The document discusses several theorems for analyzing DC networks:
1. Superposition theorem states that the current or voltage across an element is equal to the algebraic sum of currents/voltages from each independent source.
2. Thevenin's theorem states that any linear bilateral network can be reduced to a single voltage source in series with an equivalent resistance.
3. Norton's theorem states that any linear bilateral network can be reduced to a current source in parallel with an equivalent resistance.
4. Maximum power transfer theorem states that maximum power is delivered to a load when its resistance equals the Thevenin resistance of the network.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
Kirchhoff's Laws
Kirchhoff's laws quantify how current flows through a circuit and how voltage varies around a loop in a circuit
There are two laws
Kirchhoff’s Current Law (KCL) or First Law
Kirchhoff’s Voltage Law (KVL) or Second Law
Kirchhoff’s Current Law (KCL) or First Law
The total current entering a junction or a node is equal to the charge leaving the node as no charge is lost
Kirchhoff’s Voltage Law (KVL) or Second Law
According to Kirchhoff’s Voltage Law,
The voltage around ya loop equals to the sum of every voltage drop in the same loop for any closed network and also equals to zero.
Put differently, the algebraic sum of every voltage in the loop has to be equal to zero and this property of Kirchhoff’s law is called as conservation of energ.
Kirchhoff's Laws
Kirchhoff's laws quantify how current flows through a circuit and how voltage varies around a loop in a circuit
There are two laws
Kirchhoff’s Current Law (KCL) or First Law
Kirchhoff’s Voltage Law (KVL) or Second Law
Kirchhoff’s Current Law (KCL) or First Law
The total current entering a junction or a node is equal to the charge leaving the node as no charge is lost
Kirchhoff’s Voltage Law (KVL) or Second Law
According to Kirchhoff’s Voltage Law,
The voltage around ya loop equals to the sum of every voltage drop in the same loop for any closed network and also equals to zero.
Put differently, the algebraic sum of every voltage in the loop has to be equal to zero and this property of Kirchhoff’s law is called as conservation of energ.
An electric circuit is a path in which electrons from a voltage or current source flow. The point where those electrons enter an electrical circuit is called the "source" of electrons.
An electric circuit is a path in which electrons from a voltage or current source flow. The point where those electrons enter an electrical circuit is called the "source" of electrons.
MLDM provides an original scientific position in Europe on problems related to pattern recognition, machine learning, classification, modelling, knowledge extraction and data mining. These issues have a strong employability potential for students trained in the field of modelling, prediction or decision support, as well as in the area of the Web, image and video processing, health informatics, etc.
For graphs of mathematical functions, see Graph of a function. For other uses, see Graph (disambiguation). A drawing of a graph. In mathematics graph theory is the study of graphs, which are mathematical structures used.In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any acyclic connected graph is a tree. A forest is a disjoint union of trees.
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The Parallel RLC Circuit is the exact opposite to the series circuit we looked at in the previous tutorial although some of the previous concepts and equations still apply.
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Discrete Mathematics - Relations. ... Relations may exist between objects of the same set or between objects of two or more sets. Definition and Properties. A binary relation R from set x to y (written as x R y o r R ( x , y ) ) is a subset of the Cartesian product x × y .
Propositional calculus (also called propositional logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages.
Propositional calculus (also called propositional logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components. Logical connectives are found in natural languages.
In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
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students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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2. DC Network Theorem
Superposition theorem
Current through, or voltage across, an element in a linear bilateral
network is equal to algebraic sum of the currents or voltages
produced independently by each source
• used to find the solution to networks with two or more sources not in series or
parallel
• does not require the use of a mathematical technique such as determinants
• each source is treated independently, and the algebraic sum is found to
determine unknown quantity
• No. of networks to be analyzed = No. of sources
• requires that sources to be removed and
replaced without affecting the final result
• voltage source must be set in zero (short circuited)
• removing a current source requires that its terminals be opened (open circuit)
3. DC Network Theorem: Superposition theorem
• Fig. 4.1.1 (c) represents when RHS battery acts alone
• Fig. 4.1.1 (b) represents when LHS battery acts alone
• II = II' -II“ I2= I2" - I2‘ I = l' + 1"
4. DC Network Theorem: Superposition theorem
Ex: Using superposition theorem find current through resistor R3 (4 Ω) shown
Considering the effect of E1 = 54 V
5. DC Network Theorem: Superposition theorem
Ex:
Considering the effect of E1 = 48 V
6. DC Network Theorem: Superposition theorem
Ex: Using superposition theorem find current through resistor R2 (6 Ω) shown
in Fig. 4.5 (a). Also find total power dissipated in the resistor R2 (6 Ω).
Considering the effect of the E (36 V)
Considering the effect of the I (9 A) source
Total current through the resistor R2 (6 Ω)
Power dissipated in the resistor R2 (6 Ω)
7. DC Network Theorem: Thevenin theorem
Thevenin’s theorem states that: Any two-terminal linear bilateral dc
network can be replaced by an equivalent circuit consisting
of a voltage source and a series resistor as in Fig. 4.6 (a).
Fig. 4.6(a) Fig. 4.6(b) Fig. 4.6(c)
• a mathematical technique for replacing a given network, as viewed from two
output terminals,
• by a single voltage source with a series resistance
• It makes the solution of complicated networks (particularly, electronic
networks) quite quick and easy.
• Application of this extremely useful
8. DC Network Theorem: Thevenin theorem
Let, it is required to find current flowing through load resistance RL as in Fig.
Fig. 4.7(a) Fig. 4.7(b) Fig. 4.7(c)
We will proceed as under:
1. Remove RLfrom the circuit terminals A and B and redraw the circuit as
shown in Fig. 4.7. Obviously, the terminals have become open-circuited. ·
2. Calculate the open-circuit voltage Voc which appears across terminals A and
B when they are open i.e. when RL is removed.
Voc = drop across R2 = lR2 , where I is the circuit current when A and B are open
9. DC Network Theorem: Thevenin theorem
Fig. 4.7(a) Fig. 4.7(b) Fig. 4.7(c)
• Now, imagine the battery to be removed from the circuit, leaving its internal resistance r
behind and redraw the circuit, as shown in Fig. 4.7
• When viewed inwards from terminals A and B, the circuit consists of two parallel paths:
one containing R2 and the other containing (Rt + r)
• Equivalent resistance of-the network, as viewed from these terminals
This resistance is also called, *Thevenin resistance RTh.
10. DC Network Theorem: Thevenin theorem
Steps to follow
1. Remove that portion of the network across which the Thevenin’s equivalent
circuit is to be found.
2. Mark the terminals of the remaining two terminal network.
3. calculate RTH by first setting all sources to zero (Voltage sources are
replaced by SC and current sources by OC) and then finding the resultant
resistance between the two marked terminals (internal resistances of the
sources must remain when the sources are set to zero).
4. Calculate VTH by first returning all sources to their original position and
finding the open circuit voltage (VOC) between the terminals. It is the same
voltage that would be measured by a voltmeter placed between the marked
terminals.
5. Draw the Thevenin equivalent circuit with the portion of the circuit previously
removed between the terminals of the equivalent circuit.
11. DC Network Theorem: Thevenin theorem
Ex: Find the Thevenin equivalent circuit for
the network in Fig. 4.8 (a). Then find the
current through R L for value of 2 Ω and 100 Ω.
Solution:
• Steps 1 and 2 produce the network of Fig.
• RL has been removed and the two “holding”
terminals have been defined as a and b.
• Step 3: Replacing the voltage source E1 with SC yields network, where
• For determining RTh
12. DC Network Theorem: Thevenin theorem
Ex: Find the Thevenin equivalent circuit for
the network in Fig. 4.8 (a). Then find the
current through R L for value of 2 Ω and 100 Ω.
Solution:
• Steps 3
• OC voltage ETh = voltage drop across 6 Ω resistor.
• Applying VDR,
• For determining RTh
13. DC Network Theorem: Thevenin theorem
Ex: Find the Thevenin equivalent circuit for
the network in Fig. 4.8 (a). Then find the
current through R L for value of 2 Ω and 100 Ω.
Solution:
•
•
14. DC Network Theorem: Norton theorem
Norton’s theorem states that
Any two terminal linear bilateral dc network can be replaced by an
equivalent circuit consisting of a current source and a parallel resistor
• Every voltage source with a series internal
resistance has a current source equvalent.
• Current source equivalent of Thevenin’s network
can be determined by Norton’s theorem.
•
15. DC Network Theorem: Norton theorem
• Steps leading to proper values of IN and RN are:
• Remove that portion of the network across which the Thevenin’s equivalent
circuit is to be found.
• Mark the terminals of the remaining two terminal network.
• calculate RN by first setting all sources to zero (Voltage sources are replaced
by SC and current sources by OC) and then finding the resultant resistance
RN between the two marked terminals (internal resistances of the sources
must remain when the sources are set to zero). Since RN=RTH, the
procedure and value obtained using the approach described for Thevenin’s
theorem will determine the proper value of RN.
• Calculate IN by first returning all sources to their original position and finding
the short circuit current (ISC) between the terminals. It is the same current
that would be measured by an ammeter placed between the marked
terminals.
• Draw the Norton equivalent circuit with the portion of the circuit previously
removed between the terminals of the equivalent circuit.
16. DC Network Theorem: Norton theorem
Ex: Find the Norton equivalent circuit for the network in the shaded area.
• Identifying the terminals of interest
• Determining RN for the network
•
Determining IN,
• Norton equivalent circuit
17. Network Theorem: Maximum power transfer theorem
The maximum power transfer theorem states that
• A load will receive maximum power from a linear bilateral
dc network when its total resistive value is exactly equal to
the Thevenin resistance of the network as seen by the
load.
• For the network of Fig. the maximum power
will be delivered to the load when RL = RTH
• Determining RN for the network
18. Network Theorem: Maximum power transfer theorem
Proof:
• Circuit current I = E/( RL + RTH)
• Power consumed by the load PL = I2
RL = E2
RL/( RL + RTH)2
• For PL to be maximum, dPL/dRL = 0
• Differentiating and equating to zero, we have RL = RTH.
• So, Max. power Pmax = E2
RL/4R2
L = E2
/4RL = E2
/4RTH
•
19. Network Theorem: Maximum power transfer theorem
Ex: For the network shown in Fig. determine the value of R for maximum
power to R, and calculate the power delivered under these conditions.
• Determining RTH or RN
•
• Determining ETH