Sub Code Sub Name Credits Semester : 19ECE301 : Control Theory (Pre-Requisite(s): Signals and Systems) :4 :V Branch Faculty : ECE : Ms. Bhavana V Assistant Professor, Dept. of ECE Lecture 15-18 Reduction of Multiple Subsystems Signal Flow Graph Ms. Bhavana V,Assistant Professor, Dept. of ECE 1 Signal-flow graphs • Signal-flow graphs are an alternative to block diagrams. This approach was coined by S.J Mason • Block diagrams, consists of blocks, signals, summing junctions, and pickoff points, etc • A signal-flow graph consists only of branches, which represent systems, and nodes, which represent signals. • The main advantage of SFG, it doesn't require any reduction process because of availability of gain formula which gives the transfer function of control system without any transformation. Ms. Bhavana V,Assistant Professor, Dept. of ECE 2 Signal-flow graphs • SFG is a graphical representation of the block diagram or representation of the relationships between the variables of a set of linear algebraic equations. Ms. Bhavana V,Assistant Professor, Dept. of ECE 3 Signal-flow graphs 1. The SFG is applicable only to linear time invariant systems 2. The signal flow along the branch is the direction of arrow associated with the branch 3. The signal gets multiplied by the branch gain or branch transmittance when it travels through branch. Ms. Bhavana V,Assistant Professor, Dept. of ECE 4 Signal-flow graphs 4. The value of the variable represented by the node is given by the algebraic sum of all the signals entering that node. Ms. Bhavana V,Assistant Professor, Dept. of ECE 5 Signal-flow graphs 5. The value of the variable represented by any node is available to all the branches leaving that node. Ms. Bhavana V,Assistant Professor, Dept. of ECE 6 Numerical A system is described by the following set of equations . Construct the signal flow graph of the system. Ms. Bhavana V,Assistant Professor, Dept. of ECE 7 Solution Note : The SFG should contain five nodes, since there are five variables. Ms. Bhavana V,Assistant Professor, Dept. of ECE 8 Terminologies used in SGF • Source Node: The node having only outgoing branches is known as the Source Node or Input Node In Fig A, the source node is ๐ฅ1 • Sink Node/ Output Node: The node having only incoming branches is known as sink node or output node. In Fig A, the source node is ๐ฅ5 = output node, after transmitting ๐ฅ5 through a branch of unity gain . • Chain Node: A node having incoming and outgoing branches is known as chain node. In Fig A, the chain nodes = ๐ฅ2 , ๐ฅ3 , ๐ฅ4 Ms. Bhavana V,Assistant Professor, Dept. of ECE 9 Terminologies used in SGF • Forward Path : It is a path from input node to the output node. ---- In forward path no node should be traversed more than once. ---- In Fig A, • Feedback Loop: A path which originates from a particular node and terminating on the same node , travelling through at least one other node without tracing any other node twice is called feedback loop or simply the loop. Ms. Bhavana V,Assistant Professor, Dept. of ECE 10 Terminologies used in SGF • Self Loop: A loop which originates and terminates on the same node is called the self loop. • Path Gain: The product of all the branch gains in a forward path is called the path gain. • Loop Gain: It is the path gain of a loop Ms. Bhavana V,Assistant Professor, Dept. of ECE 11 Terminologies used in SGF Non Touching Loops: Two loops are said to be non touching if they don’t share a common node. Ms. Bhavana V,Assistant Professor, Dept. of ECE 12 Masons Gain Formula The gain formula which gives the relationship between input variable and output variable of SFG is known as Mason’s gain formula. Where ๐ถ(๐) σ๐ ๐พ=1 ๐๐พ โ๐พ ๐ ๐ = = ๐ (๐) โ N= Number of forward paths between R(S) and C(S) ๐๐พ =gain of the ๐พ ๐กโ forward path โ = Determinant of the SFG , given by โ = 1 – [ sum of all the individual feedback loop gains including the self loop] + [ sum of the gain products of all combinations of two non touching loops ] – [ sum of the gain products of all combination of three non touching loops ] + ………… Ms. Bhavana V,Assistant Professor, Dept. of ECE 13 Masons Gain Formula โ๐พ = 1 – [ sum of all the individual feedback loop gains including the self loop which are not present in the ๐พ ๐กโ forward path] + [ sum of the gain products of all combinations of two non touching loops which are not present in ๐พ ๐กโ forward path ] – [ sum of the gain products of all combination of three non touching loops which are not present in the ๐พ ๐กโ forward path ] + ………… Ms. Bhavana V,Assistant Professor, Dept. of ECE 14 Numerical For the signal flow graph shown below ,obtain the transfer function ๐ฅ5 ๐ฅ1 using masons gain formula Ms. Bhavana V,Assistant Professor, Dept. of ECE 15 Solution Ms. Bhavana V,Assistant Professor, Dept. of ECE 16 Solution (Contd…) โ = 1 – [ sum of all the individual feedback loop gains including the self loop] + [ sum of the gain products of all combinations of two non touching loops ] – [ sum of the gain products of all combination of three non touching loops ] + ………… โ๐ = 1 – [ sum of all the individual feedback loop gains including the self loop which are not present in the 1st forward path] + [ sum of the gain products of all combinations of two non touching loops which are not present in 1st forward path ] – [ sum of the gain products of all combination of three non touching loops which are not present in the 1st forward path ] + ………… โ๐ = 1 – [ sum of all the individual feedback loop gains including the self loop which are not present in the 2nd forward path] + [ sum of the gain products of all combinations of two non touching loops which are not present in 2 nd forward path ] – [ sum of the gain products of all combination of three non touching loops which are not present in the 2 nd forward path ] + ………… Ms. Bhavana V,Assistant Professor, Dept. of ECE 17 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 18 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 19 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 20 Numerical For the signal flow graph shown below ,obtain the transfer function ๐ฆ7 ๐ฆ1 and ๐ฆ6 ๐ฆ1 using masons gain formula Ms. Bhavana V,Assistant Professor, Dept. of ECE 21 Solution Ms. Bhavana V,Assistant Professor, Dept. of ECE 22 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 23 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 24 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 25 Numerical For the block diagram shown below, construct the SFG. Find the transfer function ๐ถ(๐) ๐ (๐) using Masons Gain Formula Ms. Bhavana V,Assistant Professor, Dept. of ECE 26 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 27 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 28 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 29 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 30 Numerical For the network shown below, construct the SFG. Find the transfer function ๐0 (๐) ๐๐ (๐) using Masons Gain Formula Ms. Bhavana V,Assistant Professor, Dept. of ECE 31 Solution Ms. Bhavana V,Assistant Professor, Dept. of ECE 32 Solution (Contd…) Ms. Bhavana V,Assistant Professor, Dept. of ECE 33 Solution (Contd…) ๐0 (๐) 10 = ๐๐ (๐) 10๐ 2 + 21๐ + 10 Ms. Bhavana V,Assistant Professor, Dept. of ECE 34