DR. GYURCSEK ISTVÁN Circuit Theorems Sources and additional materials (recommended) q Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3 q Ch. Alexander, M. Sadiku: Fundamentals of Electric Circuits, 6th Ed., McGraw Hill NY 2016, ISBN: 978-0078028229 q Simonyi K.: Villamosságtan. AK Budapest 1983, ISBN:9630534134 q Dr. Selmeczi K. Schnöller A.: Villamosságtan 1. MK Budapest 2002, TK szám: 49203/I q Dr. Selmeczi K. Schnöller A.: Villamosságtan 2. TK Budapest 2002, ISBN:9631026043 q Zombory L.: Elektromágneses terek. MK Budapest 2006, (www.electro.uni-miskolc.hu) 1 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 2 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Linearity Property (A) Conditions of linearity (Mathematics) Homogeneity / = 0(1) ( $ / = 0(( $ 1) Additivity / + = 0 1 +, / - = 0 1 - / + + / - = 0 1 + + 1 - (B) Linear circuit (no independent source internally) Linear relationship bw. v S input (excitation) and i output (response) Homogeneity Additivity! = # $ % & ( $! = # $ (( $ % & )! + = # $ % &+,! - = # $ % &-! + +! - = # $ % &+ + # $ % &- = # $ % &+ + % &- (C) OUTCOMES IN CIRCUIT ANALYSIS 1 [HOMOGENEITY] [NEXT SLIDE EXAMPLE]; 2 [ADDITIVITY] [SUPERPOSITION PRINCIPLE] (D) WARNING! The power relation is nonlinear!! + 3 + =! - + $ 4 2! - 3 - =! - - $ 4 567! + +! - 3 +- =! + +! - - $ 4 =! - + $ 4 +! - - $ 4 + 2 $! + $! - $ 4 3 +- 3 + + 3-3 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calc. Example with Homogeneity Find I 0 in the circuit.! : #$$%&' ( ) = 1, 8 : ( 9 = ( / + ( ) = 3, > : (? = ( 9 + ( < = 5, - :. / = 3 + 5 3 ( ) = 8. : :. 9 = ( 9 3 2 +. / = 14. @ : A%B (? DEFG = ( H = 5 3 5 = 15, 5 : ( / =. / 4 = 2, ; : ( < =. 9 7 = 2, I : Bh%$ ( ) DEFG = 5 3 ( ) = 3, 4 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 5 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Superposition Principle Steps to apply q Turn off independent sources except one and find output. q Repeat it for each of independent sources. q Find total by adding all the contributions. Example Find v by using superposition.! =! # +! % 12( # 6 = 0 ( # = 0.5 /! # = 4( # = 2 1 or! # = 4 4 + 8 6 = 2 1 ( 5 = 8 4 + 8 3 = 2 /! % = 4( 5 = 8 1! =! # +! % = 2 + 8 = 10 1 6 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 7 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Source Transform Source transformation q Another tool for simplifying circuits q Active equivalent transformation q For real generators only! (/ 0, ) / 78 = / 9 = / $ 78 = & 9 6 /!h#$#%&% () & +, = $. /, 12342% () & +, = &.!h#$#%&% 5) $ +, = $., 12342% 5) $ +, = &. 6 / & 9 = $ 78 / 8 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calculation Example v 0 =?! = 2 2 + 8 & 2 = 0.4 * +, = 8! = 3.2. /0 +, = 2 34 &! = 2 & 8 2 + 8 & 2 = 3.2. 9 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 10 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Thevenin s Theorem ' "# = ( )*! "# =! %& 11 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Thevenin s Theorem Determining R Th q CASE 1 no dependent sources q CASE 2 dependent sources also! "# =! %&! "# = ' ( ) ( Another way (later on)! "# = ' *+ ),+ 12 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calculation Example Find the current through the 6, 16 and 36 ohms load.! "# = 4 12 + 1 = 4 * 12 16 + 1 = 4 Ω 32 + 4/ 0 + 12 / 0 / 1 = 0 / 1 = 2 3, / 0 = 0.5 3 7 "# = 12 / 0 / 1 = 12 * 2.5 = 30 7 89 32 7 "# 4 + 2 = 7 "# 12 7 "# = 30 7 < = = 7 "#! "# +! = = 30 4 +! = = 3, 1.5, 0.75 3 13 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 14 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Norton s Theorem +, = ( )' +, = * "# * "# = % &'! "#! "# = % &' ( )' = * "# +, =!, 15 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calculation Example Find the Norton equivalent circuit.! " = 5 8 + 4 + 8 = 5 ) 20 25 = 4 Ω 16 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calculation Example! " = 2 %, 20! ( 4! " 12 = 0! ( = 1 % =! -. = / 0 17 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calculation Example Alternatively & ' = ( )* + )*! " = 2 % 25! - 4! " 12 = 0! - = 0.8 % 5 67 = ( )* = 5! - = 4 ( & ' = ( )* + )* = 4 4 = 1 % 18 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 19 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Maximum Power Transfer! = # $ % & = ' () % () + % & $ % & +! $ % () + % $ & 2% & % () + % & $ % () + % & 2% & = ' +% () & % () + %. = ' () & % () + % / = 0 & % () = % & + $! $ +% == ' () $ + % () % & & +% & % () + % / & < 0 >?@#>A>!BCDE! 123 = ' () $ 4% () 6 =! 123! 7 = $ ' () 8 4% () $ = 0.5 = 50% ' () 8 2% () 20 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Calculation Example Find R L for maximum power transfer. Find the maximum power.! "# = 2 + 3 + 6 12 = 5 + 6, 12 18 = 9 Ω 12 + 181 2 121 3 = 0, 1 3 = 2 6 1 2 = 2 3 6 12 + 61 2 + 31 3 + 2, 0 + 7 "# = 0 7 "# =22 V! ; =! "# = 9 Ω, < =>? = 223 4, 9 = 13.44 B 21 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
q Linearity Property q Superposition Principle q Source Transformation q Thevenin s Theorem q Norton s Theorem q Maximum Power Transfer q Applications: Practical Sources 22 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Real Voltage and Current Sources! " =! $ % " % $ + % " ' " = ' $ % ( % ( + % " 23 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.
Questions 24 gyurcsek.istvan@mik.pte.hu 2018. 10. 13.