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# DC Network - Comprehending Theorems

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A Comprehending Overview of Theorems with examples will let you understand the basics instincts on solving theorem questions.

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### DC Network - Comprehending Theorems

1. 1. Topic DC Network Presenter : Y. Aakash (@wisdom_wieldr) about.me/yaakash
2. 2. Contents • Kirchoff laws • Nodal Analysis • Superpostion Theorem • Thevenin Theorem • Norton Theorem • Maximum Transfer Theorem
3. 3. Kirchhoff's Laws 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.
4. 4. 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).
5. 5. • The current entering any junction is equal to the current leaving that junction. i1 + i4 =i2 + i3   0I
6. 6. 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”
7. 7. 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 stated as: The sum of all the voltages around the loop is equal to zero. v1 + v2 + v3 - v4 = 0   RIVemf
8. 8. Nodal Analysis 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.
9. 9. Steps to Solve Nodal Analysis :  Note all connected wire segments in the circuit. These are the nodes of nodal analysis.  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.
10. 10.  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 unknown voltage.
11. 11. Steps of Nodal Analysis + – V 500W 500W 1kW 500W 500W I2 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.
12. 12. Currents and Node Voltages R2V1 V2 2 21 R VV  R1 V1 1 1 R V
13. 13. KCL at Node 1 R2 R1 I1 V1 V2 12 121 1 R V R VV I    KCL at Node 2 R3 V2 V3V1 0 432 32212     R VV R V R VV
14. 14. KCL at Node 3 2 323 54 I R V R VV   R5 R4 I2 V2 V3
15. 15. Superposition Theorem • It is used to find the solution to networks Containing two or more Voltage sources. • 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
16. 16. How to Apply Superposition Theorem 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 circuit). • 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).
17. 17. For Example : Solution: • 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 6 12 A4 2 i1 i2 i3 i4
18. 18. • To calculate the branch current, the node voltage across the 3Ω resistor must be known.Therefore 120 V 3 6 4 2 i' 1 i' 2 i' 3 i' 4 v1 42 v 3 v 6 120v 111    = 0 where v1 = 30V The equations for the current in each branch,
19. 19. 6 30120  = 15 A i'2 = 3 30 = 10 A i' 3 = i' 4 = 6 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 3 6 12 A4 2 i1 " i2 " i3 " i4 " i'1 =
20. 20. To determine the branch current, solve the node voltages across the 3Ω dan 4Ω resistors as shown in Figure 4 The two node voltages are 3 6 12 A4 2 v3 v4 + - + - 263 4333 vvvv   12 4 v 2 vv 434   = 0 = 0
21. 21. • By solving these equations, we obtain v3 = -12V v4 = -24V Now we can find the branches current,
22. 22. To find the actual current of the circuit, add the currents due to both the current and voltage source,
23. 23. Thevenin's theorem 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
24. 24. Thevenin’s Theorem TheThevenin equivalent circuit provides an equivalence at the terminals onlyThis theorem achieves two important objectives: • Provides a way to find any particular voltage or current in a linear network with one, two, or any other number of sources • We can concentration on a specific portion of a network by replacing the remaining network with an equivalent circuit
25. 25. Calculating the Thevenin equivalent • Sequence to proper value of RTh and ETh • Preliminary • 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.
26. 26. • Mark the terminals of the remaining two-terminal network. (The importance of this step will become obvious as we progress through some complex networks) • RTh: 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) • ETh: 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)
27. 27. • Conclusion: • 5. Draw theThévenin equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit. This step is indicated by the placement of the resistor RL between the terminals of theThévenin equivalent circuit Insert Figure 9.26(b)
28. 28. Another way of Calculating the Thévenin equivalent • 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
29. 29. Example: Solution : • 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, 25 V 20 + - v13 A 5 4 + - vab a b
30. 30. • 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. 03 20 v 5 25v 11   25 V 20 + - v23 A 5 4 + - vab a b isc Figure 2
31. 31. 0 4 v 3 20 v 5 25v 222   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 circuit current isc is The Thevenin resistance RTh is Figure 3 shows the Thevenin equivalent circuit for the Figure 1.
32. 32. Norton theorem 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 resistor. Any two linear bilateral dc network can be replaced by an equivalent circuit consisting of a current and a parallel resistor.
33. 33. Calculating the Norton equivalent • The steps leading to the proper values of IN and RN • Preliminary • 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 • RN: • 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
34. 34. Norton’s Theorem • 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 terminals. • Conclusion: • Draw the Norton equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit
35. 35. Example : Step 1: Source transformation (The 25V voltage source is converted to a 5 A current source.) 25 V 20 3 A 5 4 a b 20 3 A5 4 a b 5 A
36. 36. 48 A 4 a b Step 3: Source transformation (combined serial resistance to produce theThevenin equivalent circuit.) 8 32 V a b Step 2: Combination of parallel source and parallel resistance
37. 37. • Step 4: Source transformation (To produce the Norton equivalent circuit.The current source is 4A (I =V/R) 32V/8 Norton equivalent circuit. 8Ω a b 4 A
38. 38. 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 Req= 2RL
39. 39. Resistance network which contains dependent and independent sources L 2 Th R4 V  2 L L 2 Th R2 RV pmax = = • Maximum power transfer happens when the load resistance RL is equal to theThevenin equivalent resistance, RTh.To find the maximum power delivered to RL,
40. 40. 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
41. 41. Thank you