Signals and Network Analysis

1. Signals and Network Analysis Target group 3rd year ECED Students 2 Lecture Hours 3 Tutorial Hours Contact hours: School of Computing and Electrical Engineering Eeng-3121

2. Chapter one INTRODUCTION Basic definition and representation of networks Network is an interconnection of electrical elements such as resistors, inductors, apacitors, transmission lines, voltage ources, current sources, and switches. Excitation – source of electrical energy to be connected to the network Response – output signal produced by the network in response to the input (excitation)

3. Chapter one INTRODUCTION Analysis and Synthesis Network Analysis If the network and the excitation are given, and the problem is to find current or voltage through/across elements of the network or to find their relation with other voltages or currents, the process to find the solution is called network analysis. Eg. Find the current across each resistor Network synthesis If the input and the output are given explicitly or implicitly (in the form of a function relating the two) given and the problem is to create a network that meets the given specifications, the procedure followed is called network synthesis. Eg. Synthesize a network whose voltage gain is 60

4. Chapter one INTRODUCTION Network components Passive elements are those that have no energy sources. This includes resistances, inductances, capacitances and coupled inductances (transformers) Active elements are gyrators, dependent sources, transistors, op-amps, GIC (general impedance converters), NIC (negative impedance converters), FDNR (frequency dependent negative resistance) etc. There are separate energy sources in these elements, without which they can not function. Active and passive Linear and non-linear linear elements are those that have a linear response (current or voltage) to the input (current); or elements that have linear relationship between current and voltage through/across them. Lumped and distributed Network elements usually exist in lumped manner; electrical parameters are constant. However, many network elements consist of R, L and C in a distributed manner in space Transmission line is an example where its electrical parameters are distributed along the line so that voltage and current values vary along the line

5. Chapter one INTRODUCTION Types of networks Depending on type of components that makeup the network, a network can be classified as active and passive, linear and non-linear, lumped and distributed, continuous and discrete Examples

6. Chapter one INTRODUCTION Types of networks … Depending on the number of terminals, a network can also be classified as 1-port, 2-port, … n-port. A pair of terminals such that current entering one of the terminals is the same as current leaving the other terminal is called port One-port Two-port

7. Chapter one INTRODUCTION Mathematical equations (time-domain and transformed) In laplace domain Voltage V(s) Current I(s) Generalized network element (resistor, inductor, capacitor) The relation V(s) = Z(s)I(s) is called impedance description of the device (element) Z(s) is called impedance of the element. Similarly, I(s) = Y(s)V(s) is called admittance description. (Y = 1/Z) Y(s) is called admittance of the element. Time-domain and Laplace domain equations

8. Chapter one INTRODUCTION Network function It is the ratio of zero-state response to the input in Laplace domain. There are two kinds of network functions Deriving point function The excitation and response are measured at the same set of terminals (port). There are two deriving-point functions: Deriving-point impedance (Zdp) and Deriving-point admittance (Ydp). Transfer function Measurements are taken at different ports.

9. Poles and zeros of a network function Chapter one INTRODUCTION Poles and zeros of a network function Critical frequencies Network functions are rational functions of s. In general, a network function can be written as: By factorizing each polynomial, H(s) can be written as shown. Where zi are roots of P(s) and pi are roots of Q(s). Roots of P(s) are called zeros of H(s) since H(s) becomes zero at those values (frequencies). Similarly, roots of Q(s) are calld poles of H(s) since H(s) becomes infinite at those values (frequencies) When the degree of P(s) is greater than the degree of Q(s), then H(s) goes to infinite as s goes to infinity; and hence, H(s) is said to have pole at infinity. Similarly, when the degree of P(s) is less than the degree of Q(s), then H(s) goes to zero as s goes to infinity; and hence, H(s) is said to have zero at infinity.

10. Poles and zeros of a network function Chapter one INTRODUCTION Poles and zeros of a network function … • H(s) can have three types of critical frequencies (poles and zeros) • Poles and zeros at s = 0 and s = ∞ • Poles and zeros at real frequency (σ). In the form of (s+σ1) during factorization. • Poles and zeros at complex frequency (σ + jω). in the form of (s - σ1 - jω) during factorization. For network functions, these complex poles and zeros exist in conjugates so that the product of these terms [(s - σ1 - jω) x (s - σ1 + jω)] gives the term (s2 - 2 σ1s+ σ21+ω2). • For poles and zeros on the jω –axis, σ is zero so that the above term is reduced to (s2+ω2) Example Find its pole-zero plot for the following transfer function

11. Poles and zeros of a network function Chapter one INTRODUCTION Partial fraction expansion (PFE) of network function and residues When the denominator Q(s) is factorized, H(s)can be written as: Partial fraction expansion is done by breaking up H(s) in to sum of smaller functions each containing a pole. The partial fraction expansion of H(s) is given as: Ko = (1/s)H(s) |s = ∞ K1 = (s - p1)H(s) |s = p1 K2 = (s - p2)H(s) |s = p2 ... The constants Ki are called residues of the poles. The term Kos exists only if H(s) has a pole at infinity. When H(s) contains jω –axis poles and zeros, it can be written as: Partial fraction expansion of H(s) will be

12. Poles and zeros of a network function Chapter one INTRODUCTION Partial fraction expansion of network function and residues … Example Find partial fraction expansion of the following admittance function Constants 2/3 and -2/3 are residues of poles at s = -1 and s = -4

13. Poles and zeros of a network function Chapter one INTRODUCTION Continued fraction expansion (CFE) Continued fraction expansion of a network function is accomplished by long division where the division continues by taking the reciprocal of remainder functions. The polynomials qi(s) are quotients of the sunsquent divisions. Example Express the following impedance function using continued fraction expansion