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 R eq of below circuit: The dependent source monitors port voltage v x 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 R eq = v / i By Ohm’s Law: So, Req can have variety of different values depending upon k.

R eq 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, R eq = R 0< k < 1: increasing value of k in given range, increases the R eq of the circuit. k < 0 : decreasing k below zero, makes R eq 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) = I c / I B. I E = I C + I B. I C is constant in the circuit regardless of the value of V CE. 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 i x and v x are not known but are found by calculations of different equations Dependent sources can’t be suppressed to find R eq ; this would invalidate constraint between controlled source and controlling signal. However, independent sources can be suppressed to find R eq 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 v x / i x 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 v oc and i sc are zero, then we’ve to move towards Method 2. If there is no independent source in circuit; obviously v oc and i sc 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 v 2 depends upon v 1 and n, regardless of load, so secondary can be modeled as dependent voltage source of value nv 1. Likewise, i 1 depends upon load and n, regardless of v 1, so primary can be modeled as dependent current source of value ni 2.

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