# Mathematically representation of two loops where one is much faster than the other

In this post, akellyirl wrote:

As a rule of thumb for design simplicity, the $$\G_2H_2\$$ loop is designed to be stable with significantly higher bandwidth than the final loop bandwidth so that its phase delay can be neglected in the design of the outer loop; i.e. its phase/magnitude contribution to the overall loop is negligible at crossover.

Is it possible to express that idea mathematically?

• $G=\dfrac{G1G^{’}}{G1G^{’}H1+1}$ approaches 1/H1 Jun 10 '21 at 16:32
• @TonyStewartEE75 if so then shouldn't he have said G or the closed loop gain of the inner loop NOT just G2H2 is much larger than the outer loop? Jun 10 '21 at 18:04
• More GBW is possible with cascaded linear loops because with G1G2 , Without H2 Then H1 must be bigger meaning more attenuation vs s thus less BW. It’s a generalization for the assumption of high phase margin but not a guarantee. Jun 10 '21 at 18:30
• @TonyStewartEE75 from your formula of G. You'll need G to be large to be able to simplify it to 1/H1. What does G` relate to the inner system bandwidth? Jun 10 '21 at 18:41
• G’ is assumed to be much larger BW than you need, but permits more BW than without internal compensation H2. so In general, the higher the gain, the lower the BW of the integrator to linearize the loop so it is a 1st order system at 0dB gain open loop in order to achieve decent phase margin Jun 10 '21 at 18:48