1Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
Chapter 2
Circuit Analysis Techniques
2Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
• 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.
Objectives
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ELCT708: Electronics for Biotechnology
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’svoltage law (KVL)
we can analyze almost any circuit by obtaining a set of simultaneous equationsthat are then solved to obtain the required values of current or voltage.
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ELCT708: Electronics for Biotechnology
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).
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ELCT708: Electronics for Biotechnology
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
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ELCT708: Electronics for Biotechnology
Steps to Determine Node Voltages Method :
Step 1Select a node as the reference node.
The reference node is commonly called the ground since it is assumed to have zero potential.
This is the reference Node
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ELCT708: Electronics for Biotechnology
Steps to Determine Node Voltages Method ( Cont.)
Step 2Assign 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 v1and v2.
8Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
Steps to Determine Node Voltages Method( Cont.)
Step 3Apply 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. Atby applying KCL gives
node 1
node 2
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ELCT708: Electronics for Biotechnology
Steps to Determine Node Voltages Method( Cont.)
Step 4Use Ohm’s law to express the branch currents in terms of node voltages.
The key idea to bear in mind is that, sinceresistance is a passive element, by the passive sign convention, currentmust always flow from a higher potential to a lower potential.
10Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
Steps to Determine Node Voltages Method( Cont.)
Step 5Solve the resulting simultaneous equations to obtain the unknownnode voltages.
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ELCT708: Electronics for Biotechnology
Node Voltage Equations (Resistors)
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ELCT708: Electronics for Biotechnology
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 asR
VbVaIab
−=
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ELCT708: Electronics for Biotechnology
ExampleCalculate the node voltages
Solution
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ELCT708: Electronics for Biotechnology
At node 1
Multiply by 4
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ELCT708: Electronics for Biotechnology
At node 2
Multiply by 12
16Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
Example
Find the voltage at node 1, 2 & 3
Solve this example and handle it to me
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ELCT708: Electronics for Biotechnology
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 = Number of nodes - 1
18Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
Nodal Analysis with Voltage Sources
note that:
• A current source produces constant current in a give direction.
• A Voltage source maintains the voltage constant between its terminals.
• No need to consider Va a circuit variable.
IsVa Vb
Is is leaving Va
- Is is leaving Vb
+-
Vs
Va
Va = Vs
19Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
If a voltage source is connected between the reference node and a non-reference node.
simply set the voltage at the non-referencenode equal to the voltage of the voltage source
v1 = 10 V
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ELCT708: Electronics for Biotechnology
If the voltage source (dependent or independent) is connected between two non-reference 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.
21Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
Steps to Determine Nodal Analysis with Voltage Sources.
Step 1Choose a reference node (ground, node 0)
Step 2 Define unknown node voltages (those not connected to ground by voltage sources). Va, Vb, ……
Step 3Write KCL equation at each unknown node.
How? Each current involved in the KCL equation will either comefrom 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).
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ELCT708: Electronics for Biotechnology
Step 4 Apply KCL to the supper node
Super-node
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ELCT708: Electronics for Biotechnology
Step 5Apply KVL to the supper node
Step 6Solve the set of equations (N linear KCL equations for N unknown node voltages).
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ELCT708: Electronics for Biotechnology
Example
• 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.
• Solve for Va and Vb in terms of circuit parameters.
0R
VVRV
RVV
3
ba
2
a
1
1a =−++−
node voltage set�
Va Vb
← reference node
R4V1 R2
+- IS
R3R1
S4
b
3
ab IRV
RVV =+−
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ELCT708: Electronics for Biotechnology
Example Find the node voltage
Solution
Apply KCL at the super node
Multiply by 4
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ELCT708: Electronics for Biotechnology
Apply KVL to the loop
………………………………………….1
…………………….……2
From 1 and 2
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ELCT708: Electronics for Biotechnology
ExampleCalculate the power absorbed bythe 6 ohm resistor using nodal analysis
Apply nodal KCL at V1
062
1 211 =��
���
� −−��
���
�− vvv
0126121 =−��
���
�+��
���
� − vvv………………1
28Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
At Node V2
0476221 =−��
���
�−��
���
� − vvv
0476212 =+��
���
�+��
���
� − vvv………………………………….2
Solve for 1 & 2
62
6 RiP =
Wvv
P 246
1446
6
221 ==��
���
� −=
29Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
ExampleFind Io using the Node-Voltage
��
���
� −+��
���
�=63
4 211 vvv
214 ii +=
0463
211 =−��
���
� −+��
���
� vvv
…..........................................................1
At node V1
…
30Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
V1 = 10.838 VV2 = 8.516 V
At node V2
43 iiIo +=
��
���
� −+��
���
�=��
���
� −6
1246
2221 vvvv
06
1246
2212 =��
���
� −+��
���
�+��
���
� − vvvv………………………………..2
Solve for 1 and 2
31Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
ExampleUse nodal analysis, find vo
SolutionAt Node V1
521 += ii
521
011 +��
���
� −=��
���
� vvv
521
40 01 +��
���
� −=��
���
� vv
32Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
521
40 01 +��
���
� −=��
���
� vv
Multiply by 2
( ) ( ) 1080 01 +−= vv
( ) 7001 =− vv
33Dr.-Eng. Hisham El-SherifElectronics and Electrical Engineering Department
ELCT708: Electronics for Biotechnology
At Node V0
32 5 ii =+
( )��
���
� −−+��
���
�=+��
���
� −8
204
52
0001 vvvv
Multiply by 8
2074 01 −=− vv
Solving for v0 and v1
V0 = 30v