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REVIEW OF NETWORK THEOREMS

REVIEW OF NETWORK THEOREMS. 07, July, 2014 Lecture 4. OHM’S law. KIRCHHOFF’S LAWS. Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits

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REVIEW OF NETWORK THEOREMS

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  1. REVIEW OF NETWORK THEOREMS 07, July, 2014 Lecture 4

  2. OHM’S law ECE203 - Network Analysis - K.JeyaPrakash - Kalasalingam University

  3. KIRCHHOFF’S LAWS • Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits • Corollaries of the Maxwell equations in the low-frequency limit • Accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits ECE203 - Network Analysis - K.Jeya Prakash - Kalasalingam University

  4. KIRCHOFF’S current LAW • Also called Kirchhoff's first law, Kirchhoff's point rule, or Kirchhoff's junction rule (or nodal rule) • 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 network of conductors meeting at a point is zero. • Based on principle of conservation of electric charge The current entering any junction is equal to the current leaving that junction. i1 + i4 =i2 + i3 ECE203 - Network Analysis - K.Jeya Prakash - Kalasalingam University

  5. Kirchhoff's voltage law • Also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule • The directed sum of the electrical potential differences (voltage) around any closed network is zero, • or: More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop, • or: 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. • Based on the conservation of energy whereby voltage is defined as the energy per unit charge. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  6. KIRCHHOFF’S voltage law • The sum of all the voltages around the loop is equal to zero. v1 + v2 + v3 - v4 = 0 ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  7. Limitations of Kirchhoff's laws • Applicable only to lumped network models • KCL is valid only if the total electric charge, Q , remains constant in the region being considered • KVL is based on the assumption that there is no fluctuating magnetic field linking the closed loop. • KCL and KVL only apply to circuits with steady currents (DC). However, for AC circuits having dimensions much smaller than a wavelength, KCL, KVL are also approximately applicable. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  8. VOLTAGE DIVIDER • Series circuit – Voltage divider • Same current flows; Voltage drops proportional to value of resistors/impedance; Different voltage from single source; So called voltage divider • Power in series circuit: Sum of powers in each resistor in series ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  9. Current divider • Current from source divides in all branches of parallel circuit; So called current divider • Power in parallel circuit: Sum of powers in each resistor in parallel ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  10. NETWORK THEOREMS • Important fundamental theorems of network analysis. They are the • Superposition theorem • Thévenin’s theorem • Norton’s theorem • Maximum power transfer theorem • Substitution theorem • Millman’s theorem • Reciprocity theorem • Tellegen’s theorem ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  11. Superposition theorem (1/5) • 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. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  12. SUPERPOSITION theorem (2/5) • Used to find the solution to networks with two or more sources that are not in series or parallel. • The current through, or voltage across, an element in a network is equal to the algebraic sum of the currents or voltages produced independently by each source. • Since the effect of each source will be determined independently, the number of networks to be analyzed will equal the number of sources. • A circuit is linear when superposition theorem can be used to obtain its currents and voltages ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  13. SUPERPOSITION THEOREM (3/5) • 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 ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  14. Superposition theorem (4/5) • Superposition theorem can be applied only to voltage and current • It cannot be used to solve for total power dissipated by an element • Power is not a linear quantity • Follows a square-law relationship • 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 • Diode and transistor circuits will have both dc and ac sources • Superposition can still be applied ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  15. SUPERPOSITION THEOREM (5/5) 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). • When this theorem is applied to an ac circuit, it has to be remembered that the voltage and current sources are in the phasor form and the passive elements are impedances. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  16. Thévenin’s theorem (1/7) • For DC Circuits • Any two-terminal, linear, bilateral, active dc network can be replaced by an equivalent circuit consisting of an equivalent voltage source(Thévenin’s Voltage Source) and an equivalent series resistor (Thévenin’s Resistance) • For AC Circuits • Any two-terminal, linear, bilateral, active ac network can be replaced by an equivalent circuit consisting of an equivalent voltage source(Thévenin’s Voltage Source) and an equivalent series impedance (Thévenin’s Impedance) ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  17. Thévenin’s theorem (2/7) • DC Circuits • AC Circuits ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  18. Thévenin’s theorem (3/7) • Thévenin’s theorem can be used to: • Analyze networks with sources that are not in series or parallel. • Reduce the number of components required to establish the same characteristics at the output terminals. • Investigate the effect of changing a particular component on the behaviour of a network without having to analyze the entire network after each change. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  19. Thévenin’s theorem (4/7) • Procedure to determine the proper values of RTh and Eth • 1. Remove the portion of the network across which the Thévenin’s equivalent circuit is to be found • 2. Mark the terminals of the remaining two-terminal network. (The importance of this step will become obvious as we progress through some complex networks.) ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  20. Thévenin’s theorem (5/7) • 3. 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.) ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  21. Thévenin’s theorem (6/7) • 4. 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.) ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  22. Thévenin’s theorem (7/7) • 5. Draw the Thévenin equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  23. NORTON’S theorem (1/5) • Dual of Thévenin's theorem • For DC Networks • Any two-terminal, linear, bilateral, active dc network can be replaced by an equivalent circuit consisting of an equivalent current source(Norton’s Current Source) and an equivalent parallel resistor (Norton’s Conductance) • For AC Circuits • Any two-terminal, linear, bilateral, active ac network can be replaced by an equivalent circuit consisting of an equivalent current source(Norton’s Current Source) and an equivalent shunt admittance (Norton’s Admittance) ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  24. Norton's theorem (2/5) • DC Circuits • AC Circuits ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  25. NORTON’S THEOREM (3/5) • Procedure • 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 ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  26. NORTON’S THEOREM (4/5) • 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 for Thévenin’s theorem will determine the proper value of RN. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  27. Norton’s theorem (5/5) • 4. Calculate INby first returning all the sources to their original position and then finding the short-circuit current between the marked terminals. • 5. Draw the Norton equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit. ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  28. Thévenin – NORTON EQUIVALENT • Possible to find Norton equivalent circuit from Thévenin equivalent circuit • Use source transformation method • ZN = ZTh • IN = ETh/ZTh ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  29. MAXIMUM POWER transfer THEORM • DC Circuits A load will receive maximum power from a linear bilateral dc network when its load resistive value is exactly equal to the Thévenin’s resistance RL = RTh • AC Circuits A load will receive maximum power from a linear bilateral ac network when its load impedance is complex conjugate of the Thévenin’s impedance ZL = ZTh* ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  30. = pmax = MAXIMUM Power transfer theorem Resistance network which contains dependent and independent sources ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  31. Reciprocity theorem • The current I in any branch of a linear bilateral passive network, due to a single voltage source E anywhere in the network, will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured • The location of the voltage source and the resulting current may be interchanged without a change in current ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

  32. RECIPROCITY THEOREM ECE203 - Network Analysis - K20-07-2014.Jeya Prakash - Kalasalingam University

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