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LINEAR CIRCUIT ANALYSIS EE-111 ENGR. IMRAN AZIZ Chapter 5: Transformers and Amplifiers • Dependent Sources • Circuit Analysis with Dependent Sources • The Ideal Transformer • Amplifiers Introduction: • Both amplifiers and transformers are examples of two-port • A hi-fi system is more sophisticated two-port which takes weak input signal and provides amplified output • Both input and output port exhibit individual i-v characteristics • Distinguishing feature of two port is relationship between input and output signals, called the transfer characteristics. • This inter-dependence between the ports is modeled with the help of dependent sources. 5.1 Dependent Sources: • Dependent sources are indispensable ingredient in modeling of transformer and amplifier • A dependent source acts much like an independent source except that its voltage/current is being controlled by some other voltage/current in the circuit. • This can also release or absorb power • Resistance Transformation: • A dependent source is unable to initiate any voltage/current in a circuit; an independent source is required to do that. • Then what is the role of Dependent Source? • Dependent source can be regarded as a generalized concept of resistance • Resistance imposes constraint between voltage and current of same branch • Dependent source impose constraint between different branches • This is the reason we’ll often get quite unexpected results based on our experience of independent sources. • We’ll understand it through an example: • We wish to find the Req of below circuit: • The dependent source monitors port voltage vx being fed from left and drives the right side by multiplying it with k. • Here we can’t suppress the source, because suppressing the source would mean to make k = 0 (short circuit) • Proper way is to add the test voltage source at open terminals • Then Req = v / i • By Ohm’s Law: • So, Req can have variety of different values depending upon k. • Req can have infinite or even negative values due to the presence of dependent source in the circuit, which indirectly affects the voltage across R. • The role of dependent source can be more appreciated by taking some specific element values; let v = 1V and R = 1 Ohm. • Now we’ll examine circuit behavior for different values of k. • k = 0: Short circuit. i = 1 A, Req = R • 0< k < 1: increasing value of k in given range, increases the Req of the circuit. • k < 0 : decreasing k below zero, makes Req approach to zero. • k > 1: Negative resistance behavior. Positive resistance: test source is delivering power Negative resistance: test source is receiving power • Transistor Modelling: • An important application of dependent source is transistor modeling. • Current Gain (Beta) = Ic / IB. • IE = IC + IB. • IC is constant in the circuit regardless of the value of VCE. • Circuit symbol and model of npn BJT • In circuits, having transistors, we can simply replace it with its model and use the circuit analysis techniques to calculate the desired values. • Voltage Divider Circuit: • These equations are repeatedly used when dealing with transistors. 5.2 Circuit Analysis with Dependent Sources • We’ll use the same techniques of circuit analysis as before. • A few things shall be taken care of: • In general, values of controlling signals i.e ix and vx are not known but are found by calculations of different equations • Dependent sources can’t be suppressed to find Req; this would invalidate constraint between controlled source and controlling signal. • However, independent sources can be suppressed to find Req because their values are independent of rest of circuitry. • Thevenin’s and Nortan’s Equivalent: • Nodal / Loop analysis can be used to find the Thevenin’s or Norton’s Equivalent of one-port with dependent sources (Method 1): • Generally open circuit values of vx / ix shall be different from short circuit values. • Method 2 can also be used to find Req by suppressing all the independent sources and applying a test voltage in circuit. Then Where v is voltage of test source. • Source transformation techniques are also applicable in circuits with dependent sources but we should avoid tempering controlling signals. • Concluding Remarks: • When looking for Thevenin’s / Norton’s Equivalents, its good practice to pause and try developing a strategy to minimize the computational effort • Try to analyze that which method is easier for specific circuit. • When looking for Req, if voc and isc are zero, then we’ve to move towards Method 2. • If there is no independent source in circuit; obviously voc and isc are zero, then we also have to move towards Method 2. 5.3 The Ideal Transformer • Transformer is our first example of two port device. • Two coils, called primary and secondary, are wound around magnetic core • Primary coil plays role of i/p port while secondary of o/p port • N1 and N2 are no. of turns in windings, then, the turn ratio of transformer is • The function of dots is to identify signal polarities. • If v1 and v2 are chosen to be positive at dotted terminal, then they are in phase, otherwise, they are out of phase. • Likewise, currents are in phase if one enters in dotted terminal and other leaves the dotted terminal. • Dots can be omitted when phase relationship is unimportant. • Ideally transformer dissipates no energy. i.e power absorbed via primary equals the power released by secondary. • n > 1:Output voltage is greater than input voltage. • n < 1:Output voltage is less than input voltage • n = 1:Provides electrical isolation and suppress dc component. • Circuit Model of Ideal Transformer: • As v2 depends upon v1 and n, regardless of load, so secondary can be modeled as dependent voltage source of value nv1. • Likewise, i1 depends upon load and n, regardless of v1, so primary can be modeled as dependent current source of value ni2. THE END