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FUNDAMENTALS OF ELECTRICAL ENGINEERING [ ENT 163 ] LECTURE #4 CIRCUIT THEOREMS HASIMAH ALI Programme of Mechatronics, School of Mechatronics Engineering, UniMAP. Email: [email protected] CONTENTS Introduction Linearity Property Superposition Source Transformation Thevenin’s Theorem Northon’s Theorem Maximum Power Transfer INTRODUCTION 1. Disadvantage of Kirchhoff’s laws – tedious computation for lage , complex circuit. 2. To handle complexity – Thevenin’s and Northon’s theorems (applicable to linear circuits) 3. Related concept – superposition, source transformation and maximum power transfer. LINEARITY PROPERTY A linear circuit is one whose output is linearly related (or directly proportional) to its input. 1. Consist of linear elements, linear dependent sources and independent sources. 2. Property of linearity: a) Homogeneity property • If input/ excitation is multiplied by a constant, then the output/ response is multiplied by the same constant • E. g. from Ohm’s law • If multiply by a constant k v iR kv kiR LINEARITY PROPERTY b) Addivitity • Response to a sum of inputs is the sum of the responses to each input applied separately. • E.g. Voltage-current relationship v1 i1R and v2 i2 R • Applying : (i1 i2 ) v (i1 i2 ) R i1R i2 R v1 v2 Therefore we can say that a resistor is a linear element (voltage –current relationship satisfies both properties). LINEARITY PROPERTY • • • Linearity principle can be illustrated by considering circuit below. The linear circuit has no independent sources inside it. It is excited by a voltage source vs , which serve as the input. The circuit is terminated by a load R. Suppose vs= 10V and output i=2A. Linearity principle would gives i=0.2A when vs=1V and i=1mA when vs=5mV. LINEARITY PROPERTY Example: For the circuit in Figure , find I when vs=12V and vs=24V. SUPERPOSITION 1. Use to determine the value of a specific variable (voltage or current); based on linearity. The superposition principle states that the voltage across (or current through )an element in a linear circuit is the algebraic sum of the voltage across ( or current through) that element due to each independent source acting alone. 1. Steps to apply superposition principle: a) Turn off all independent sources except one source. Find the output (voltage or current ) due to that active source using previous techniques. b) Repeat step(a) for each of the other independent sources. c) Find the total contribution by adding algebraically all the contributions due to the independent sources. Disadvantage – may involve more work. SUPERPOSITION Keep in mind: • To consider only one independent source, every voltage sources are replaced by 0 V (short circuit) and current sources by 0 A (open circuit). • Dependent sources are left intact because they are controlled by circuit variables. SUPERPOSITION Example: Use the superposition theorem to find v in the circuit in Figure shown. Solution SUPERPOSITION SUPERPOSITION Example: For the circuit shown, use the superposition theorem to find i. Solution: SUPERPOSITION SUPERPOSITION SOURCE TRANSFORMATION • Source transformation is another tool for simplifying circuits (like seriesparallel and wye – delta ). A source transformation is the process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice verse. • Basic to these tools is the concept of equivalence. vs is R vs is R SOURCE TRANSFORMATION • Source transformation also applies to dependent source. Note: 1. The arrow of the current source is directed toward the positive terminal of the voltage source. 2. Source transformation is not possible when R=0 (ideal voltage source) and R=∞ (ideal current source) SOURCE TRANSFORMATION Example: Use source transformation to find vo in the circuit given. Solution: 1. Transform the current and voltage sources. SOURCE TRANSFORMATION 2. Combine the 4Ω and 2Ω resistors in series and transform the 12-V voltage source. 3. Combine the 2-A and 4-A current sources to get 2-A source. 4. Use current division to get i. THEVENIN’S THEOREM • Usage: to avoid analyzing entire circuit for every chnaes in variable element . • Provides a technique by which the fixed part of the circuit is replaced by an equivalent circuit. Thevenin’s theorem states that a linear two – terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTH in series with a resistor RTH, VTH is the open circuit voltage at the terminals and RTH is the input or equivalent resistance at the terminals when the independent sources are turned off. THEVENIN’S THEOREM Let’s consider the figure: • We would like to replace the original linear two-terminal circuit by its equivalent Thevenin circuit. • The circuits are said to be equivalent if they have same voltage-current relation at their terminals. • If the terminals a-b are made opencircuited (by removing the load), no current will flows, so that the terminals a-b must be equal to the voltage source VTH , since the two circuits are equivalent, Therefore: VTh voc THEVENIN’S THEOREM • With the load disconnected and terminals a-b open-circuited, we turn off all independent sources. • The input resistance of the dead circuit at the terminals a-b must be equal to RTH since the two circuits are equivalent, RTh Rin THEVENIN’S THEOREM In finding the Thevenin resistance RTh, we need to consider two cases: Finding VTh and RTh • If the network has no dependent sources, we turn off all independent sources. RTh is the input resistance of the network looking between terminals a and b. • If the network has dependent sources, we turn off all independent sources. We apply a voltage source vo at the terminals a and b determine the resulting current io. THEVENIN’S THEOREM •Then RTh= vo/io as shown below. Alternatively, we may insert a current source io at terminals a-b and find the terminal voltage vo. Again RTh= vo/io (either of the two approaches will give the same result.) THEVENIN’S THEOREM • Important of Thevenin’s theorem- helps simplify a circuit (a large circuit may be replaced by a single independent voltage source and a single resistor) • The current IL through the load and the voltage VL across the load are easily determined once the Thevenin equivalent of the circuit at the load’s terminals is obtained. : V Th IL RTh RL VL RL I L RL VTh RTh RL Note: A negative RTh value shows that the circuit is supplying power (circuit with dependent source) THEVENIN’S THEOREM Example: Find the Thevenin equivalent circuit shown in Figure below, to the left of the terminals a-b. Then find the current through RL=6, 16, and 36Ω. Solution: THEVENIN’S THEOREM THEVENIN’S THEOREM THEVENIN’S THEOREM Exercise: Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit shown. Then find I. NORTHON’S THEOREM Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short circuit consisting of a current through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off. We find RN in the same way we find RTh RN RTh NORTHON’S THEOREM To find IN: • Determine the short-circuit current flowing from terminal a to b in both circuits. • Since the two circuits are equivalent, I N isc VTh IN RTh Source transformation NORTHON’S THEOREM To determine the Thevenin or Northon equivalent circuit, we must find: • The open – circuit voltage vc across terminals a and b, since Vth=voc. • The short – circuit current Isc at terminals a and b , since IN=iSC • The equivalent or input resistance Rin at terminals a and b when all independent sources are turned off, yields: voc RTh RN isc NORTHON’S THEOREM Example: Find the Northon equivalent of the circuit below. Solution: NORTHON’S THEOREM NORTHON’S THEOREM NORTHON’S THEOREM Exercise: Find the Northon equivalent circuit for the circuit shown. MAXIMUM POWER TRANSFER • Practically, a circuit is designed to provide power to a load. • Certain application, e. g., communication – desire to maximize the power delivered to a load. • Thevenin equivalent – useful in finding the maximum power a linear circuit can deliver to a load. • By assuming that the load resistance, RL can be adjusted, the power delivered to the load is: 2 VTh RL p i RL RTh RL 2 Where VTh and RTh are fixed MAXIMUM POWER TRANSFER • Relationship of power and load resistance: Where maximum power occurs when RL=RTh. MAXIMUM POWER TRANSFER • Maximum power theorem stated that the maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (RL=RTh) • Therefore when RL=RTh, maximum power is equal to: pmax 2 Th V 4 RTh MAXIMUM POWER TRANSFER Example: Find the value of RL for maximum power transfer in the circuit shown. Find the maximum power. Solution: MAXIMUM POWER TRANSFER FURTHER READING 1. Fundamentals of Electric Circuits, 2nd Edition,McGrawhill Alexander, C. K. and Sadiku, M. N. O. 2. Electric Circuit, 8th Edition, Pearson, Nillson and Riedel. 3. Circuits,Brooks/ Cole, A. Bruce Carlson. 4. http://www.scribd.com/word/full/2031941?access_key=key1vybbz6deqeinoosecmm