Chapter 8 – Methods of Analysis Lecture 8 by Moeen Ghiyas 13/08/2015 1

Introduction Current Sources... (Voltage Sources already done) Source Conversions Current Sources in Parallel & Series Branch Current Analysis Method

The step-by-step procedure learnt so far cannot be applied if the sources are not in series or parallel. In this chapter, we will try to learn analysis methods required to solve networks with any number of sources in any arrangement Source Conversion Method Branch-current analysis Mesh analysis Nodal analysis

All the methods can be applied to linear bilateral networks The term linear indicates that the characteristics of the network elements (such as the resistors and capacitor / inductor in steady state condition) are independent of the voltage across or current through them. The second term, bilateral, refers to the fact that there is no change in the behaviour or characteristics of an element if the current through or voltage across the element is reversed.

The current source is often referred to as the dual of the voltage source, where duality implies interchange ability A voltage source or battery supplies a fixed voltage, and the source current can vary; but the current source supplies a fixed current to the branch in which it is located, while its terminal voltage may vary as determined by the network to which it is applied

The interest in the current source is due primarily to semiconductor devices such as the transistor, being current-controlled devices.

Or simply we can say, A current source determines the current in the branch in which it is located and the magnitude and polarity of the voltage across a current source are a function of the network to which it is applied.

Example - Find the voltage V s and the currents I 1 and I 2 for the network of fig Solution:.Applying KCL

An ideal voltage or current source should have no internal resistance, but that’s not the case in reality For the voltage source, if R s = 0 or is so small compared to any series resistor that it can be ignored, then we have an “ideal” voltage source.

For the current source, if R s = ∞ or is large enough compared to other parallel elements that it can be ignored, then we have an “ideal” current source

Source conversions are equivalent only at their external terminals The internal characteristics of each are quite different.

We want the equivalence to ensure that the applied load of will receive the same current, voltage, and power from each source and in effect not know, or care, which source is present.

In fig below, if we solve for the load current I L, we obtain If we multiply this by a factor of 1, which we can choose to be Rs /Rs, we obtain

Example - Convert the voltage source to a current source, and calculate the current through the 4Ω load for each source.

Example - Determine the current I 2 in the network Solution:

If two or more current sources are in parallel, they may all be replaced by one current source having the magnitude and direction of the resultant, which can be found by summing the currents in one direction and subtracting the sum of the currents in the opposite direction

Current sources of different current ratings are not connected in series, just as voltage sources of different voltage ratings are not connected in parallel

Networks with two isolated voltage sources cannot be solved using the approach learnt so far However, augmenting Reduce & Return approach with source conversion techniques may provide solution at times (as already learnt) But, there is no linear dc network for which a solution cannot be found by Branch Current Analysis Method (Only method not restricted to bilateral networks)

Five steps Step 1 – Assign a current direction -arbitrary to each branch Since there are three distinct branches (cda, cba, ca), three currents of arbitrary directions (I 1, I 2, I 3 ) are assigned Step 2 – Indicate the polarities for each resistor

Step 3 KVL – The best way to determine how many times Kirchhoff’s voltage law will have to be applied is to determine the number of “windows” in the network.

Step 4 KCL – Apply Kirchhoff’s current law at the minimum number of nodes that will include all the branch currents. The minimum number is one less than the number of independent nodes of the network. Step 5 – Simultaneous solution of linear equations

Example – Apply the branch-current method to the network. Step 1, Assign arbitrary current directions; Step 2, we draw polarities

Step 3 – Apply KVL in all loops

Step 4 – Apply KCL at node a (one less than total nodes) Step 5 – Simultaneous solution of 3 equations for 3 unknowns

Using determinants

Introduction Current Sources Source Conversions Current Sources in Parallel & Series Branch Current Analysis Method 13/08/

13/08/