Karadeniz Technical University Department of Electrical and Electronics Engineering 6080 Trabzon, Turkey Chapter 4- Block Diagram Reduction Bu ders notları sadece bu dersi alan öğrencilerin i kullanımına açık olup, üçüncü sahıslara verilmesi, herhangi bir yöntemle çoğaltılıp başka yerlerde kullanılması, yayınlanması Prof. Dr. İsmail H. ALTAŞ ın yazılı iznine tabidir. Aksi durumlarda yasal işlem yapılacaktır. Chapter 4-2
HATIRLATMALAR QUIZ 2-6Mayıs 2008 Salı ünü saat 5:30 da Halis Duman Amfisinde yapılacaktır. 6Mayıs 2008 Salı günü saat 3:00 5:00 de arası A ve B gruplarına birlikte toplu ders yapılacaktır. Ödev 2 nin bugün teslim edilmesi gerektiğini unutmayınız. Chapter 4-3 BLOCK MANIPULATION RULES ain Block E a (s) (s) θ m (s) Summing Junction Pick-off Point R(s) E(s) (s) Chapter 4-4
BLOCK MANIPULATION RULES COMBININ SERIAL BLOCKS 2 2 COMBININ PARALLEL BLOCKS 2 m 2 Chapter 4-5 BLOCK MANIPULATION RULES CLOSIN A FEEDBACK LOOP U E ± =E E=U ± =H H E= E=U ± H =U±H U = ( U±H) =U±H m H=U ( m ) H =U = U m H H Chapter 4-6
BLOCK MANIPULATION RULES CLOSIN A FEEDBACK LOOP U(s) E(s) (s) H(s) (s) E(s)=U(s)-H(s)(s) (s)=(s)e(s) (s)=(s) [ U(s)-H(s)(s) ] (s)=(s)u(s)-(s)h(s)(s) (s) [ (s)h(s) ] =(s)u(s) (s) (s)= U(s) (s)h(s) U(s) (s) (s)h(s) This is a rule that is used extensively. (s) Chapter 4-7 BLOCK MANIPULATION RULES U E 2 ± H U 2 ± H 2 Chapter 4-8
Chapter 4-9 BLOCK MANIPULATION RULES MOVIN A SUMMIN JUNCTION AHEAD OF A BLOCK / Z - Z MOVIN A SUMMIN JUNCION PAST A BLOCK Z Z Chapter 4-0 MOVIN A PICKOFF POINT AHEAD OF A BLOCK BLOCK MANIPULATION RULES MOVIN A PICKOFF POINT PAST A BLOCK /
BLOCK DIARAM REDUCTION (s) 2 (s) E a (s) R sls L K i a a - H (s) K b T d (s) 3 (s) B s B m sj m 4 (s) With T d (s)=0, first combine the inner forward path. s s θ m (s) E a (s) 2 3 (s) 4 (s) θ m (s) H (s) Chapter 4- BLOCK DIARAM REDUCTION Next combine the feedback loop. E a (s) 2 3 ( s) H ( s) 4 (s) 2 3 θ m (s) The final series combination is E a (s) 234 ( s) H ( s) 2 3 θ m (s) Note: The defined values of the components may be substituted in to get the final transfer function in terms of system parameters. Chapter 4-2
BLOCK DIARAM REDUCTION iven a control system represented in the block diagram shown. Determine the relationship (s)/r(s). H2 R(s) 2 3 (s) H (a) Chapter 4-3 BLOCK DIARAM REDUCTION H2 R(s) 2 3 (s) (a) H H2 R(s) 2 3 (s) H Chapter 4-4
BLOCK DIARAM REDUCTION H2 R(s) 2 3 (s) H H2 R(s) 2-2H 3 (s) Chapter 4-5 BLOCK DIARAM REDUCTION R(s) - 23-2H23H2 (s) R(s) 23 (s) - 2H23H223 Chapter 4-6
Basic definitions Signal Flow raphs The block diagram reduction method works well for relatively simple block diagrams, but it gets very confusing for more complicated models. A signal flow graph represents the same information as the block diagram, however it leads to a set of rules that allow a systematic approach to finding the overall input/output transfer function. Chapter 4-7 Signal Flow raphs DEFINITION: - It is a graphical tool for control systems analysis and design - It consists of nodes and branches - The relationship between the inputs(s) and output(s) are determined by Mason s gain formula Chapter 4-8
Signal Flow raphs PROPERTIES OF FLOW RAPHS: Each branch is unilateral (one direction) Each node transmits the sum of all entering signals along each output branch A forward path is the path travelled by the signal in a forward direction A loop is formed when the signal travels and returns to its original source Special nodes: Source node - no inputs Sink node - no outputs Chapter 4-9 Signal Flow raphs The main steps are as follows: a) construct the signal flow graph either from a block diagram or from the basic physical connection of system components (the transfer functions of the components must be known). b) Identify and calculate the various paths and loops in the signal flow graph. c) With the results from b), apply a formula, Mason s formula, to determine the overall transfer function. Chapter 4-20
Signal Flow raphs Construction Nodes, branches and transmission elements x t 2 x 2 node branch node x 2 = t 2 x x t 2 x 2 t 2 2 (s) Chapter 4-2 Signal Flow raphs Summation node Construction Distribution node x 2 x t 24 t 34 t 4 x 4 x t 2 x 2 t 3 t 4 x 3 x 3 x 4 x 4 = t 4 x t 24 x 2 t 34 x 3 x 2 = t 2 x x 3 = t 3 x x 4 = t 4 x Chapter 4-22
Signal Flow raphs Construction. A SINLE BRANCH V T2 V2 V,V 2 are called nodes and T 2 is called a branch This single branch represents the equation V 2 = T 2 V Note: V = V 2 /T 2 (each branch is unilateral) Chapter 4-23 Signal Flow raphs Construction 2. SUM OF TWO BRANCHES V T3 V3 T23 V2 V3 = T3V T23V2 Chapter 4-24
Signal Flow raphs Construction 3. PARALLEL BRANCHES T2b V T2a V2 V2 = (T2a T2b) V V T2 V2 V 2 = T 2 V V = T 2 V 2 T2 Chapter 4-25 Signal Flow raphs Construction 4. CASCADED BRANCHES V T2 V2 T23 V3 5. NODE ELIMINATION V3 = T 2 T 23 V V3 = T3V T23V2 and V4 = T34V3, then V 4 = (T 34 T 3 )V (T 34 T 23 )V 2 V T3 V3 T34 V4 V T34 T3 V4 V2 T23 T34 T23 Chapter 4-26
Signal Flow raphs Construction Write down and label the nodes from input to output, representing all the important signals. Draw in all the branches connecting the nodes and write down their transmission functions. Check for any additional nodes and branches required in the feedback paths. Chapter 4-27 Signal Flow raphs Example Servomotor System (s) 2 (s) 3 (s) 4 (s) E a (s) x x 2 x 3 x 4 θ m (s) K i R a s L a B m s J m s x5 - H (s) K b T d (s) E a (s) x x 2 2 T d (s) x 3 x 4 3 4 θ m (s) -H Chapter 4-28
Signal Flow raphs Source node: only has outgoing branches. Sink node: only has incoming branches. Path: continuous unidirectional succession of branches (passes through no node more than once). Forward path: a path from input to output. Feedback path or loop: originates and terminates at the same node. Non-touching paths: paths with no common nodes. Path gain or loop gain: product of branch gains or transmission functions along the path. Chapter 4-29 T = Signal Flow raphs FP ( - Loops not touching FP) T = M k = k k L p ; p the number of forward paths. k where = ( all loop gains) loop gain products of non touching pairs loopgain productsof non touching groups of three - Loops loopgain productsof non touching groups of four = M k k Chapter 4-30
Signal Flow raphs... continued T = M k k k k = L p ; p the number of forward paths. where M = k k k = th forward path gain defined only using loops not touching the k th forward path. Chapter 4-3 Signal Flow raphs. Identify all forward paths and write the path gains M k. 2. Identify all loops and write the loop gains. 3. Identify all non touching loop pairs and write down the loop gain products. 4. Do the same for groups of 3, 4, non touching loops. 5. Calculate as defined. 6. Identify all loops not touching forward path k, and repeat steps 2 5 to calculate k. 7. Apply Mason s formula to calculate the overall transfer function. Chapter 4-32
Signal Flow raphs Servomotor System E a (s) x θ m E a ( s) ( s) x 2 2 T d (s) x 3 x 4 3 4 Example θ m (s) Forward paths: -H M E a x x 2 x 3 x 4 θ m ain = 2 3 4 Feedback loops: L x x 2 x 3 x 4 x Loop gain = - 2 3 H Non touching loop pairs: none Chapter 4-33 Signal Flow raphs Servomotor System...continued then, = 2 3 H Loops not touching forward path : none then, = Apply Mason s formula. θ T = E m a () s = (s) M 23 = 4 H 2 3 Chapter 4-34
Signal Flow raphs Servomotor System...continued Consider the transfer function from the disturbance input, T d (s) to the output, θ m (s), with (E a = 0). The forward path is now M T d x 3 x 4 θ m ain = 3 4 The loops are not changed, so and are unchanged. Applying Mason s formula; () s M 34 T = θ m = = T ( s) H d Note: The denominator has not changed. 2 3 Chapter 4-35 Signal Flow raphs Example 6 R (s) x 2 x 2 x 3 3 4 x 4 5 C(s) R(s) -H -H 2 Forward Paths: M R R x 3 x 4 C ain = 6 4 5 M 2 R R x x 2 x 3 x 4 C ain = 2 3 4 5 Chapter 4-36
Signal Flow raphs... continued Feedback loops: L x x 2 x Loop gain = 2 H L 2 x 3 x 4 x 3 Loop gain = 4 H 2 Non touching loop pairs: L L 2 Loop gain = 2 4 H H 2 then = ( 2 H 4 H 2 ) ( 2 4 H H 2 ) = 2 H 4 H 2 2 4 H H 2 Chapter 4-37 Signal Flow raphs... continued Loops not touching forward path : L then, = ( 2 H ) = 2 H Loops not touching forward path 2 : none then, 2 = Now applying Mason s formula T C( s) = R( s) = M M 2 2 = 645 ( 2H) 234 H H H H 2 4 2 2 4 2 5 Chapter 4-38
Signal Flow raphs Example Vin A V B V2 C V3 D Vout F E V4 Vout / Vin = Vout / Vin = FP - LP ABCD - CEF Chapter 4-39 Signal Flow raphs Example C Vin A V B V2 E Vout D FP = ACE, FP2 = ABDE F LOOP # = B, LOOP #2 = EF T = Vout/Vin = = ACE [ - 0 ] ABDE [ - 0 ] - ( B EF ) ( BEF ) ACE ABDE - ( B EF ) ( BEF ) Chapter 4-40
Signal Flow raphs Example C Vin A B D E F Vout. F.P. = ACF, F.P.2 = ABDEF, I LOOP # = DI LOOP #2 = FH H T = Vout/Vin = = ACF [ - DI ] ABDEF [ - 0 ] - ( DI FH ) ( DIFH ) ACF - ACFDI ABDEF - DI - FH DIFH Chapter 4-4 Signal Flow raphs Example B Vin A C D E F. Vout H FP = () (B) () = B LOOP # = CH FP2 = () (A) (C) (D) (E) (F) () = ACDEF LOOP #2 = E T = Vout/Vin = T = Vout/Vin = B [ - ( CH E ) ( CHE ) ] ACDEF [ - 0 ] - ( CH E ) ( CHE ) B - BCH BE BCHE ACDEF - CH - E CHE Chapter 4-42