The unity gain frequency (ωu) and the gain-Bandwidth product(GBW) are often considered as identical in ideal or standard circuit theory.
However, in real-world circuits, the unity gain frequency ωu is almost always slightly lower than the GBW.
Understanding the difference between the asymptotic Bode plot and the real frequency response curve helps us to identify hidden stability hazards, such as the so-called “pole-zero doublets”.
The Ideal 1st Order System
In systems with only one unique pole, the gain bandwidth (GBW) is “almost” exactly the same as the unity gain frequency (wu), also referred as crossover frequency.
The magnitude response of a 1st order system is given by:
If we consider that the unity gain frequency is much greater than the bandwidth frequency in open-loops systems (usually high gain open-loop systems).
Therefore, this simplifies the previous equation:
As a result, the unity gain frequency is equal to the gain bandwidth
The Real-World 1st-Order Approximation
In real-life IC circuits, there are additional secondary poles and zeroes rather than the dominant pole due to parasitic capacitances, internal nodes, etc. These non-dominant poles are normally located at high-frequencies, but they cause some bending below the asymptote.
Circuits are never purely single-pole systems
The unity gain frequency or crossover frequency (wu) is slightly smaller than the first order asymptotic straight-line (wu1) of the Bode plot of one pure 1st order system. That means that for real systems UGF < GBW.
The ratio of wu/wu1 can be seen graphically on the following plot. Where the wu1 => asymptotic 1st order and wu => real unity gain frequency.
If the second non-dominant pole is far away from the unity gain frequency:
UGF ≈ GBW (wp2/wu1 → 10).
What could happen if UGF > BGW?
Normally wu < GBW. But it can be the case of the opposite and the unity gain frequency is higher than the bandwidth frequency. This could mean that a zero is located before the crossover frequency and is flattening the gain slope.
pole-zero doublet is a pole and a zero located very close to each other in frequency.
This behaviour is undesirable for several reasons…
The zero in the active passband doesn’t cancel perfectly the corresponding pole, creating a “pole-zero doublet” (a pole and a zero located very close to each other in frequency). Although theoretically, to cancel a pole with a zero is a nice feature, in the real silicon, a pole zero cancellation is never achieved due to component mismatches, IC resistor accuracy, process variation, temperature and/or aging.
Pole-zero doublets introduce long-lasting tails in the step response taking significantly longer to achieve its target value. This is not nice for ADCs or switched-capacitor circuits.
The unity-gain frequency, it may sustain the gain above 0 dB longer than intended, which can lead to a fast phase drop, degraded stability and causing severe ringing.
Strategies that push undesired zeros to very high frequencies are often preferred rather than pole-zero cancellation.
Conclusions
While standard 1st order systems assume that wu = GBW, real-life parasitics yield wu < GBW. Detecting that wu> GBW could help to detect and avoid pole-zero doublets.
Sources
[1] Franco Maloberti. Analog Design for CMOS VLSI Systems. Chapter 5.5
[2] Sergio Franco. Design with operational amplifiers and analog integrated circuits. Chapter 8.3
[3] Razavi. Design of Analog Integrated Circuits 2nd edition. Chapter 10.19
