Electric Circuits - Information Engineering Chapter 2D PDF

ELECTRIC CIRCUITS Information Engineering Chapter 2d Electric Network fundamentals and permament regimes Prof. Piergiorgio Sonato Department of Industrial Engineering University of Padova  Index Chapter 2d Chapter 2d:...

ELECTRIC CIRCUITS Information Engineering Chapter 2d Electric Network fundamentals and permament regimes Prof. Piergiorgio Sonato Department of Industrial Engineering University of Padova  Index Chapter 2d Chapter 2d: Electric Networks fundamentals and permanent regimes – 2.6 Linear networks 2.6.1 Principle of Superimposition 2.6.2 Equivalent Voltage Source Theorem – Thévenin’s Theorem 2.6.3 Equivalent Current Source Theorem – Norton’s Theorem – 2.7 Reduced analysis methods 2.7.1 Loop current analysis method 2.7.2 Node voltage analysis method – 2.8 Other methods 2.8.1 Compensation principle 2  2.6 Linear Circuits 2.6.1 Principle of Superimposition The superimposition method is applicable only to LINEAR circuits – The current and the voltage on each branch is the algebraic sum of the currents and voltages obtained as an effect of each individual source acting in the network When one source is switched on, the others sources must be switched off: – The voltage source switched off must be substituted with short circuits – The current source switched off must be substituted with open circuits The sources can be operated not only individually, but also in subset, provided they are acting no more than one time R1 R2 R1 R2 R1 R2 + + E R3 J = E R3 + R3 J I Ri = I Ri _ E + I Ri _ J VRi = VRi _ E + VRi _ J 3  2.6.2 Equivalent Voltage Source Theorem–Thévenin’s Theorem Statement In a LINEAR CIRCUIT where it has been identified one branch, for example at terminals A-B The behaviour of the network upstream to the branch A-B can be represented by a Real Voltage Source called Equivalent Voltage Source A I A I + Ri + + + E I V  E0 V B B 4  2.6.2 Equivalent Voltage Source Theorem–Thévenin’s Theorem Demonstration On insert, on the original circuit between A and B , 2 voltage sources in series having the same value of E0, but opposite sign. Demonstration: superimposition principle 1. The circuit with all the sources switched ON, except E0 having the + sign on the right side that is switched OFF gives the current between A and B: I’ 2. The circuit with the last source E0 having the + sign on the right side switched ON ad all the others switched OFF gives the current between A and B: I’’ E0 E0 I A + + For the Superimposition Principle it is always: + + V I=I’+I” E J if I' = 0 B  I = I” E0 E0 A + I' = 0 A I" + + + This condition is fulfilled if E0 is the voltage + E J V' + Ri V'' VAB without the branch A-B and therefore in OPEN CIRCUIT configuration B B 5  2.6.3 Equivalent Current Source Theorem–Norton’s Theorem Statement In a LINEAR CIRCUIT where it has been identified one branch, for example at terminals A-B The behaviour of the network upstream to the branch A-B can be represented by a Real Current Source called Equivalent Current Source A I A I + + + Ri E J V  J0 V B B 6  2.6.3 Equivalent Current Source Theorem–Norton’s Theorem Demonstration On insert, on the original circuit between A and B, 2 current sources in parallel having the same value of J0, but opposite sign. Demonstration: superimposition principle 1. The circuit with all the sources switched ON, except J0 having the current flowing in the same direction of I switched OFF, gives the current between A and B: I’ 2. The circuit with the source J0 having the current flowing in the same direction of I ON ad all the others switched OFF gives the current between A and B: I’’ A I + For the Superimposition Principle always: + E J V J 0 J0 I=I’+I” if I' = 0 B  I = I” A + I' = 0 A I" + + + + This condition is fulfilled if J0 is the current E J V' J0 Ri V'' J0 + flowing in the branch A-B with VAB=0 and B B therefore in SHORT CIRCUIT configuration 7  2.7 Reduced analysis methods For the analysis of a network the equations that can be written: – From KCL  n-1 linearly independent equations – From KVL  ℓ-(n-1) linearly independent equations – From characteristic equations of bipoles in a generic branch between two nodes X-Y ℓ equations in the form VK-xy=f(IK) By introducing “auxiliary variables” it is possible analyse a network reducing the number of unknown parameters – Loop current analysis method Auxiliary variable: loop current Jℓ – n-1 equations are satisfied for definition: the KCL equations – Node voltage analysis method Auxiliary variable: node voltage Vx (x is the generic node) – ℓ-(n-1) equations are satisfied for definition: the KVL equations 8  2.7 Reduced analysis methods 2.7.1 Loop current analysis method Auxiliary variable: loop current Jℓ – ℓ-(n-1) currents circulating in each loop/mesh belonging to one set of linearly independent loops or meshes – The unknown parameters are the ℓ-(n-1) loop currents – The n-1 KCL equations are automatically satisfied – The ℓ-(n-1) KVL equations must be verified on the set of linearly independent loops or meshes – After having solved the ℓ-(n-1) KVL equation system: The branch currents are obtained as algebraic sum of the loop currents in each branch The branch voltages are then derived from the characteristic equations of the bipoles inserted in each branch 9  2.7 Reduced analysis methods 2.7.1 Loop current analysis method - example R1 = 5Ω E1 = 200 V I1 I2 R2 = 6Ω E2 = 150 V E1 R1 R2 E2 R3 = 2Ω E3 = -120 V + + R4 = 10 Ω E4 = 50 V I5 J1 1 R5 = 8Ω I4 R5 2 J2 R6 = 3Ω + : network tree E4 R4 R6 I6 J3 1,2,3 : set of lin. ind. loop eq. E3 3 J1, J2, J3 : mesh current + ℓ-(n-1)=3 : lin. Indep. Eq. Sys. I3 R3  E1 − E4 = R1J1 + R5( J1 − J2 ) + R4 (J1 − J3 )  J1 = 0.33 A I1=J1=0.33 A   I2=J2=-9.8 A  − E2 = R2J2 + R5(J2 − J1) + R6(J2 − J3 )  J 2 = −9.8 A I3=J3=-6.4 A I4=J1-J3=6.77 A E + E = R J + R (J − J ) + R (J − J )  J = −6.4 A I5=J1-J2=10.13 A  3 4 3 3 4 3 1 6 3 2  3 I6=J3-J2=3.4 A 10  2.7 Reduced analysis methods 2.7.1 Loop current analysis method - peculiarities Voltage Sources have the positive sign in a loop equation if the positive terminal is in the same direction of a positive Jℓ current Voltage Sources have the negative sign in a loop equation if the positive terminal is in the opposite direction of a positive Jℓ current + In the i loop equation: +Ek Ji In the j loop equation: - Ek Ek Jj In the case in which in one branch there is a Current Source: – If the branch belongs to the network tree: Ji Jk Jj J i − J j = J k – If the branch belongs to the network co-tree: The loop current is imposed by the Current Source ad therefore the equations of that loop is unnecessary The overall system of equations has a reduced number of unknown and equations11  2.7 Reduced analysis methods 2.7.2 Node voltage analysis method Auxiliary variable: node Voltage Vx – n-1 node voltages are defined on n-1 nodes – On the n-th node the voltage is defined to be Vn=0 – The unknown parameters are the n-1 node voltages – The ℓ-(n-1) KVL equations are automatically satisfied – The n-1 KCL equations must be verified on the set of linearly independent nodes of the network – After having solved the n-1 KCL equation system: The voltages on each branch are the difference between the node voltages at each branch The currents on each branch are then derived from the characteristic equations of the bipoles inserted in each branch 12  2.7 Reduced analysis methods 2.7.2 Node voltage analysis method – equation ex. Ek Rk T + S K’ Ik Vk =VST =VS −VT =VSK' +VK'T =−Rk ⋅ Ik +Ek Ek VT −VS Ik = + Rk Rk After having identified the branch equation for each branch it will be necessary to write the n-1 KCL node equations of the network 13  2.7 Reduced analysis methods 2.7.2 Node voltage analysis method – Millmann formula Extreme case in which the network has only two nodes: A E V − VS R1 R2 Ri Rn Ik = k + T I1 + I2 + Ii + In + Rk Rk E1 E2 Ei En Ei VA B If it is considered: VB =0 I  i = − Ri Ri n n The KCL at node A: Ei n n Ei n VA   Ri i =1 Gi Ei  Ii =  R −  R = 0 The VA is therefore: VA = n 1 = n i =1 i =1 i =1 i i =1 i  R  Gi i =1 i i =1 14  2.8.1 Compensation principle - 1 Applicable to a LINEAR NETWORK If is known the currents in the network and in particular in the branches where flows Ik and In It is possible to determine the current in the branch k due to the topological variation in the branch n without solving again the full set of equations in the modified circuit It is requested to determine the current Ik’ and ∆Ik , if Ik e In are known – When a resistor RAB is inserted in the branch n Ik Ik’ A A + + + + V’ E J V In E J In’ RAB B B 15  2.8.1 Compensation principle - 2 It has to be inserted a Voltage Source in such a way that: Ik A 0 + + Ik E J V In A B + RAB + In Ik’ V E J A E + + + V’ B E J RAB In’ This is a circuit having the same currents as the original one it has been obtained B 16  2.8.1 Compensation principle - 3 The circuit that has been identified can be solved with the superimposition method Ik Ik’ A A + E + J + RAB V In = E + J V’ RAB In’ E B B + + The variation of the current in the branch k Ik” is obtained as the solution of a network in A which only one Voltage Source E is ON + RAB V” In” I k = I k '+ I k " E ∆I k = I k '− I k = − I k " + B 17