Understanding the Concept:
The gain crossover frequency ($\omega_{gc}$) is a critical metric used in frequency domain stability analysis to determine a system's relative stability margins (such as the phase margin). It identifies the specific frequency boundary where the open-loop system transfer function magnitude drops to absolute unity.
Step 1: Define the condition mathematically.
By definition, the gain crossover frequency $\omega_{gc}$ is the frequency at which the absolute magnitude of the loop transfer function $G(j\omega)H(j\omega)$ equals exactly 1:
\[
|G(j\omega_{gc})H(j\omega_{gc})| = 1
\]
Step 2: Convert to decibel (dB) log-magnitude scale.
To express this condition on a standard logarithmic scale (as used in Bode plots), apply the decibel conversion formula:
\[
\text{Gain in dB} = 20 \log_{10} |G(j\omega)H(j\omega)|
\]
Substitute the unity crossover condition ($|G(j\omega_{gc})H(j\omega_{gc})| = 1$) into this formula:
\[
\text{Gain at } \omega_{gc} = 20 \log_{10}(1)
\]
Since the logarithm of 1 to any base is exactly 0:
\[
\text{Gain at } \omega_{gc} = 20 \times 0 = 0\text{ dB}
\]
Thus, the gain crossover frequency is the specific frequency where the open-loop magnitude curve crosses the $0\text{ dB}$ reference line.