LINEAR CIRCUIT ANALYSIS EE-111 ENGR. IMRAN AZIZ

CHAPTER 3: CIRCUIT ANALYSIS TECHNIQUES Circuit Solution by Inspection Nodal Analysis Loop Analysis Linearity and Superposition Source Transformations

3.1 Circuit Solution by Inspection As circuits are made of elements, each branch current and branch voltage must satisfy two set of laws Element Laws Connection Laws This section is devoted for circuit analysis through rapid use of Ohm’s Laws and Kirchhoff’s Laws

The set of equations are used in power systems, such as Power supplies, electric motors and power amplifiers where Currents are in the range of amperes. Exercise 3.2 and Exercise 3.3 are your homework.

These circuits look intimidating at first, but once you start solving them, you’d realize that it only takes Ohm’s Law and Kirchhoff’s Laws to solve them. Key idea is to follow step by step approach, instead of looking in the circuit on whole and getting worried. This is how engineer’s deal practically with more sophisticated tasks.

DC Biasing: Devices like diodes and transistors require prescribed DC voltage and current to operate properly. Resistors need to be added with them to fulfill their requirements

3.2 Nodal Analysis Circuit is known completely once all branch voltages and currents are known. Two approaches are used to find these unknown values: Nodal Method Loop Method To keep things simple, this chapter focuses only on resistors and independent sources

The Nodal Method: Experiences indicate that beginners mostly get stuck due to failure in circuit labelling.

Example 3.8

Checking: Verification of results is as important as their derivation A good engineer must know how to solve the problem and how to verify it In case of nodal analysis: Node voltage should be used to find branch currents Branch currents should fulfill KCL

Supernodes: A special case occurs when two nodes are connected by a voltage source; in this case current can’t be expressed in terms of voltage source directly. This is overcome by taking source and its adjacent nodes as generalized node or supernode.

3.3 Loop Analysis As nodal analysis is used to find voltages in circuit, loop analysis is used to find currents in circuit. The Loop Method: Inadequate labelling will cause you to get stuck.

Checking: Calculated mesh currents are used to find branch currents Branch currents are used to find branch voltages. Then it will be verified that branch voltage do satisfy KVL

Supermeshes: A special case occurs when two or more mesh currents pass through same current source;

3.4 Linearity and Superposition A circuit is said to be linear if it satisfies the following properties: The Scaling Property: If there is single source in a circuit, then branch currents and node voltages are linearly proportional to the source. Which means; multiplying a constant value to source, multiplies all currents and voltages with same constant In particular, setting a source to zero, makes all currents and voltages zero The Additive Property: If there are more than one sources in a circuit, each branch current and node voltage is algebraic sum of the contributions of each source acting alone (with all other sources set to zero) All the circuits with resistors and sources are linear circuits Circuits consisting of diodes and transistors are highly non-linear

Example of Linearity: For fig. (a) which shows linear relation For fig. (b) which also show linear relation For linearity, voltage v should be algebraic sum of both v1 and v2 This can be verified by applying nodal analysis

Superposition Principle: Superpose means to calculate the contributions of individual sources and then add them algebraically. Alternative of nodal and loop analysis methods Nodal and Loop methods allow to find all voltages and currents in a circuit, while with the help of superposition specific voltage or current can be found. Superposition allows to find individual contributions of sources which can be compared

Superposition Principle: When only independent sources are used in a circuit; we can apply this principle through following steps: Label the current / voltage to be found and indicate reference direction/ polarity Find the contribution of individual source, with all other sources set to zero, or suppressed. Add all contributions algebraically Clearly, superposition principle is direct consequence of linearity property.

Concluding Observation: Power Calculation: Superposition principle is not applicable to resistive power because it is quadratic function and hence nonlinear. Concluding Observation: On keen observation of the circuits, it will be revealed that superposition principle can often be solved by applying simple voltage and current divider formulas. We have to search for hidden series/parallel combination As a reminder, voltage divider is applied on series resistances and current divider is applied on parallel resistances.

3.5 Source Transformations Source transformation exploits the equivalence between voltage source with series resistance and current source with parallel resistance. This manipulation simplifies the circuit analysis

Voltage Source to Current Source Transformation

Current Source to Voltage Source Transformation

Source Transformations: Source transformations are only applicable to sources having dynamic resistance, i.e practical sources A 12V car battery having 0.05 Ohm internal resistance can also be presented as 12/0.05 = 240A current source with 0.05 Ohm parallel resistance These circuits have equivalent voltage at terminals but not equivalent in energetic point of view. In no load conditions, voltage source would dissipate no energy internally. But current source will dissipate a power of 2.88kW. Source transformation is also sometimes used for circuit analysis; polarity and current directions should be taken care of.

Analysis Techniques Comparison: We’ve studied four circuit analysis techniques Nodal Analysis Loop Analysis Superposition Method Source Transformation Node and loop analysis methods are used when complete state of circuit is sought Quite often we’re interested in finding specific voltage/current, which can be found by Superposition method along with voltage/current division OR node/loop analysis Certain circuits can be solved using source transformations; if not a complete circuit, at least a part of it can be. Develop the habit of pausing and making strategy to solve the circuit