Interview questions & answers
Q1. What is a signal flow graph and how is it different from a block diagram?
A signal flow graph (SFG) is a directed graph where nodes represent system variables and directed branches represent the transmission (gain) between variables, while a block diagram uses functional blocks and summing junctions connected by signal lines. The SFG for a second-order system G(s) = ω²n/(s² + 2ζωns + ω²n) can be drawn with 3 nodes and feedback branches, while its block diagram needs a summer, two integrators, and gain blocks. SFGs are preferred for systematic gain computation using Mason's formula because they avoid the graphical reduction rules needed for block diagrams.
Follow-up: What are the basic elements of a signal flow graph?
Q2. State Mason's gain formula and define each term.
Mason's gain formula gives the overall transfer function T = (Σk Pk Δk)/Δ, where Pk is the path gain of the k-th forward path, Δ = 1 − (sum of all individual loop gains) + (sum of products of non-touching loop gains taken two at a time) − … is the graph determinant, and Δk is the cofactor of the graph with all loops touching the k-th path removed. For a forward path with gain G1G2 and a feedback loop with gain −G1G2H, Δ = 1 + G1G2H and Δ1 = 1, giving T = G1G2/(1 + G1G2H), identical to the block diagram reduction result. Mason's formula avoids step-by-step block diagram algebra and is directly applicable to multi-loop systems.
Follow-up: What does it mean for two loops to be 'non-touching' in Mason's formula?
Q3. How do you identify forward paths in a signal flow graph?
A forward path is any continuous directed path from the input node to the output node that visits each node at most once, and its path gain is the product of all branch transmittances along the path. In an SFG with nodes 1→2→3→4→5 where node 5 is output, a direct path 1→2→3→5 (skipping 4) is a valid forward path if that branch exists, and its gain is the product of the three branch gains. Identifying all forward paths is the first step in applying Mason's formula; most textbook problems have 1–3 forward paths, while practical system SFGs may have 5–10.
Follow-up: Can a forward path and a loop share a node in a signal flow graph?
Q4. What is a self-loop in a signal flow graph and how does it enter the Mason's formula?
A self-loop is a branch that starts and ends at the same node, with its loop gain equal to the branch transmittance, representing a direct feedback from a variable to itself such as a local integral or derivative term. In an SFG modeling a PID controller, a self-loop of gain −KP at the error node represents the proportional feedback directly on that node. Self-loops are included as individual loops in the Δ calculation just like any other loop, added in the (−1)^1 × L term of the graph determinant.
Follow-up: What is the loop gain of a self-loop with transmittance a, and how does it affect the node variable?
Q5. What is the graph determinant Δ and how do you compute it for a 2-loop system?
The graph determinant Δ = 1 − (L1 + L2) + L1·L2 for two non-touching loops with gains L1 and L2, or Δ = 1 − (L1 + L2) for two touching loops since their product term is zero. For a unity feedback system with forward path G(s) and single feedback loop with gain −G(s): Δ = 1 − (−G(s)) = 1 + G(s), and T = G(s)/Δ = G(s)/(1+G(s)), the standard closed-loop formula. The key step is correctly identifying which loops share nodes (touching) versus loops that share no node (non-touching).
Follow-up: How does Δ simplify when all loops in the SFG touch each other?
Q6. How do you convert a block diagram to a signal flow graph?
To convert a block diagram to an SFG, assign a node to each signal (block input, summer output, system output), draw a directed branch for each block with its transfer function as the transmittance, and replace each summer with a node where incoming branches carry signed gains (+1 for positive, −1 for negative). A unity feedback block diagram with G(s) and H(s) in the feedback path becomes a 4-node SFG: input R, error E, plant output C, feedback signal B, with branches G(s) from E to C, H(s) from C to B, −1 from B to E, and 1 from R to E. Once drawn correctly, the SFG gives the same transfer function as block diagram reduction but in one formula application.
Follow-up: What node does the error signal correspond to in the SFG of a standard closed-loop control system?
Q7. What is the cofactor Δk and why is it needed in Mason's formula?
The cofactor Δk is the graph determinant of the SFG obtained by removing all loops that touch the k-th forward path, ensuring that only loops independent of the k-th path contribute to its effective transmission. For a two-forward-path system where path P2 touches all loops, Δ2 = 1 because there are no remaining non-touching loops after removing all those touching P2. When Δk = 1 for all forward paths (common in simple systems), Mason's formula reduces to T = (ΣPk)/Δ, simplifying computation.
Follow-up: Give an example where Δk ≠ 1 for a forward path.
Q8. How does the SFG method apply to electronic amplifiers, such as a two-stage op-amp?
A two-stage amplifier with gain A1 and A2 in cascade with feedback β can be modeled as an SFG with nodes for input, output of stage 1, output of stage 2, and feedback, where the forward path gain is A1A2 and the feedback loop gain is −A1A2β. Applying Mason's formula directly: T = A1A2/(1 + A1A2β), which is the classic feedback amplifier gain formula used for a TL071 op-amp with a resistive voltage divider in the feedback path. The SFG approach is more systematic than circuit algebra for multi-loop amplifiers like cascode or differential pairs with multiple feedback paths.
Follow-up: What happens to the gain of a feedback amplifier as loop gain A1A2β becomes very large?
Q9. What is the significance of the loop gain in stability analysis using SFG?
The loop gain of each feedback loop in the SFG determines the system's stability: if the sum of loop gains evaluated at s = jω results in a graph determinant Δ(jω) = 0, the system is on the verge of instability (poles on the imaginary axis). In a phase-locked loop (PLL) SFG, the loop gain LG = KvKd/s (in rad/s) determines whether the PLL locks and how fast it acquires frequency, with stability requiring the phase margin to be positive at the gain crossover frequency. The loop gain expression directly from the SFG is the open-loop transfer function used in Bode and Nyquist stability criteria.
Follow-up: How does Mason's gain formula relate to the characteristic equation of a control system?
Q10. What is the difference between a touching and non-touching loop in an SFG?
Two loops are touching if they share at least one common node, meaning their signals interact, while non-touching loops have completely disjoint sets of nodes, meaning they operate independently in the graph. In the SFG of a two-joint robot arm, the inner velocity loop and outer position loop share the velocity node, making them touching loops, so their product does not appear in the Δ formula. Correctly identifying non-touching pairs is the most common error in applying Mason's formula, and a systematic approach is to list all loops, then check every pair for shared nodes.
Follow-up: What is the maximum number of non-touching loop combinations considered in Mason's formula?
Q11. Apply Mason's gain formula to find T(s) for a system with forward path gain G1G2G3, one feedback loop G1G2H1, and one feedback loop G2G3H2.
The two loops have gains L1 = −G1G2H1 and L2 = −G2G3H2; to check if they touch: L1 involves nodes with G1 and G2, L2 involves nodes with G2 and G3 — they share the G2 node, so they are touching and their product is zero in Δ. Therefore Δ = 1 − (L1 + L2) = 1 + G1G2H1 + G2G3H2, and with one forward path P1 = G1G2G3 that touches both loops, Δ1 = 1, giving T = G1G2G3/(1 + G1G2H1 + G2G3H2). This is the standard result used in verifying state-space model transfer functions for two-stage process control loops.
Follow-up: How would the answer change if H1 and H2 loops were non-touching?
Q12. What is the relationship between the characteristic equation and the graph determinant?
The characteristic equation of a closed-loop system is Δ(s) = 0, where Δ is the graph determinant from Mason's formula, and its roots are the closed-loop poles that determine stability and transient response. For a system with Δ = 1 + G(s)H(s), setting Δ = 0 gives 1 + G(s)H(s) = 0, the standard characteristic equation solved by root locus or Routh criterion. The Routh-Hurwitz array is applied to the polynomial obtained by substituting s-domain expressions into Δ(s) = 0 to count poles in the right-half plane.
Follow-up: How do you obtain the closed-loop poles from the characteristic equation of a second-order system?
Q13. How does an SFG represent state-space equations?
A state-space model ẋ = Ax + Bu, y = Cx + Du is directly represented as an SFG with integrator branches (gain 1/s) for each state variable, feedback branches from states to derivatives using A matrix elements, input branches using B, and output branches using C, with direct feedthrough D as a branch from input to output. The SFG for a second-order system with A = [[0,1],[−ω²n, −2ζωn]] has two integrators in series and two feedback branches from position and velocity states to the acceleration node. Mason's formula applied to this SFG reproduces the transfer function Y(s)/U(s) = ω²n/(s² + 2ζωns + ω²n) without algebraic matrix inversion.
Follow-up: What is the physical meaning of the integrator branches (1/s) in the state-space SFG?
Q14. What are the limitations of signal flow graphs compared to bond graphs or block diagrams?
SFGs require all system equations to be written in explicit input-output form with all variables defined before drawing, making them unsuitable for bilateral (energy-exchanging) systems like electromechanical coupling where power flows in both directions, which bond graphs handle naturally. For a DC motor with back-EMF coupling both electrical and mechanical domains, the SFG requires separate electrical and mechanical equations with the coupling terms as cross-branches, while a bond graph represents the same system with a single gyrator element. SFGs also become unwieldy for systems with more than 8–10 nodes and 15+ branches, where MATLAB/Simulink block diagram tools are more practical.
Follow-up: When would you prefer a block diagram reduction over Mason's formula for finding transfer functions?
Q15. How do you use SFG to find the transfer function of a PID controller in a closed-loop system?
The PID controller C(s) = Kp + Ki/s + Kd·s is represented in the SFG as three parallel forward branches (Kp, Ki/s, Kd·s) from the error node to the plant input node, summed at a node before the plant G(s), with unity or sensor H(s) feedback closing the outer loop. The total forward path gains are P1 = Kp·G(s), P2 = (Ki/s)·G(s), P3 = Kd·s·G(s), all sharing the same feedback loop L = −G(s)·C(s)·H(s), so Δ1 = Δ2 = Δ3 = 1. The transfer function T = (P1+P2+P3)/Δ = C(s)G(s)/(1 + C(s)G(s)H(s)), confirming the standard PID closed-loop formula used in tuning a Siemens S7-1200 PLC PID block.
Follow-up: How does adding integral action (Ki) in the PID controller affect the type number of the closed-loop system?
Common misconceptions
Misconception: The graph determinant Δ in Mason's formula is the same as the system's characteristic polynomial.
Correct: Δ is the graph determinant (a ratio expression in s), and setting Δ = 0 gives the characteristic equation, but Δ itself includes the denominator polynomials from branch transmittances and is not a polynomial until multiplied through.
Misconception: All loops in a signal flow graph must be touching.
Correct: Loops are non-touching if they share no common node, and their products appear with alternating signs in Δ; most multi-loop systems have at least some non-touching loop pairs.
Misconception: A forward path cannot pass through the same branch twice but can revisit nodes.
Correct: A valid forward path must pass through each node at most once and each branch at most once; revisiting any node would create a loop within the path, which is not a forward path.
Misconception: Mason's gain formula and block diagram reduction always give different results for the same system.
Correct: Mason's formula and block diagram reduction are equivalent methods that give identical transfer functions; Mason's formula is simply a systematic algebraic shortcut that avoids the graphical manipulation steps of block diagram reduction.