This document covers Signal Flow Graph Examples.

1. By Prof. Hitesh Dholakiya
Signal Flow Graph
Examples
Control Engineering
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2. Steps to find transfer function from Signal flow graph
❖ Find Forward paths. (E.g. 𝑭𝟏, 𝑭𝟐, … … , 𝑭𝒏)
❖ Find Single Loops. (E.g. 𝑳𝟏, 𝑳𝟐, … … , 𝑳𝒌)
❖ Find Two Non-Touching Loops (E.g. 𝑳𝟏
′
= 𝑳𝟏 × 𝑳𝟐)
❖ Find ∆
∆= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 + ෍ 𝑻𝒘𝒐 𝑵𝒐𝒏 𝑻𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝑳𝒐𝒐𝒑𝒔 − … … … …
❖ Find ∆𝟏, ∆𝟐, … … , ∆𝒏 based on forward path
∆𝟏= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 𝑵𝒐𝒕 𝒕𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝒑𝒂𝒕𝒉 𝑭𝟏
∆𝟐= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 𝑵𝒐𝒕 𝒕𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝒑𝒂𝒕𝒉 𝑭𝟐
❖ Transfer Function
𝑪(𝒔)
𝑹(𝒔)
=
𝑭𝟏∆𝟏 + 𝑭𝟐∆𝟐 + … … + 𝑭𝒏∆𝒏
∆
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3. R C
-1
-1 -1
1 1
𝐺1 𝐺2 𝐺3
𝐺4 𝐺5
𝐺6
𝐹1 = 1 × 𝐺1 × 𝐺2 × 𝐺3 × 1 = 𝐺1𝐺2𝐺3
Find Forward Path
𝐹2 = 1 × 𝐺4 × 𝐺5 × 1 = 𝐺4𝐺5
Find Single Loops
𝐿1 = 𝐺1 × 𝐺2 × 𝐺3 × −1 = −𝐺1𝐺2𝐺3
𝐿2 = 𝐺2 × −1 = −𝐺2
𝐿3 = 𝐺3 × −1 = −𝐺3
𝐿4 = 𝐺6
𝐿5 = 𝐺4 × 𝐺5 × −1 = −𝐺4𝐺5
Find Two Non Touching Loops
𝐿1
′
= 𝐿1 × 𝐿4 = −𝐺1𝐺2𝐺3 × 𝐺6 = −𝐺1𝐺2𝐺3𝐺6
𝐿2
′
= 𝐿2 × 𝐿4 = −𝐺2 × 𝐺6 = −𝐺2𝐺6
𝐿3
′
= 𝐿2 × 𝐿5 = −𝐺2 × −𝐺4𝐺5 = 𝐺2𝐺4𝐺5
𝐿4
′
= 𝐿3 × 𝐿4 = −𝐺3 × 𝐺6 = −𝐺3𝐺6
Find ∆
∆= 1 − ෍ 𝑆𝑖𝑛𝑔𝑙𝑒 𝐿𝑜𝑜𝑝𝑠 + ෍ 𝑇𝑤𝑜 𝑁𝑜𝑛 𝑇𝑜𝑢𝑐ℎ𝑖𝑛𝑔 𝐿𝑜𝑜𝑝𝑠
∆= 1 − −𝐺1𝐺2𝐺3 − 𝐺2 − 𝐺3 + 𝐺6 − 𝐺4𝐺5 + (−𝐺1𝐺2𝐺3𝐺6 − 𝐺2𝐺6 + 𝐺2𝐺4𝐺5 − 𝐺3𝐺6)
Find ∆𝒊
∆𝟏= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 𝑵𝒐𝒕 𝒕𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝒑𝒂𝒕𝒉 𝑭𝟏
∆𝟏= 𝟏 − 𝑮𝟔
∆𝟐= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 𝑵𝒐𝒕 𝒕𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝒑𝒂𝒕𝒉 𝑭𝟐
∆𝟐= 𝟏 + 𝑮𝟐
𝑪(𝒔)
𝑹(𝒔)
=
𝑭𝟏∆𝟏 + 𝑭𝟐∆𝟐
∆
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4. R C
1 1 1
𝐺1 𝐺2 𝐺3
𝐺4
𝐻1
−𝐻1
−𝐻2
Find Forward Path
𝐹1 = 1 × 𝐺1 × 𝐺2 × 𝐺3 × 1 × 1 = 𝐺1𝐺2𝐺3
𝐹2 = 𝐺4 × 1 = 𝐺4
Find Single Loops
𝐿1 = 𝐺1 × 𝐺2 × 𝐻1 = 𝐺1𝐺2𝐻1
𝐿2 = 𝐺2 × −𝐻1 = −𝐺2𝐻1
𝐿3 = 𝐺2 × 𝐺3 × −𝐻2 = −𝐺2𝐺3𝐻2
No Two Non Touching Loops
Find ∆
∆= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 + ෍ 𝑻𝒘𝒐 𝑵𝒐𝒏 𝑻𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝑳𝒐𝒐𝒑𝒔
∆= 𝟏 − 𝑮𝟏𝑮𝟐𝑯𝟏 − 𝑮𝟐𝑯𝟏 − 𝑮𝟐𝑮𝟑𝑯𝟐
Find ∆𝒊
∆𝟏= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 𝑵𝒐𝒕 𝒕𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝒑𝒂𝒕𝒉 𝑭𝟏 = 𝟏
∆𝟐= 𝟏 − ෍ 𝑺𝒊𝒏𝒈𝒍𝒆 𝑳𝒐𝒐𝒑𝒔 𝑵𝒐𝒕 𝒕𝒐𝒖𝒄𝒉𝒊𝒏𝒈 𝒑𝒂𝒕𝒉 𝑭𝟐 = 𝟏 + 𝑮𝟐𝑮𝟑𝑯𝟐 + 𝑮𝟐𝑯𝟏 − 𝑮𝟏𝑮𝟐𝑯𝟏
𝑪(𝒔)
𝑹(𝒔)
=
𝑭𝟏∆𝟏 + 𝑭𝟐∆𝟐
∆
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