Electrical Engineering Lecture 12 PDF

Introduction to Electrical Engineering EE 103 Lecture 12 R e l a x !!!  KCL holds for closed curves or surfaces i 1 (t) i 2(t) i1 (t ) + i2 (t ) + i3 (t ) = 0...

Introduction to Electrical Engineering EE 103 Lecture 12 R e l a x !!!  KCL holds for closed curves or surfaces i 1 (t) i 2(t) i1 (t ) + i2 (t ) + i3 (t ) = 0 i (t) 3 Algebraic sum of the currents entering a closed curve or a surface is zero at every instant of time. Loop or Closed Path: A loop or closed path in a circuit is a contiguous sequence of branches which begins and ends on the same node and touches no other node more than once. KVL: Algebraic sum of the voltage drops around any closed path is zero at every instant of time Circuit Analysis: Mesh or Loop analysis (application of KVL) (Maxwell 1881) Nodal analysis (application of KCL) (1901, 1960) Loop analysis R1 R2 R3 I2 R4 + + V _ 1 V2 _ R6 I1 I3 R5 Loop analysis R1 I 1 R2 I R3 2 R4 + + V _ 1 V2 _ R6 I 3 R5 Loop analysis 1 4 2 I2 + + 2 28 V _ 12V _ 8A I1 I3 1 Nodal Analysis : Computation of all node voltages of a ckt by applying KCL. Assign a node to be the reference node - arbitrarily ? Leave the node at which a voltage source is directly connected with respect to the reference node Leave also the reference node  G5 _ Va + Vx Vb Vc G2 G4 + G1 G3 Is Vd _ Reference Node Reference node is also called ground  v1 v2 +  iy gmvx iy G1 G2 vx G4 G6 _ v3 + vin _ G5 Ref Node Systems: Relation between its input and output Issue of Linearity  IA 60  VB 1 + (VB − Vs 2 ) = I s 2 120 60 Is1 VB 120  Vs2 or 2 VB = 40 I s1 + Vs 2 3  IA 60  2 VB = 40 I s1 + Vs 2 3 Is1 VB 120  Vs2 1 1 2 IA = (VB − Vs 2 ) = (40 I s1 + Vs 2 − Vs 2 ) 60 60 3 2 1 = I s1 − Vs 2 2 3 180 VB = 40 I s1 + Vs 2 3 2 1 I A = I s1 − Vs 2 2 input, 2 output system 3 180 Linearity: For any linear circuit, any output voltage or current, denoted by the variable y, is related linearly to the independent sources. y = a1u1 + a2u2 + + amum u1 um = voltage and current values of independent sources a1 am = properly dimensioned constants  iB Linear Vs1 circuit is1 containing no independ- ent Vsn ism sources VA  iB va = 1vs1 +  n vsn + 1is1 + +  mism iB = 1vs1 +  n vsn +  1is1 + +  mism Linear Vs1 circuit is1 containing no independ- ent Vsn ism sources VA 2 output, m+n input system Consequence of : y = a1u1 + a2u2 + + amum Superposition theorem: In any linear circuit containing more than one independent source, any output (voltage or current) in the circuit may be calculated by adding together the contributions due to each independent source acting alone, with remaining independent sources deactivated.  y = a1u1 + a2u2 + + amum Additivity property of linear networks If all sources are multiplied by a constant, the response is multiplied by the same constant – homogeneity property of linear circuit Example-1 Compute Vout Compute power consumed by the o/p resistor + 6 12  24  Vout Vs1 Vs2 _  + 8 4 6 12  24  V = 1 Vs1 = Vs 2 8+6 out 7 Vout Vs1 Vs2 _ 4.8 2 V = 2 Vs 2 = Vs 2 4.8 + 12 out 7 4 2 Vout = V + V = Vs1 + Vs 2 1 out 2 out 7 7 Vout2 1  16 16 4 2  P= =  (Vs1 ) + (Vs1Vs 2 ) + (Vs 2 )  2 24 24  49 49 49   Example-2 50  200  40  Is1 Vout V3 Vs2 V3 Compute Vout  50  200  V31 V3 − V3 1 1 + = is1 40  200 40 Is1 1 V1 Vout 3 or V31 200 I s1 V = 1 (6 − 5 ) 3 Therefore, 500 − 250 V = 50 I s1 + V = 1 1 I s1 6 − 5 out 3  50  200  V32 − Vs 2 V32 − V32 + =0 40  200 40 2 Vout V32 Vs2 V32 or 1 Vout = V =2 Vs 2 6 − 5 3 Therefore, 500 − 250 1 Vout = Vout1 + Vout2 = I s1 + Vs 2 6 − 5 6 − 5 Network theorems Thevenin’s Theorem (1883) : Thevenin’s Theorem: A iL Resistances and Arbitrary independent VL Network sources B Rth A iL Arbitrary Voc VL Network B Thevenin’s Theorem : Given an arbitrary two-terminal linear network N, for almost all such N, there exists an equivalent two-terminal network consisting of an resistance (impedance), Rth in series with an independent voltage source, voc (t). The voltage source, voc(t), called the open circuit voltage, is what appears across the two terminals of N when no other network is attached. Rth called the thevenin equivalent resistance, is the equivalent resistance of N when all independent sources are deactivated Norton’s Theorem: Norton’s Theorem: Given an arbitrary two-terminal linear network N, for almost all such N, there exists an equivalent two-terminal network consisting of an resistance (impedance), Rth in parallel with an independent current source, isc(t). The current source, isc (t), called the short circuit current, is what flows through a short circuit of the two terminals of N. R thcalled the thevenin equivalent resistance, is the equivalent resistance of N when all independent sources are deactivated Neq Resistances and Independent isc Rth sources Find Thevenin and Norton equivalent circuits seen at the terminal A-B VA VB i R 5 k 20 k 20 k is3 Vs1 Vs2  VB = −20kis 3 VA VB i Applying KCL at node - A 5 k 20 k 20 k is3 VA − Vs 2 VA − Vs1 Vs1 Vs2 + =0 20k 5k or, VA − Vs 2 + 4VA − 4Vs1 =0 20k or 4Vs1 + Vs 2 VA = 5 Therefore, 4Vs1 + Vs 2 Voc = VAB = + 20kis 3 5  VA VB Rth i 5 k 20 k 20 k is3 Vs1 Vs2 A 5 k 20 k A B 20 k A 24 k 4 k B 20 k B  Thevenin and Norton’s theorem for circuits containing active elements All controlling voltages and currents should be within the two terminal network whose Thevenin or Norton’s equivalents are sought Find Theven equivalent circuit seen at the terminal A-B A i 4k  50  Vd Vd  = 101 Rth B  voc = isc = 0 Applying KCL at node - A 1 1 − 101 −i + + =0 i=− 1 50 4 10 3 200 A i 4k  50  Vd 1 Rth = = −200 1V Vd Rth 1 − 200 B Maximum Power Transfer Theorem R th i + Any load v + v network oc - - pL = v  i vvoc − v 2 pL = Rth voc − v i= dpL = 0  RL = Rth Rth dv