# How to determine the critical delay for a closed-loop system?

Using the open-loop transfer function we can easily determine its Nyquist plot, and from it we can get the critical gain such that the system is stable. Now, let's consider a system with delay $t_d$. Then, the new transfer function will have the factor $e^{-st_d}$ . According to the theory, we can just "ignore" this delay and draw the Nyquist and from it we can deduct the maximum delay before the system gets unstable. How can we get that delay? Can we use the phase margin to know it?

A delay can be included in the conversion chain as shown below for a buck converter. Here the delay is incurred to the comparator propagation time which is significant at a high switching frequency.

The thing is that you end up with a transfer function now including an exponential term. You can rewrite the exponential term using a Padé approximant fitting the delay versus frequency with the precision you want. A first-order version is given by $e^{-t}\approx \frac{1-\frac{s\tau}{2}}{1+\frac{s\tau}{2}}$ in which you recognize a RHP zero and a LHP pole tuned at the same frequency depending on $\tau$ your delay. We can show that the new stability criterion is no longer the phase margin but the delay margin. The maximum acceptable delay in the loop is defined as $\tau_{max}=\frac{\phi_m}{\omega_c}$ in which $\phi_m$ is the phase margin measured at $\omega_c$ the crossover angular frequency. Please look at this document for more details on delay and modulus margins. Just as a side note, a transfer function including RHP zeros or RHP poles or delays is a non-minimum-phase function and the Bode criteria may fail to predict stability. Nyquist is the way to go in this case.

• If $\phi_m$ is $\infty$, can we say that there isn't any critical delay and we can add any delay we want? – Miguel Duran Diaz Nov 18 '17 at 16:48
• I am not that familiar with infinite $\phi_m$ systems. Where did you see such a network, just for my curiosity? – Verbal Kint Nov 18 '17 at 16:59
• It is a two-phase buck converter with a fourth order filter output with a controller $k_c=0.8$ gives me a Nyquist such that $\phi_m$ is infinite. (Actually I'm not sure if it is a "real" buck converter, because it is a work for the course Control Systems in my university). However, for the same system using $k_c=\frac{100}{s}$ gives me a finite $\phi_m$ so we have a critical delay. – Miguel Duran Diaz Nov 18 '17 at 18:13

Find the highest frequency F where the total gain of the feedback loop (forward gain * feedback factor) has magnitude =1. Let the total phase shift of the loop be =X (radians) at that frequency.

Adding delay T you introduce phase lag 2Pi * F * T at frequency =F. Solve the smallest positive T from equation 2N*Pi = X - 2Pi * F * T . N is an integer. test different values for N, positive and negative

The found T is the limit case, the slightest amount more gain will cause continuous oscillation with growing amplitude.