## Introduction

1.1 Linear Networks

A network is said to be linear if it is both homogenous and additive. Consider a system shown in Fig. 1.1.

The system produces outputs Y_{1}, Y_{2} for the inputs X_{1}, X_{2}, respectively. For this system to be linear it has to satisfy the condition,_{}

For a zero input, the output of the system must be zero. If both these conditions are satisfied, the system is said to be a linear system.

Example

For the transfer characteristics given in Fig. 1.2, Y_{1} and Y_{2} are outputs for the inputs X_{1} and X_{2}, respectively

When X_{1} X_{2} is applied to the system, the output is supposed to be Y_{1} Y_{2,}

which is equal to 4.5, but the actual output obtained from its characteristics is for X_{1} X_{2} = 1 is 2.5. Hence, the system is not linear.

Linear systems are important because of the fact that the response of the system to one input is sufficient to find response of the same for all possible inputs. Generally, impulse response is used to characterize a system. All practical systems are non-linear.

**1.2 A Simple Non Linear System **

Fig 1.3: A Simple non linear system

A simple non linear system, shown in Fig. 1.3, consists of a resistor R1 and a diode D1 connected in series with a voltage source V1. Applying kirchoff’s voltage law in this loop we get,

The current through the diode I is given by the equation,

It can be seen from Eq.(1.6) that the system is non linear. The voltage is non linearly related to the current in the loop.

**1.2.1 Incremental linearity**

Even though the actual input and output of a system are related in a non linear fashion, an incremental change in input can always be related to incremental change in the output in a linear way as long as the incremental change is small.

For obtaining the absolute value of current and voltage of an element, first the operating point has to be found. This has to be followed by deriving an incremental equivalent circuit. The absolute values of current and voltage can then be found by adding quiescent value of current/voltage with incremental value of current/voltage.

Total voltage across the diode = Quiescent voltage Incremental voltage.The incremental equivalent circuit is as shown below.

Figure 1.8 Incremental equivalent circuit

**1.2.2 Notion of incremental linearity in case of a network with more than one non linear element**

Consider the network shown in Fig. 1.9. It has three non linear elements E1,E2,E3. V_{1 , }V_{2} , V_{3} are voltages across the elements E1, E2, E3 respectively.I_{1 , } I_{2, }I_{3} are currents through the elements E1, E2, E3 respectively.

Now, when small signals v_{A} and v_{B} are added to the sources V_{A} and v_{B}, respectively, the currents will change from I_{1} , I_{2} to I_{1} i_{1,} I_{2} i_{2} respectively. The voltages V_{1}, V_{2}, V_{3} will change to V_{1} v_{1,} V_{2} v_{2,} V_{3} v_{3} respectively. Replacing the elements E1,E2,E3 with voltage sources equal to V1 v_{1,} V_{2} v_{2, }V_{3} v_{3} respectively, we get an equivalent circuit as shown in the Fig. 1.10.

Applying the Eqs.( 1.19, 1.20), the circuit the sources V_{1}, V_{2}, V_{3} can be elliminated.

In order to find the incremental equivalent network, the quiescent currents I_{1}, I_{2}, I_{3} must be eliminated. For this purpose, current sources are carefully added across each branch such that the node voltages are not modified.

The incremental equivalent network of the circuit is shown in Fig. 1.14. It can be seen that, like absolute node voltages and branch currents, the incremental node voltages and branch currents also follow Kirchoff's current law and Kirchoff's voltage law.

**1.2.3 Incremental equivalents**

The characteristics of an independent voltage source is as shown in Fig. 1.15. For a small change in current, the voltage supplied by the source does not change. Hence, the incremental equivalent of an independent voltage source is a short circuit.

The characteristics of an independent current source is as shown in Fig. 1.16.For a small change in voltage, the current supplied by the source does not change. Hence, the incremental equivalent of an independent current source is an open circuit.

Definition for a small signal Consider a non linear element which has following

**1.2.4 Nonlinear two port network**

Consider a non linear two port network shown in Fig. 1.19 with following relations.

**1.3 Amplifiers**

Amplifiers are two port networks which are used to amplify weak or attenuated signals. An amplifier is connected to the source, V_{s} on one side which has an internal resistance, R_{S} and to a load, R_{L} on the other side. Ideally, the gain of an amplifier,