Chapter 2
Circuit Analysis Techniques
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
1
Objectives
• To formulate the node-voltage equations.
• To solve electric circuits using the node – voltage
method.
• To introduce the mesh – current method.
• To formulate the mesh-current equations.
• To solve electric circuits using the mesh-current
method.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Two Powerful Techniques for Circuit Analysis
Nodal Analysis,
Mesh Analysis,
based on a
systematic
application of
Kirchhoff’s
current law
(KCL
based on a
systematic
application of
Kirchhoff’s
voltage law
(KVL)
we can analyze almost any circuit by obtaining a set of simultaneous equations
that are then solved to obtain the required values of current or voltage.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Nodal Analysis
• So far, we have been applying KVL and KCL “as
needed” to find voltages and currents in a circuit.
• Good for developing intuition, finding things quickly…
• …but what if the circuit is complicated? What if you get
stuck?
• Systematic way to find all voltages in a circuit by
repeatedly applying KCL: Node Voltage Method
(Nodal Analysis).
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Branches and Nodes
( reminder from last part)
Branch: elements connected end-to-end,
nothing coming off in between (in series)
Node:
place where elements are joined—includes entire wire
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Steps to Determine Node Voltages Method :
Step 1
Select a node as the reference
node.
This is the reference Node
The reference node is commonly called the
ground since it is assumed to have zero
potential.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Steps to Determine Node Voltages Method ( Cont.)
Step 2
Assign voltages v1, v2, . . . , vn−1 to
the remaining n − 1 nodes. The
voltages are referenced with respect
to the reference node.
Node 0 is the reference node (v = 0), while
nodes 1 and 2 are assigned voltages v1
and v2.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Steps to Determine Node Voltages Method
( Cont.)
Step 3
Apply KCL to each of the n − 1 non
reference nodes
add i1, i2, and i3 as the currents through
resistors R1,R2, and R3, respectively. At
by applying KCL gives
node 1
node 2
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Steps to Determine Node Voltages Method
( Cont.)
Step 4
Use Ohm’s law to express the
branch currents in terms of node
voltages.
The key idea to bear in mind is that, since
resistance is a passive element, by the
passive sign convention, current
must always flow from a higher potential to
a lower potential.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Steps to Determine Node Voltages Method
( Cont.)
Step 5
Solve the resulting simultaneous
equations to obtain the unknown
node voltages.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Node Voltage Equations (Resistors)
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Example:
The voltage drop from node X to a reference node (ground) is
called the node voltage Vx.
The current through resistors can be expressed as
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
Va − Vb
Iab =
R
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Example
Calculate the node voltages
Solution
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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At node 1
Multiply by 4
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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At node 2
Multiply by 12
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Example
Find the voltage at node 1, 2 & 3
Solve this example and handle it to me
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Conclusion for Nodal Analysis
• Nodal analysis is simply writing KCL equations in a
systematic way assuming all currents leaving.
• Nodes’ voltages’ are the circuit variables.
• Currents are expressed in terms of nodes’ voltages.
• The number of variables =
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
Number of nodes - 1
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Nodal Analysis with Voltage Sources
note that:
• A current source produces constant current in a
give direction.
Is is leaving Va
Is
Va
Vb
- Is is leaving Vb
Va
• A Voltage source maintains the voltage constant
between its terminals. Va = Vs
• No need to consider Va a circuit variable.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
Vs
+
-
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If a voltage source is
connected between the
reference node and a
non-reference node.
simply set the voltage at the non-reference
node equal to the voltage of the voltage
source
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
v1 = 10 V
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If the voltage source
(dependent or independent) is
connected between two nonreference nodes.
we apply both KCL and KVL to
determine the node voltages.
V1 and v2 are called a super-nodes as they enclose a (dependent or
independent) voltage source connected between them and any elements
connected in parallel.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Steps to Determine Nodal Analysis with Voltage Sources.
Step 1
Choose a reference node (ground, node 0)
Step 2
Define unknown node voltages (those not connected to ground by
voltage sources).
Va, Vb, ……
Step 3
Write KCL equation at each unknown node.
How? Each current involved in the KCL equation will either come
from a current source (giving you the current value) or through a
device like a resistor.
If the current comes through a device, relate the current to the node
voltages using I -V relationship (like Ohm’s law).
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Step 4
Apply KCL to the supper node
Super-node
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Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Step 5
Apply KVL to the supper node
Step 6
Solve the set of equations (N linear KCL
equations for N unknown node voltages).
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Example
node voltage set
R1
+
-
V1
Va
R
3
Vb
R2
IS
R4
← reference node
•
•
•
Choose a reference node.
Define the node voltages (except reference node and the one set
by the voltage source).
Apply KCL at the nodes Va and Vb with unknown voltage.
Va − V1 Va Va − Vb
+
+
=0
R1
R2
R3
•
Vb − Va Vb
+
= IS
R3
R4
Solve for Va and Vb in terms of circuit parameters.
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Example
Find the node voltage
Solution
Apply KCL at the super node
Multiply by 4
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Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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………………………………………….1
Apply KVL to the loop
…………………….……2
From 1 and 2
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Example
Calculate the power absorbed by
the 6 ohm resistor using nodal analysis
Apply nodal KCL at V1
v1
v1 − v2
1−
=0
−
2
6
v1 − v2
v1
+
−1 = 0
6
2
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
………………1
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At Node V2
v1 − v2
v2
−
−4=0
6
7
v2 − v1
v2
+
+4=0
6
7
Solve for 1 & 2
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
………………………………….2
2
6
6
2
1
2
P =i R
v −v
P=
6
144
= 24W
6=
6
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Example
Find Io using the Node-Voltage
At node V1
4 = i1 + i2
v1
v1 − v2
4=
+
3
6
v1
v1 − v2
−4 = 0
+
3
6
…
…..........................................................1
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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At node V2
I o = i3 + i4
v1 − v2
v2
v2 − 12
=
+
6
4
6
v2 − v1
v2
v2 − 12
+
+
=0
6
4
6
………………………………..2
Solve for 1 and 2
V1 = 10.838 V
V2 = 8.516 V
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
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Example
Use nodal analysis, find vo
Solution
At Node V1
i1 = i2 + 5
v1 − v0
v1
=
+5
1
2
v1 − v0
40
=
+5
1
2
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
31
v1 − v0
40
=
+5
1
2
Multiply by 2
(80) = (v1 − v0 ) + 10
(v1 − v0 ) = 70
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
32
At Node V0
i2 + 5 = i3
v1 − v0
v0
v0 − (− 20 )
+5 =
+
2
4
8
Multiply by 8
4v1 − 7v0 = −20
Dr.-Eng. Hisham El-Sherif
Electronics and Electrical Engineering Department
Solving for v0 and v1
V0 = 30v
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