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.
• 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