Generate Nyquist plots from transfer functions. Analyze encirclements of the critical point for stability assessment.
Convert an open-loop transfer function into the complex-plane path traced by L(j omega), then judge how that path winds around the Nyquist critical point, -1 + 0j. Real and imaginary response form the horizontal and vertical coordinates, while frequency determines position along the curve. The stability interpretation applies to the loop transfer function in a negative-feedback characteristic equation of the form 1 + L(s) = 0, not automatically to an arbitrary plant transfer function. A frequent mistake is counting visible loops without accounting for direction and open-loop right-half-plane poles.
The generator substitutes s = j omega into the entered transfer function over a frequency range. Each complex value becomes one point: Re[L(j omega)] on the horizontal axis and Im[L(j omega)] on the vertical axis. Sweeping omega joins those points into the locus. For real-coefficient transfer functions, the negative-frequency branch is the complex-conjugate reflection of the positive-frequency branch across the real axis.
Nyquist stability reasoning compares the oriented winding of the complete mapped contour around -1 + 0j with the number of right-half-plane poles of L(s). Together, those quantities determine how many right-half-plane zeros 1 + L(s) has, corresponding to unstable closed-loop poles. A curve image without direction, pole count, and a valid contour is not a complete stability test.
For a transfer function with real coefficients, L(-j omega) is the complex conjugate of L(j omega). The negative-frequency locus therefore reflects the positive-frequency locus across the real axis. This symmetry does not generally follow for transfer functions with complex coefficients, and a positive-frequency-only display is only one branch of the complete contour.
No. Merely avoiding -1 rules out that specific boundary condition but does not settle the required winding count. Under negative unity feedback, stability depends on the net oriented encirclements of -1 together with the number of right-half-plane poles of L(s). Imaginary-axis poles and exact passages through -1 require separate treatment.
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