- Write the polynomial equation in descending powers of the variable (e.g., s).
- Create a Routh array by writing the coefficients of the polynomial equation in two rows. The first row corresponds to the coefficients of the even powers of the variable, while the second row corresponds to the coefficients of the odd powers of the variable. If there are missing coefficients, replace them with zeros.
- Use the first two rows of the Routh array to generate the remaining rows. To do this, compute the determinant of a 2×2 matrix consisting of the first two elements of each row. Repeat this process for each pair of adjacent rows until the entire Routh array is filled out.
- Analyze the Routh array to determine the number of roots that lie in the right half of the complex plane. This can be done by examining the number of sign changes in the first column of the Routh array. If there are no sign changes in the first column, then all the roots of the polynomial equation lie in the left half of the complex plane, and the system is stable. If there is one sign change, then there is one root in the right half of the complex plane, and the system is marginally stable. If there are more than one sign changes, then there are multiple roots in the right half of the complex plane, and the system is unstable.

The Routh-Hurwitz stability criterion is a powerful tool for analyzing the stability of control systems. It can be used to determine the stability of a wide variety of polynomial equations, including those that arise in the design and analysis of control systems. By applying this criterion, control engineers can ensure that their systems are stable and meet specific performance criteria.

**Define Frequency Response**

Frequency response refers to the behavior of a system or device as a function of frequency. In the context of signal processing and control engineering, the frequency response of a system is the relationship between the input and output signals at different frequencies. It describes how a system responds to inputs of different frequencies and is typically represented graphically as a plot of amplitude and phase versus frequency.

The frequency response is an important characteristic of many systems, as it determines how the system will behave in response to different types of signals. For example, in control systems, the frequency response can be used to determine the stability and performance of the system. In signal processing applications, the frequency response can be used to filter out unwanted signals or to amplify specific frequency components of a signal.

The frequency response is typically represented using a transfer function, which is a mathematical relationship between the input and output signals of the system. The transfer function can be obtained through various methods, including Laplace transforms, Fourier transforms, and Z-transforms, and it can be used to analyze the system’s behavior over a range of frequencies. By understanding the frequency response of a system, engineers can design and optimize control systems and signal processing algorithms to meet specific performance requirements.

**Describe and construct Bode diagram**

A Bode diagram is a graphical representation of the frequency response of a system, which shows the amplitude and phase response of the system as a function of frequency. It is a commonly used tool in control system analysis and design.

To construct a Bode diagram, the transfer function of the system is first written in terms of its magnitude and phase response. This transfer function can be obtained using methods such as Laplace transforms, Fourier transforms, or Z-transforms.

The Bode diagram is then plotted by separately graphing the magnitude and phase response of the system on logarithmic scales. The x-axis is typically logarithmic and represents the frequency, while the y-axis represents either the magnitude response (in decibels) or the phase response (in degrees).

The magnitude response is plotted as a function of frequency on a logarithmic scale, with the magnitude expressed in decibels. The magnitude plot typically has a straight-line portion called the “asymptotic magnitude curve” that indicates the system’s overall gain at low and high frequencies. At a frequency where there is a pole or zero in the transfer function, the magnitude plot will have a peak or a dip, respectively.

The phase response is also plotted as a function of frequency on a logarithmic scale, with the phase expressed in degrees. The phase plot typically has a straight-line portion called the “asymptotic phase curve” that indicates the system’s overall phase shift at low and high frequencies. At a frequency where there is a pole or zero in the transfer function, the phase plot will have a sharp change in slope.

By analyzing the Bode diagram, engineers can determine important system characteristics such as the gain and phase margins, which are used to evaluate the stability of a control system. The Bode diagram is also useful for designing compensators to adjust the system’s performance and achieve desired specifications.

**Describe the fundamental concepts and procedure of Network Synthesis**

The fundamental concepts of network synthesis include the following:

- Impedance and Admittance Functions: These are mathematical functions that describe the electrical properties of a network. The impedance function represents the relationship between voltage and current in a network, while the admittance function represents the relationship between current and voltage.
- Synthesis Techniques: There are several techniques used in network synthesis, including the image parameter method, the Foster’s reactance theorem, and the insertion loss method. Each method involves a different approach to designing a network that meets the desired specifications.
- Realizability Conditions: These are conditions that a network must satisfy to be physically realizable. They ensure that the network can be built using practical components and that it operates within the limitations of those components.

The procedure for network synthesis typically involves the following steps:

- Specify the network requirements: This includes defining the desired frequency response, gain, phase shift, and other performance specifications.
- Choose a synthesis method: Based on the network requirements, a suitable synthesis method is selected. The choice of method depends on the complexity of the network, the desired performance specifications, and other factors.
- Design the network elements: Using the chosen synthesis method, the circuit elements such as resistors, capacitors, and inductors are designed and interconnected to create the desired network.
- Verify the network performance: The performance of the network is then analyzed and verified to ensure that it meets the desired specifications. This may involve simulations or measurements using test equipment.

**Define Hurwitz Polynomial**

A polynomial is said to be Hurwitz if all its roots lie in the left half of the complex plane. In other words, a Hurwitz polynomial has only stable roots. A polynomial with complex coefficients is Hurwitz if and only if its associated Routh-Hurwitz matrix has all its elements positive. Hurwitz polynomials are important in control system theory and signal processing, as they represent stable systems or filters.

**Determine whether the given function is Hurwitz polynomial or not**

The Routh-Hurwitz criterion states that a polynomial is Hurwitz if and only if all the elements in its Routh-Hurwitz array have the same sign. The Routh-Hurwitz array is constructed by writing the coefficients of the polynomial in a table and then calculating the entries in subsequent rows based on certain formulas.

Let’s take an example polynomial:

P(s) = s^3 + 3s^2 + 2s + 1

The Routh-Hurwitz array for this polynomial is:

| 1 2 |

| 3 1 |

| 7/3 0 |

Since all the elements in the first two columns of the Routh-Hurwitz array are positive, we can conclude that all the roots of the polynomial lie in the left half of the complex plane. Therefore, the polynomial P(s) is a Hurwitz polynomial.

**Define Positive Real Functions**

A function is said to be positive real if it satisfies the following two conditions:

- It is a real, rational function of the complex variable s, where s = σ + jω (where σ is the real part and ω is the imaginary part of s).
- It has a positive real part for all values of s lying on the jω axis (i.e., when σ = 0).

In other words, a positive real function is a function that is both real and positive when evaluated on the imaginary axis. Positive real functions have important properties in the theory of control systems, signal processing, and circuit theory, and they are closely related to the concept of physical realizability of a system.

**Determine whether the given function is Positive Real Function or not**

To determine if a given function is positive real, we need to check if it satisfies the two conditions for positive real functions:

- It is a real, rational function of the complex variable s.
- It has a positive real part for all values of s lying on the jω axis.

Let’s take an example function:

H(s) = (s+1)/(s^2+2s+2)

To check if H(s) is a positive real function, we can first evaluate it on the jω axis, i.e., when s = jω. Substituting s = jω, we get:

H(jω) = (jω+1)/(-ω^2 + 2jω + 2)

The real part of H(jω) is:

Re[H(jω)] = (ω^2 – 1)/(ω^2 + 2)

Since the denominator is always positive, we only need to check the sign of the numerator to determine if H(jω) has a positive real part. The numerator is negative for ω < 1 and positive for ω > 1. Therefore, the real part of H(jω) is negative for ω < 1 and positive for ω > 1, and it changes sign at ω = 1.

**Synthesise LC, RC, and RL networks in Foster’s I form**

Foster’s I form is a way to represent a network as a series combination of inductors and shunt (parallel) combinations of capacitors. This form is particularly useful for synthesizing LC, RC, and RL networks.

To synthesize an LC network in Foster’s I form, follow these steps:

- Determine the required poles of the network, which will determine the number and values of the inductors and capacitors needed. Let’s say we want to synthesize an LC network with two poles at frequencies ω1 and ω2.
- Calculate the normalized pole frequencies α1 and α2 using the following formulas:

α1 = ω1/ω0, α2 = ω2/ω0

where, ω0 is a reference frequency that is chosen based on the desired impedance level of the network. For example, if we want to synthesize a 50-ohm LC network, we can choose ω0 = 1/√(LC), where L and C are the values of the inductor and capacitor in the network. - Determine the values of the inductors and capacitors needed for the network using the following formulas:

L1 = 1/(ω0^2C1α1(1-α1α2))

L2 = 1/(ω0^2C2α2(1-α1α2))

C3 = α1α2/(α1+α2)

where C1, C2, and C3 are the values of the capacitors in the network.

**Synthesise LC, RC, and RL networks in Foster’s II form**

Foster’s II form is a synthesis method used to realize passive networks using LC (inductor-capacitor), RC (resistor-capacitor), and RL (resistor-inductor) elements. The LC, RC, and RL networks can be synthesized in Foster’s II form as follows:

- LC Network:

To synthesize an LC network in Foster’s II form, you need to determine the characteristic impedance (Z0) and the termination impedance (Zt) of the network.

- For a series LC network:
- Choose an inductor (L) and a capacitor (C) with values that satisfy Z0 = sqrt(L/C).
- Connect the inductor and capacitor in series.
- Connect the series LC network to the termination impedance Zt.

- For a parallel LC network:
- Choose an inductor (L) and a capacitor (C) with values that satisfy Z0 = sqrt(L/C).
- Connect the inductor and capacitor in parallel.
- Connect the parallel LC network to the termination impedance Zt.

- RC Network:

To synthesize an RC network in Foster’s II form, you need to determine the characteristic impedance (Z0) and the termination impedance (Zt) of the network.

- For a series RC network:
- Choose a resistor (R) and a capacitor (C) with values that satisfy Z0 = R/sqrt(1+C^2R^2).
- Connect the resistor and capacitor in series.
- Connect the series RC network to the termination impedance Zt.

- For a parallel RC network:
- Choose a resistor (R) and a capacitor (C) with values that satisfy Z0 = R/sqrt(1+C^2R^2).
- Connect the resistor and capacitor in parallel.
- Connect the parallel RC network to the termination impedance Zt.

- RL Network:

To synthesize an RL network in Foster’s II form, you need to determine the characteristic impedance (Z0) and the termination impedance (Zt) of the network.

- For a series RL network:
- Choose a resistor (R) and an inductor (L) with values that satisfy Z0 = R.
- Connect the resistor and inductor in series.
- Connect the series RL network to the termination impedance Zt.

- For a parallel RL network:
- Choose a resistor (R) and an inductor (L) with values that satisfy Z0 = R.
- Connect the resistor and inductor in parallel.
- Connect the parallel RL network to the termination impedance Zt.

By following these steps and selecting appropriate component values, you can synthesize LC, RC, and RL networks in Foster’s II form. This form provides a systematic approach to realizing passive networks with specific impedance characteristics and termination impedances.

**Synthesise LC, RC, and RL networks in Cauer’s I form**

Cauer’s I form is a type of network synthesis that is used to realize a given transfer function using only passive components, such as inductors and capacitors. The resulting circuit is a ladder network that alternates between series and parallel LC circuits.

The steps involved in synthesizing an LC network in Cauer’s I form are as follows:

- Obtain the partial fraction expansion of the transfer function.
- Determine the number of inductors required for the network, which is equal to the number of poles of the transfer function.
- Choose the values of the capacitors such that they form a ladder network with alternating series and parallel LC circuits.

Calculate the values of the inductors using the following formula:

- L
_{i}= 1/(omega_{i}* C_{i}), where omega_{i}is the i-th pole frequency and C_{i}is the corresponding capacitor value. - Implement the LC network using the calculated values of inductors and capacitors.

The resulting LC network will have a transfer function that matches the original transfer function within a specified frequency range. The advantage of using Cauer’s I form for network synthesis is that it produces networks with a very low sensitivity to component tolerances.

**Synthesise LC, RC, and RL networks in Foster’s II form**

Foster’s II form is a type of network synthesis that is used to realize a given transfer function using only passive components, such as inductors and capacitors. The resulting circuit is a ladder network that consists of alternating series and parallel LC circuits.

The steps involved in synthesizing an LC network in Foster’s II form are as follows:

- Obtain the partial fraction expansion of the transfer function.
- Determine the number of inductors required for the network, which is equal to the order of the transfer function.
- Choose the values of the capacitors such that they form a ladder network with alternating series and parallel LC circuits.

Calculate the values of the inductors using the following formula:

- L
_{i}= 1/(2*omega*C_{i }_{i}), where omega_{i}is the i-th pole frequency and C_{i}is the corresponding capacitor value. - Implement the LC network using the calculated values of inductors and capacitors.

The resulting LC network will have a transfer function that matches the original transfer function within a specified frequency range. The advantage of using Foster’s II form for network synthesis is that it produces networks with a simple and regular structure, which makes it easier to analyze and design.