Kirchhoff's circuit laws are two equalities that deal with
the conservation of charge and energy in electrical circuits.
Two Laws of Kirchhoff's are as follows :
1. Kirchhoff's Current Law.
2. Kirchhoff's Voltage Law.
Kirchoff’s Current Law (KCL)
The principle of conservation of electric charge implies that:
“At any node (junction) in an electrical circuit, the sum
of currents flowing into that node is equal to the sum of currents
flowing out of that node, or The algebraic sum of currents in a
of conductors meeting at a point is zero.”
This law is also called Kirchhoff's first law, Kirchhoff's point
rule, Kirchhoff's junction rule (or nodal rule).
• The current entering any
junction is equal to the
current leaving that
junction. i1 + i4 =i2 + i3
Kirchhoff's voltage law (KVL)
• This law is also called Kirchhoff's second law, Kirchhoff's loop
(or mesh) rule.
• The principle of conservation of energy implies that
“The directed sum of the electrical potential
differences (voltage) around any closed circuit is zero, or the
the emf’s in any closed loop is equivalent to the sum of the
potential drops in that loop”
The algebraic sum of the products of the resistances of the
conductors and the currents in them in a closed loop is equal to
the total emf available in that loop. Similarly to KCL, it can be
The sum of all the voltages around the
loop is equal to zero. v1 + v2 + v3 - v4 = 0
In electric circuits analysis, nodal analysis, node-
voltage analysis, or the branch current method is a
method of determining the voltage (potential
difference) between "nodes“) in an electrical circuit in
terms of the branch currents.
Steps to Solve Nodal Analysis :
Note all connected wire segments in the circuit. These are the nodes of
Select one node as the ground reference. The choice does not affect the
result and is just a matter of convention. Choosing the node with most
connections can simplify the analysis.
Assign a variable for each node whose voltage is unknown. If the voltage
is already known, it is not necessary to assign a variable.
For each unknown voltage, form an equation based on Kirchhoff's
current law. Basically, add together all currents leaving from the node
and mark the sum equal to zero.
If there are voltage sources between two unknown
voltages, join the two nodes as a super node.The currents
of the two nodes are combined in a single equation, and a
new equation for the voltages is formed.
Solve the system of simultaneous equations for each
Steps of Nodal Analysis
1. Choose a reference (ground) node.
2. Assign node voltages to the other nodes.
3. Apply KCL to each node other than the reference
node; express currents in terms of node voltages.
Currents and Node Voltages
KCL at Node 1
KCL at Node 2
KCL at Node 3
• It is used to find the solution to networks Containing two or more
• The current through, or voltage across, an element in a linear
bilateral network is equal to the algebraic sum of the currents or
voltages produced independently by each source.
• For a two-source network, if the current produced by one source is
in one direction, while that produced by the other is in the opposite
direction through the same resistor, the resulting current is the
difference of the two and has the direction of the larger.
• If the individual currents are in the same direction, the resulting
current is the sum of two in the direction of either current.
• The total power delivered to a resistive element must be determined
using the total current through or the total voltage across the
element and cannot be determined by a simple sum of the power
levels established by each source
How to Apply
For applying Superposition theorem:-
• Replace all other independent voltage sources with a short
circuit (thereby eliminating difference of potential. i.e.V=0,
internal impedance of ideal voltage source is ZERO (short
• Replace all other independent current sources with an open
circuit (thereby eliminating current. i.e. I=0, internal impedance
of ideal current source is infinite (open circuit).
For Example :
• The application of the superposition theorem is, where it is
used to calculate the branch current. We begin by calculating
the branch current caused by the voltage source of 120 V. By
substituting the ideal current with open circuit, we deactivate
the current source.
120 V 3
• To calculate the branch current, the node voltage across
the 3Ω resistor must be known.Therefore
120 V 3
where v1 = 30V
The equations for the current in each branch,
= 15 A
= 10 A
3 = i'
30 = 5 A
In order to calculate the current cause by the current source, we
deactivate the ideal voltage source with a short circuit, as shown
To determine the branch current, solve the node
voltages across the 3Ω dan 4Ω resistors as shown in
The two node voltages are
• By solving these equations, we obtain
v3 = -12V
v4 = -24V
Now we can find the branches current,
To find the actual current of the circuit, add the currents due to
both the current and voltage source,
Thevenin's theorem for linear electrical networks states that
“Any combination of voltage sources, current sources, and resistors with two
terminals is electrically equivalent to a single voltage source V and a single
series resistor R.”
Any two-terminal, linear bilateral dc network can be replaced by an equivalent
circuit consisting of a voltage source and a series resistor
TheThevenin equivalent circuit provides an equivalence at
the terminals onlyThis theorem achieves two important
• Provides a way to find any particular voltage or current
in a linear network with one, two, or any other number
• We can concentration on a specific portion of a network
by replacing the remaining network with an equivalent
Calculating the Thevenin equivalent
• Sequence to proper value of RTh and ETh
• 1. Remove that portion of the network across which
theThévenin equation circuit is to be found. In the
figure below, this requires that the load resistor RL be
temporarily removed from the network.
• Mark the terminals of the remaining two-terminal network.
(The importance of this step will become obvious as we
progress through some complex networks)
Calculate RTh by first setting all sources to zero (voltage
sources are replaced by short circuits, and current sources by
open circuits) and then finding the resultant resistance
between the two marked terminals. (If the internal resistance
of the voltage and/or current sources is included in the original
network, it must remain when the sources are set to zero)
Calculate ETh by first returning all sources to their original
position and finding the open-circuit voltage between the marked
terminals. (This step is invariably the one that will lead to the most
confusion and errors. In all cases, keep in mind that it is the open-
circuit potential between the two terminals marked in step 2)
• 5. Draw theThévenin
equivalent circuit with
the portion of the circuit
replaced between the
terminals of the
equivalent circuit. This
step is indicated by the
placement of the resistor
RL between the terminals
Insert Figure 9.26(b)
Another way of Calculating the Thévenin
• Measuring VOC and ISC
• The Thévenin voltage is again determined by measuring the
open-circuit voltage across the terminals of interest, that is,
ETh = VOC. To determine RTh, a short-circuit condition is
established across the terminals of interest and the current
through the short circuit Isc is measured with an ammeter
• Using Ohm’s law:
RTh = Voc / Isc
• In order to find theThevenin equivalent circuit for the circuit shown in Figure1 ,
calculate the open circuit voltage,Vab. Note that when the a, b terminals are
open, there is no current flow to 4Ω resistor.Therefore, the voltage vab is the
same as the voltage across the 3A current source, labeled v1.
• To find the voltage v1, solve the equations for the singular node voltage. By
choosing the bottom right node as the reference node,
• By solving the equation, v1 = 32 V. Therefore, the Thevenin
voltage Vth for the circuit is 32 V.
• The next step is to short circuit the terminals and find the
short circuit current for the circuit shown in Figure 2. Note
that the current is in the same direction as the falling voltage
at the terminal.
Current isc can be found if v2 is known. By using the bottom
right node as the reference node, the equationfor v2 becomes
By solving the above equation, v2 = 16 V. Therefore, the short
current isc is
The Thevenin resistance RTh is
Figure 3 shows the Thevenin equivalent circuit for the Figure 1.
Norton's theorem for linear electrical networks states that any collection
of voltage sources, current sources, and resistors with two terminals is
electrically equivalent to an ideal current source, I, in parallel with a single
Any two linear bilateral dc network can be replaced by an equivalent circuit
consisting of a current and a parallel resistor.
Calculating the Norton equivalent
• The steps leading to the proper values of IN and RN
• 1. Remove that portion of the network across which the Norton equivalent
circuit is found
• 2. Mark the terminals of the remaining two-terminal network
• 3. Calculate RN by first setting all sources to zero (voltage sources are
replaced with short circuits, and current sources with open circuits) and
then finding the resultant resistance between the two marked terminals.
(If the internal resistance of the voltage and/or current sources is included
in the original network, it must remain when the sources are set to zero.)
Since RN = RTh the procedure and value obtained using the approach
described forThévenin’s theorem will determine the proper value of RN
• IN :
• Calculate IN by first returning all the sources to their original
position and then finding the short-circuit current between
the marked terminals. It is the same current that would be
measured by an ammeter placed between the marked
• Draw the Norton equivalent circuit with the portion of the
circuit previously removed replaced between the terminals
of the equivalent circuit
Step 1: Source transformation (The 25V voltage source is converted to a 5 A
20 3 A
20 3 A5
Step 3: Source transformation (combined serial resistance to
produce theThevenin equivalent circuit.)
Step 2: Combination of parallel source and parallel resistance
• Step 4: Source transformation (To produce the Norton
equivalent circuit.The current source is 4A (I =V/R)
Norton equivalent circuit.
Maximum power transfer theorem
The maximum power transfer theorem states that, to
obtain maximum external power from a source with a finite
internal resistance, the resistance of the load must be equal to the
resistance of the source as viewed from the output terminals.
A load will receive maximum power from a linear bilateral dc network
when its total resistive value is exactly equal to theThévenin resistance of
the network as “seen” by the load
RL = RTh
So, Req = Rth + RL
pmax = =
• Maximum power transfer happens when the load
resistance RL is equal to theThevenin equivalent
resistance, RTh.To find the maximum power delivered to
Application of Network Theorems
• Network theorems are useful in simplifying analysis of some circuits. But the
more useful aspect of network theorems is the insight it provides into the
properties and behaviour of circuits
• Network theorem also help in visualizing the response of complex network.
• The SuperpositionTheorem finds use in the study of alternating current (AC)
circuits, and semiconductor (amplifier) circuits, where sometimes AC is often
mixed (superimposed) with DC