Discussion
The Bode magnitude plot of a certain control system shows the gain to be -20 dB at the phase cross over frequency. What is the gain margin expressed as a ratio ?
Answer: C
The gain margin is 10.
Gain margin is how much the loop gain can be multiplied at the phase crossover frequency before the system reaches instability. If the gain there is -20 dB, the gain margin is +20 dB. Converting decibels to a magnitude ratio:
\[ 20 \log_{10} \text{GM} = 20 \Rightarrow \text{GM} = 10 \]
Equivalently, the loop gain at phase crossover is 0.1, and the gain margin is its reciprocal, 10. 20 is the value in decibels, not the ratio; 1 would mean the system is on the edge of instability, and 5 has no basis.
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