Gain margin example on the oven from FR-03: K slider, Bode plot, step response, comparison with the motor, and three rulers for the distance to the point −1.
Part 1: the oven from FR-03 p. 7, a plant with three poles
$$L(s)=K\cdot\frac{6}{(s+1)(s+2)(s+3)}$$
Three poles, each adds up to −90° of phase, so together the phase goes all the way down to −270°. On the way it crosses −180° (at ω = 3.32 rad/s). That is exactly where you measure the gain margin MG (margen de ganancia).
K
Wcg (phase = −180°)
3.32 rad/s
|L| at Wcg
MG = Gm
MF = Pm
Bode of the loop function L (función de lazo). Vertical line = Wcg. Red segment = MG: how far it is from 0 dB.
Closed-loop (lazo cerrado) step response (respuesta al escalón) for this K. The axis is fixed: an unstable curve runs off the chart.
Move from K = 1 to K = 10. The top curve goes up by 20 dB, and the bottom one (phase) stays put. At K = 10 the red segment disappears: at −180° the sine comes back the same size and flipped, so the loop oscillates by itself. That is why MG at K = 1 is 10 times (20 dB): that is how many times you can raise K before you reach the edge.
Part 2: comparison with your motor
Oven (3 poles) Phase crosses −180° → Wcg exists → MG is finite. Conclusion: “el sistema es estable para K < 10; para K > 10 se inestabiliza”.
Motor from the exam (2 poles) Phase only reaches −180°, never crosses it → Wcg does not exist → MG = ∞. Conclusion: “con control P el sistema es estable para cualquier K > 0; al subir K solo disminuye el MF (más oscilación)”.
Part 3: “different ways to measure the same thing”, three rulers for one distance
Plot the values of L(jω) for all ω on the complex plane (this is the Nyquist plot, FR-03 p. 10). The edge is a single point: −1. The question is always “how far is the curve from −1?”, and each way measures it in a different direction: MG: along the horizontal axis, i.e. how many times you can stretch the curve (raise K). MF: along the circle |L| = 1, i.e. how many degrees you can rotate it (extra delay). Ms: the shortest distance in any direction = radius of the circle around −1 (= 1/Ms).
Like a glass on a table: how far you can push it straight ahead (MG), how far you can turn it (MF), and the shortest distance to the edge at all (Ms). Same glass, same edge, three measurements.
The same K slider at the top controls this plot. Blue = L(jω). Red segment = MG. Green radius = the point where |L| = 1 (angle MF measured from −1). Orange circle = the closest distance 1/Ms.
MG (horizontal axis)
MF (rotation)
1/Ms (shortest)
All three shrink together as K grows, and all reach zero at K = 10. So in the exam you write each way separately and at the end: “todas las medidas coinciden: el K más pequeño tiene mayor estabilidad relativa”.