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Circuit Theory: Fundamental Analysis Methods - ET2050 HUST Lecture
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ET2050 Circuit theory ET2050: CIRCUIT THEORY Lecturer: Dao Le Thu Thao At the Faculty of Electronic Engineering Email: thao.daolethu@hust.edu.vn 3 Chapter 2: Fundamental methods to analyzing circuits • Scope: Studying the simple case: U, I relationship in the instant time, Ohm’s Law is being satisfied u(t)=R.i(t) 4 Chapter 2: Fundamental methods to analyzing circuits • Two groups: • G1: be applied to the linear circuits = the concepts of the equivalent circuits • G2: The systematic methods are Nodal analysis and Mesh analysis 5 Chapter 2: Fundamental methods to analyzing circuits • G1.1: The equivalent components - A chain of series resistors is equivalent to a single resistor whose resistance is the sum of series resistances. π π π‘Δ = ΰ· π π π=1 - Voltage division: π 2 π£2 = .π£ σ π π 6 Chapter 2: Fundamental methods to analyzing circuits • G1.1: The equivalent components - A set of parallel resistors is equivalent to a single resistor whose resistance is inverse of the sum of the inverses of the parallel resistances. (states again in terms of conductance) π πΊπ‘Δ = ΰ· πΊπ π=1 - Current division: πΊ2 π2 = .π σ πΊπ States, 2 parallel resistors π 1 π2 = .π π 1 + π 2 7 Chapter 2: Fundamental methods to analyzing circuits • G1.1: Examples Calculate the equivalent resistance for the following circuit: 8 Chapter 2: Fundamental methods to analyzing circuits • G1.1: Examples 9 Chapter 2: Fundamental methods to analyzing circuits • G1.1: The equivalent components What is happen when we connect a chain of source or set of sources 10 Chapter 2: Fundamental methods to analyzing circuits • G1.1: Examples • Voltage division and current division 11 Chapter 2: Fundamental methods to analyzing circuits • Source transformation method • States: Transform the voltage source into a current source and vice versa, provided the transformed source provides the same electrical energy to the rest of the circuit as the source being transformed • Transform voltage source to current source ππ ππ π π π π π= = . = ππ π π + π π‘ π π π π + π π‘ π π + π π‘ An ideal voltage source in series with internal resistance Ri can be replaced by a current ππ source with current source ππ = in parallel with resistance Ri π π 12 Chapter 2: Fundamental methods to analyzing circuits • Source transformation method • Transform current source to voltage source π π π π‘ π π‘ π π‘ π’ = ππ = ππ π π . = π’π π π + π π‘ π π + π π‘ π π + π π‘ π π‘ = ∞ ππ = π’π = ππ π π An ideal current source connected in parallel with the internal resistance Ri can be replaced by a voltage source with electromotive force ππ = ππ . π π connected in series with the resistor Ri 13 Chapter 2: Fundamental methods to analyzing circuits • G1.1: Example Use the source transformation method to calculate the voltage u Given is1= 5 mA; is2= 10 mA; e5=e3= 3V; R1=1,8kβ¦; R2=1,2kβ¦; R4=1kβ¦; R5=1,5kβ¦. 14 Chapter 2: Fundamental methods to analyzing circuits • Stacking method • Statement: In a linear circuit with many independent sources (voltage source, current source), the current (or voltage) at any branch is equal to the sum of the currents (or voltage) created by each separate source on the circuit. that branch, then other sources are eliminated (short circuit in voltage source, open circuit in current source). 15 Chapter 2: Fundamental methods to analyzing circuits • Stacking method β’ π = πππ + πππ β’ Stept 1: Open circuit in current source, consider voltage source es ππ 10 πππ = = = 5ππ΄ 2π 2. 103 β’ Stept 2: Short circuit of voltage source, consider current source is Principle of current division πππ −π ππ = ⇒ πππ = − = −10ππ΄ ππ 2π 2 β’ Stacking principle: π = πππ + πππ = 5ππ΄ − 10ππ΄ = −5ππ΄ 16 Chapter 2: Fundamental methods to analyzing circuits • G1.2: The Thevenin – Norton theorem: techniques: Thevenin method and Norton method i Black Box v=f(i) 17 Chapter 2: Fundamental methods to analyzing circuits • The Thevenin theorem: • Method: Replace circuit part A with a voltage source equivalent to the parameters eS and internal resistance Ri, but still ensure that the power supplied to the remaining circuit part remains unchanged. • A linear two-pole circuit can be replaced by a practical voltage source consisting of an ideal voltage source es in series with an internal resistor Ri with es equal to the open circuit voltage of the two poles (u12hm), Ri equal to the corresponding resistance equivalence of the two poles when the sources in the two poles are dissipated. • Hα» mαΊ‘ch: π’ = ππ = π’βπ π’ = π π . π + ππ π π’12βπ π π = = • NgαΊ―n mαΊ‘ch: 0 = π π . π + ππ (π = −π12ππ ) π12πππ π12πππ 18 18 Chapter 2: Fundamental methods to analyzing circuits • The Thevenin method - Step1: Given any linear circuit, rearrange it in the form of 2 one-port networks, two networks have connected a pair of wires. One one-port is simplified, and the other is left untouched (temporary forget). - Step 2: Define an opened circuit voltage Uoc as the voltage now appearing across the opened terminals. - Step 3: Turn off every independent source to form an inactive network. Leave dependent source unchanged. So, we can replace the inactive network as an equivalent resistor ( RTh) or the ratio U/I - Step 4: Connect an independent voltage source with value Uoc in series with the resistor of the inactive network. v = VT – i.RT i subcircuit V=f(i) 19 Chapter 2: Fundamental methods to analyzing circuits • The Norton theorem: • Method: Replace circuit part A with a current source equivalent to the parameters is and internal resistance Ri but still ensure that the power supplied to the remaining circuit part remains unchanged. • A linear two-pole circuit can be replaced by a practical current source consisting of an ideal current source is connected in parallel with an internal resistor Ri with is equal to the two-pole shortcircuit current (i12ngm), Ri equal to the equivalent resistance of the two poles when the sources in the two poles are dissipated. 20 Chapter 2: Fundamental methods to analyzing circuits • The Norton method - Step1: Given any linear circuit, rearrange it in the form of 2 one-port networks, two networks are connected by a pair of wires. One one-port is simplified, the other is left untouched (temporary forget). - Step 2: Define a current Isc as the current flow when short the terminals. - Step 3: Turn off every independent source to form an inactive network. Leave dependent source unchanged. So, we can replace the inactive network as an equivalent resistor ( RTh) or the ratio U/I - Step 4: Connect an independent current source with value Isc in parallel with the inactive network. i subcircuit i = iN – v/RN V=f(i) 21 Chapter 2: Fundamental methods to analyzing circuits • G1.2: The Thevenin – Norton theorem → replacing by a Thevenin circuit R1=4β¦ R3= 6β¦ i ES =24V R2=12β¦ u 22 Chapter 2: Fundamental methods to analyzing circuits • G1.3: Superposition - The proportionality principle of linear components gives a constant factor scaling combination. - The overall response of the circuit containing several sources is the sum of the response of each individual source with the others turned off / killed R1 u1 ES i2 R2 IS 23 Chapter 2: Fundamental methods to analyzing circuits • Example: 24 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis β’ The nodal analysis is a method of establishing a circuit equation with the variable being the potential of the nodes in the circuit, using the KCL law. β’ Method: i2 βStep1: Retain current source, convert the voltage source into the equivalent current source βStep 2: Consider any peak as a ground (zero potential) βStep 3: Establish the circuits equation with the variable being the potential of (Nn-1) remaining nodes in the circuit, using the KCL law ((Nn-1) equations) 25 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis β’ Convert the voltage source : ππ 1 = is3 i1 R1 π1 π = π1 . πΊ1 ; ππ 5 = 5 = π5 . πΊ5 π 1 π 5 β’ Choose node C as ground: : π’πΆ = 0 β’ Apply KCL law β Node A: R3 i3 i2 + e - 1 B A i5 i4 e5 R4 R2 + R5 C σππππ ππ = σππππ ππ π ο³ −π1 + π2 + π3 = ππ 1 + ππ 3 is3 π’πΆπ΄ π’π΄πΆ π’π΄π΅ − + + = ππ 1 + ππ 3 π 1 π 2 π 3 −(π’πΆ −π’π΄ ). πΊ1 + (π’π΄ −π’πΆ ). πΊ2 + (π’π΄ −π’π΅ ). πΊ3 = ππ 1 + ππ 3 πΊ1 + πΊ2 + πΊ3 . π’π΄ − πΊ3 . π’π΅ = ππ 1 + ππ 3 πΊ1 = 1 1 1 1 1 ; πΊ2 = ; πΊ3 = ; πΊ4 = ; πΊ5 = π 1 π 2 π 3 π 4 π 5 A is1 i1 i2 G1 G2 i3 B G3 i4 i5 is5 G4 C G5 26 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis β’ Apply KCL law: is3 i1 R1 −π3 + π4 + π5 = −ππ 3 − ππ 5 + e - 1 R3 i3 i2 β Node B: B A e5 + R5 C − π’π΄ − π’π΅ . πΊ3 + π’π΅ − π’πΆ . πΊ4 + π’π΅ − π’πΆ . πΊ5 = −ππ 3 − ππ 5 is3 A 1 1 1 1 1 πΊ1 = ; πΊ2 = ; πΊ3 = ; πΊ4 = ; πΊ5 = π 1 π 2 π 3 π 4 π 5 i4 R4 R2 π’π΄π΅ π’π΅πΆ π’π΅πΆ − + + = −ππ 3 − ππ 5 π 3 π 4 π 5 −πΊ3 . π’π΄ + πΊ3 + πΊ4 + πΊ5 . π’π΅ = −ππ 3 − ππ 5 i5 is1 i1 i2 G1 G2 i3 B G3 i4 i5 is5 G4 C G5 27 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis πΊ1 + πΊ2 + πΊ3 −πΊ3 is3 i1 π’π΄ −πΊ3 π + ππ 3 . π’ = π 1 −ππ 3 − ππ 5 πΊ3 + πΊ4 + πΊ5 π΅ R1 i2 πΊπ . π’π = ππ + e - 1 β’ Comment : B A i3 R3 i4 R4 R2 i5 e5 + R5 C β Solve the equations : π’π΄ , π’π΅ → Calculate the current in the branches : β The branch has no source conversion : π3 = (π’π΄ −π’π΅ ). πΊ3 β The branch has source conversion : π2 = π’π΄ . πΊ2 π1 R1 π’π΄ π1 π’πΆ = 0 π1 . π 1 + π’π΄ = π1 π4 = π’π΅ . πΊ4 π π’π΅ 5 π5 . π 5 − π’π΅ = π5 π1 − π’π΄ π1 = π 1 π5 + π’π΅ π5 = π 5 π5 R5 π’πΆ = 0 28 28 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis πΊ1 + πΊ2 + πΊ3 −πΊ3 is3 π’π΄ −πΊ3 π + ππ 3 . π’ = π 1 −ππ 3 − ππ 5 πΊ3 + πΊ4 + πΊ5 π΅ β’ Comment: β Conduction matrix : β Gkk = Σ sum of conductances connected to node k β Gkl = Σ sum of the conductances connecting node k to node l (always negative). A is1 i1 i2 G1 G2 i3 B G3 i4 i5 is5 G4 C G5 β Current source matrix : iskk = Σ sum of current sources connected to node k. β Entering the node → positive. β Going out the node → negative. β Number of equations: (Nn-1) → Suitable for circuits with a small number of nodes. β Nodal analysis Do not use when the circuit has mutual inductance. 29 29 29 Chapter 2: Fundamental methods to analyzing circuits • Using Nodal potential analysis: => how to select a reference node R1 ES R3 R2 R4 IS 30 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis – supernode based on using the extended version of KCL applying for a region (not only a node). => Total of currents entering a region = Total of currents leaving that region. R1 ES E3 A R2 B R4 IS 31 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal potential analysis - Normalization E3 R1 B A1 A2 ES R2 R4 iS E3 C 32 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Nodal analysis – appearance of dependent sources R1 ES A B R2 R4 C iS= g.u2 iS 33 Chapter 2: Fundamental methods to analyzing circuits • G2.1. Exercise 34 Chapter 2: Fundamental methods to analyzing circuits • G2.2. Mesh current analysis β’ The mest current method is a method of establishing circuit equations with the variable being the conventional mesh current flowing in closed loops, using the KVL law. β’ Method: βStep 1: Retain voltage source, convert the current source into the equivalent voltage source βStep 2: Select (Nb-Nn+1) meshes and select the direction of current in the corresponding mesh βStep 3: Establish the circuit equations with the selected mesh current, using the KVL law (Nb-Nn+1 equation) 35 Chapter 2: Fundamental methods to analyzing circuits • G2.2. Mesh current analysis Example: Establish circuit equations using the mesh current method i1 β’ Convert the current source : π3 = π 3 . ππ 3 β’ Choose (Nb-Nn+1) mesh currents R1 is3 i2 + e - 1 β’ Apply KVL: R3 i3 i4 e5 R4 R2 + R5 C β Mesh 1: ΰ· π’π = ΰ· ππ ο³ π’π 1 + π’π 2 = π1 πππ β i5 B A R1 πππ β ππ£1 . π 1 + ππ£1 . π 2 + ππ£2 . π 2 = π1 (π 1 + π 2 ). ππ£1 + π 2 . ππ£2 = π1 + - iv1 A R2 e3 R3 + - iv2 e1 R4 B e5 + iv3 R5 C 36 Chapter 2: Fundamental methods to analyzing circuits • G2.2. Mesh current analysis is3 π1 π 2 ππ£1 0 π 1 + π 2 . ππ£2 = π3 π 2 π 2 + π 3 + π 4 π 4 π5 π 4 + π 5 ππ£3 π 4 0 i1 + - R3 i3 β The branch has no source conversion : π3 = ππ 3 + ππ£1 −ππ£1 −ππ£2 π3 = ππ 3 − ππ£2 or −π3 +π4 +π5 + ππ 3 = 0 π3 = ππ 3 − ππ£2 −ππ£3 +ππ£3 π3 = ππ 3 − ππ£2 R5 C π1 = ππ£1 π2 = ππ£1 + ππ£2 π5 = ππ£3 π4 = −(ππ£2 + ππ£3 ) β The branch has source conversion : Node A: −π1 +π2 +π3 − ππ 3 = 0 e5 + R4 β Solve the groups of equations: ππ£1 , ππ£2 , ππ£3 → Calculate the current in the branches . i4 R2 e1 i5 B A i2 π . ππ£ = ππ β’ Comment: R1 ππ 3 i1 A i2 i3 B R3 i5 i4 iv2 37 Chapter 2: Fundamental methods to analyzing circuits is3 • G2.2. Mesh current analysis π1 π 2 ππ£1 0 π 1 + π 2 . ππ£2 = π3 π 2 π 2 + π 3 + π 4 π 4 π5 π 4 + π 5 ππ£3 π 4 0 i1 R1 i2 + e - 1 B A i3 R2 i4 R4 + e5 R5 C β’ Comment: β Circular impedance matrix R mesh R3 i5 β Voltage source matrix : β Rkk = Σ sum of resistance in the kth mesh. ekk = Σ total voltage source in the kth mesh β Rkl = Σ sum of common impedance (mutual β Positive: The voltage source is in the same inductance) between loop k and loop l. βͺ Positive: imesh k và imesh l same direction. βͺ Negative: ivong k và ivong l reverse direction. direction as the mesh current β Negative: The voltage source is in the reverse direction as the mesh current β Number of equations: (Nb–Nn+1) → Suitable for circuits with a small number of meshes 38 Chapter 2: Fundamental methods to analyzing circuits G2.2. Mesh current analysis • R1 E1 R3 R2 R4 E2 39 Chapter 2: Fundamental methods to analyzing circuits G2.2. Mesh current analysis – the appearance of the current source • R1 ES R3 R2 R4 iS 40 Chapter 2: Fundamental methods to analyzing circuits G2.2. Mesh current analysis – the supermesh • R1 R3 i3 ES iS R4 41 Chapter 2: Fundamental methods to analyzing circuits G2.2. Mesh current analysis – appearance of dependent source 2.i3 R1 R3 i3 ES iS R4 42 Chapter 2: • Practice: 43 THANK YOU ! 44
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