Circuit ElementsCircuit Analysis with dependent sourcesSimple Resistive CircuitsResistor in SerialThe current-divider circuitDelta-to-Wye (Pi-to-Tee) Equivalent CircuitsTechniques of Circuit AnalysisThevenin’s TheroremTechniques of Circuit AnalysisSolution of Linear System of Equations Using Cramer’s RuleFor two unknown variablesThe Mesh-Current MethodCircuit Elements Ideal voltage source is a circuit element that maintains a prescribed voltage across its terminals regardless of the current flowing in those terminals. Ideal current source is a circuit element that maintains a prescribed current through its terminals regardless of the voltage across those terminals. An independent source establishes voltage or current in a circuit without relying on voltages or currents elsewhere in the circuit. The value of the voltage or current supplied is specified by the value of the independent source alone. A dependent source establish a voltage or current whose value depend on the value of a voltage or current elsewhere in the circuit. (a) An ideal independent voltage source (b) An ideal independent current source (c) An ideal dependent voltage-controlled voltage source (d) An ideal dependent voltage-controlled current source (e) An ideal dependent current-controlled voltage source (f) An ideal dependent current-controlled current source Resistor constrains its voltage and current to be proportional to each other. Ohm’s Law establishes the proportionality of voltage and current in a resistor. iRv = The power absorved by a resistor RvRivip22=== A node is appoint where two or more circuit elements join. A close path is a loop traced through connecting elements, starting and ending at the same node and encountering intermediate nodes only once each Kirchhoff’s current law (KCL) states that the algebraic sum of all the current at any node in a circuit equal zero. We must assign an algebraic sign corresponding to every current at node. Assigning a positive sign to a current leaving a node requires assigning a negative sign to a current entering a node. Kirchhoff’s voltage law (KVL) states that the algebraic sum of all the voltages around any close path in a circuit equal zero. We must assign an algebraic sign to each voltage in the loop. Assigning a positive sign to a voltage rise requires assigning a negative sign to a voltage drop. + - Vs (a) is (b) + - Vs=µV(c) Vs=ρix (d) + - is=αVx (e) is=βix (f)Example: 1 a) Find the current ig and ia in the following circuit b) Find the voltage vg c) Verify that the total power developed equals the total power dissipated. R130R290igR380 a) From node (a) : 6.1+=agii from path (I) 192)9030(6.180 =+=ai therefore Aia4.280192== Aig46.14.2 =+= b) VVg144)6.1(90 == c) WPdis768)120(6.1)80(4.222=+=∑ WPdev768)192)(4( ==∑ Therefore ∑∑=devdisPP Example 2 The current i0 in the following figure is 4A. a) Find i1 b) Find the power dissipated in each resistor c) Verify that the total dissipated in the circuit equals the power developed by the 180V source 108525180V 70 a) Vv 100)25)(4(0== Vv 801001802=−= 1.6 A Ia + vg - i0 i1 i2 i3 i4 ig + v1 - + v2 _ + v0 - a IAvi 10880822=== Aiii 6410,4323=−==+ Vvvv 140)10(680321=+=+= Avi 2701407011=== Aiii 862314=+=+= Aiig128444=+=+= b) Wp 320)5(825==Ω Wp 400)25(4225==Ω Wp 280)70(2270==Ω Wp 360)10(6210==Ω Wp 800)8(1028==Ω c) WPdis2160800360280400320 =++++=∑ WiPgdev2160)12(180180 === Circuit Analysis with dependent sources Example: For the following circuit a. Find the current i1 b. Find the voltage v 5 V 8 V1.8 K6K1 V54 K a. Using KCL at node a 11123130 iiii =+= Using KVL in pole I )31(6154511ii +−= Therefore Aiµ2524061== b. Using KVL inpole II 30 i1 (f) i1 +- v a i2 I)1025(6000)1025)(30(1800866 −−+−= xvx Therefore Vv 2865.435.1−=−+= HW Problem: 2.14 (page 57) 2.27 (page 61)Simple Resistive Circuits Resistor in Serial R4R9 R8R1V1R6R3R2R5R7 V1 Req In general, if k resistors are connected in series, the equivalent single resistor value is the sum of the k resistances, kkiieqRRRRR +++==∑=....211 Resistor in Parallel: R3 R5V1 R4R1 R2 V1 Req In general form, if k resistor is connected in parallel, the equvalent single resistor value is given in the following form: kkiieqRRRRR1....1111211+++==∑= If two resistor connected parallel, in the special case: 212121111RRRRRRReq+=+= !RRRRReq+=11 Voltage-divider circuit: R1VsR2 21iRiRvs+= ! 21RRvis+= + V1 - + V2 -isvRRRiRv21111+== and svRRRiRv21222+== The current-divider circuit R1is R2 siRRRRRiRiv21212211+=== siRRRi2121+= and siRRRi2112+= The Wheatstone Bridge: The Wheatstone bridge circuit is used to precisely measure resistances of medium values (the range 1Ω to 1MΩ). VR2R1R3 Rx In the figure, R1,R2,and R3 are known resistors and Rx is the unknown resistor. To find the value of Rx, we adjust the variable resistor R3 until there is no current in the galvanometer. Then Rx can be calculated as 31ii = and xii =2 xxRiRi =33 and 2211RiRi =, and then xRiRi231= 213RRRRx= ! 312RRRRx= + v - i2 i1 i1 i2 ix i3 igDelta-to-Wye (Pi-to-Tee) Equivalent Circuits In the ∆-connected circuit, the equivalent resistance can be computed as : 21)(RRRRRRRRRcbabacab+=+++= 32)(RRRRRRRRRcbacbabc+=+++= 31)(RRRRRRRRRcbaacbcd+=+++= Y-connected resistors can be computed from ∆-connected circuit as: cbacbRRRRRR++=1 cbaacRRRRRR++=2 cbabaRRRRRR++=3 The ∆-connected resistors can be computed from Y-connected circuit as : 1133221RRRRRRRRa++= 2133221RRRRRRRRb++= 3133221RRRRRRRRc++= Ra Rb Rc a b c a b c R1 R2 R3Techniques of Circuit Analysis Thevenin’s Therorem Any two-terminal bilateral dc network can be replaced by an equivalent circuit consisting of a voltage source and a series resistor. To find the Thevenin voltage VTh and the Thevenin resistance RTh: 1. Mark the terminals of the remaining two-terminal network. 2. Calculate RTh by first setting all sources to zero(voltage sources are replaced by short circuits and current sources by open circuit) and then finding the result resistance between the two marked terminals 3. Calculate VTh by first replacing the voltage and current sources and then finding the open-circuit voltage between the marked terminals. 4. Draw the Thevenin equivalent circuit with the marked terminals Example: Find the Thevenin circuit of a and b terminals for for