GATE 2020 EE Q.13: solution and answer

GATE 2020 · EEQ.13 · 1 mark · MCQ

Which of the options is an equivalent representation of the signal flow graph shown here?

GATE 2020 EE question 13: Signal flow graph with five nodes on a horizontal line: input node x_0, then x_1, x_2, x_3, output node x_4. — Figure from the GATE 2020 EE paper. It is part of the question.

(A)
Figure — input $1$ P $a (d + c)$ Q $1$ output, feedback branch $e$ from Q to P
(B)
Figure — input $1$ P $(a + c) d$ Q $1$ output, feedback branch $e$ from Q to P
(C)
Figure — input $1$ P $a (\frac{d}{1 - c d})$ Q $1$ output, feedback branch $e$ from Q to P
(D)
Figure — input $1$ P $a (\frac{c}{1 - c d})$ Q $1$ output, feedback branch $e$ from Q to P

Answer and solution

ANSWER

Option (C) is correct. The $d$ – $c$ pair is a forward branch with positive feedback around it, so it collapses to $\frac{d}{1 - c d}$ . In series with $a$ this gives $G = a (\frac{d}{1 - c d})$ from $x_{1}$ to $x_{3}$ , and branch $e$ is left as feedback around $G$ . Both graphs give $T = \frac{a d}{1 - c d - a d e}$ .

Official answer: CSource: Official answer key, GATE 2020 EE (IIT Delhi)

Read the arrowheads. Enlarged, the figure shows the arrowhead on arc $c$ at its top pointing left, so $c$ runs from $x_{3}$ back to $x_{2}$ . The arrowhead on arc $e$ at its bottom also points left, so $e$ runs from $x_{3}$ back to $x_{1}$ . Both are feedback branches, which means $c$ is not in parallel with $d$ .
Remove node $x_{2}$ using its node equations. Signals entering $x_{2}$ come from $a$ and $c$ , and its only outgoing branch is $d$ : $x_{2} = a x_{1} + c x_{3}, x_{3} = d x_{2}$ Substituting gives $x_{3} = a d x_{1} + c d x_{3}$ , so $x_{3} = \frac{a d}{1 - c d} x_{1}$ Seen as a rule, this is a forward branch $d$ with feedback branch $c$ around it. Their loop gain $c d$ is positive (signal-flow-graph gains never change sign), so the pair reduces to one branch $\frac{d}{1 - c d}$ in series with $a$ .
Series rule. The single branch from $x_{1}$ to $x_{3}$ therefore has gain $G = a (\frac{d}{1 - c d})$
What is left. Nodes $x_{1}$ and $x_{3}$ both survive, and their equations are unchanged: $x_{1} = x_{0} + e x_{3}$ and $x_{4} = x_{3}$ . So branch $e$ still runs from $x_{3}$ back to $x_{1}$ and wraps around $G$ . That is exactly the graph drawn in option (C).
Check with Mason's gain formula on the original graph. There is one forward path, $P_{1} = 1 \cdot a \cdot d \cdot 1 = a d$ . There are two loops: $L_{1} = c d$ (through $x_{2} \to x_{3} \to x_{2}$ ) and $L_{2} = a d e$ (through $x_{1} \to x_{2} \to x_{3} \to x_{1}$ ). They share nodes $x_{2}$ and $x_{3}$ , so there is no non-touching pair. That gives $Δ = 1 - c d - a d e$ . The forward path touches both loops, so $Δ_{1} = 1$ . $T = \frac{a d}{1 - c d - a d e}$
Option (C): $T = \frac{G}{1 - G e} = \frac{\frac{a d}{1 - c d}}{1 - \frac{a d e}{1 - c d}} = \frac{a d}{1 - c d - a d e}$ This matches. The other options do not: (A) gives $\frac{a ( c + d )}{1 - a ( c + d ) e}$ , (B) gives $\frac{( a + c ) d}{1 - ( a + c ) d e}$ , and (D) gives $\frac{a c}{1 - c d - a ce}$ . None of these equals $T$ . The answer is (C).

KEY IDEA

In a signal flow graph, a forward branch $d$ with a feedback branch $c$ around it forms a loop of gain $c d$ . Branch gains simply multiply, with no minus sign, so the pair reduces to $\frac{d}{1 - c d}$ and not $\frac{d}{1 + c d}$ . Removing a node such as $x_{2}$ never touches the branches between the nodes that remain, so $e$ from $x_{3}$ to $x_{1}$ survives unchanged.

WATCH OUT FOR

Not checking which way arrow $c$ points. If you read $c$ as a second forward branch alongside $d$ , you add them and get $a (d + c)$ , which is option (A). There are two smaller slips. One is using the block-diagram negative-feedback form $\frac{d}{1 + c d}$ . The other is putting $c$ in the numerator instead of $d$ , which gives option (D). In a signal flow graph the numerator must be the forward branch, and the loop gain always goes in as $1 - (loop gain)$ .

From the GATE 2020 EE paper. Every question of this paper · Download the full paper · All 10 Block diagrams & SFG questions, with notes

GATE 2020 EE Q.13: solution and answer

Answer and solution

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