Side-by-side comparison
| Parameter | Routh Hurwitz | Nyquist Stability Criterion |
|---|---|---|
| Domain | Algebraic (polynomial coefficients) | Frequency domain (polar plot) |
| Input required | Closed-loop characteristic equation | Open-loop transfer function G(s)H(s) |
| Handles open-loop unstable plants | No — method fails if plant is open-loop unstable | Yes — encirclement rule handles RHP poles |
| Gain/Phase Margin | Not directly available | Directly readable from the Nyquist plot |
| Delay systems | Cannot handle e^(-sT) directly | Handles time delay naturally in frequency response |
| Result type | Number of RHP roots (exact count) | Stable / unstable via encirclement count N=Z-P |
| Computation effort | Low — table construction only | Higher — requires polar plot or Bode data |
| Marginal stability detection | Row of zeros indicates j-omega axis roots | Plot passes through -1+j0 point |
| Typical exam/GATE usage | Finding range of K for stability | Relative stability, gain margin problems |
| Real IC / system example | PID controller loop with known plant TF | LM741 op-amp feedback network, RF amplifier |
Key differences
Routh Hurwitz counts right-half-plane roots algebraically — you get an exact number without plotting anything, and it is fast for finding the critical gain K. Nyquist goes further: it gives gain margin (typically 6 dB for a well-designed loop) and phase margin (45°–60° is standard) directly from the polar plot. Crucially, Nyquist can handle open-loop unstable plants like an inverted pendulum where Routh fails completely. For time-delay systems with e^(-0.1s) terms, Nyquist is the only practical classical option.
When to use Routh Hurwitz
Use Routh Hurwitz when you have the closed-loop characteristic polynomial and need the range of gain K that keeps all roots in the left half-plane. For example, finding the maximum proportional gain of a third-order PID-controlled DC motor before oscillation starts.
When to use Nyquist Stability Criterion
Use Nyquist when the plant itself is open-loop unstable or when you must determine gain and phase margins numerically. For example, analysing the stability margin of a RF power amplifier feedback network designed around the LM7171 op-amp.
Recommendation
For most university exam problems where the plant is stable and the question asks for the range of K, choose Routh Hurwitz — it is faster and needs no plotting. Switch to Nyquist only when gain or phase margin numbers are explicitly asked for.
Exam tip: Examiners expect you to form the Routh array correctly, identify the row-of-zeros special case, and state the number of RHP roots — not just say "stable"; for Nyquist, state N, Z, P explicitly using N=Z-P.
Interview tip: Interviewers at core companies like BHEL or L&T often ask you to compare the two on open-loop unstable systems — be ready to say Routh cannot handle them and explain the Nyquist encirclement argument in one clear sentence.