Understanding the trade-off between gain and bandwidth is essential for designing efficient CMOS circuits. The gain-bandwidth trade-off impacts the performance of amplifiers, filters, and other circuits.
The science of Analog IC Circuit design is not 100% exact
Accurate models are really complex and can’t be handled by humans,
The designer need to manage a balancing act between various performance factors like power, speed, bandwidth, noise, silicon area, and stability among others. In IC design, trade-offs are inevitable, and the specific requirements of each application determine how these competing properties are prioritized.
Generic Feedback Circuit
Let’s consider a standard feedback system as drawn in the figure below:
Where A is the open-loop gain (gain without feedback) and β is the feedback factor, which is a fraction of Vout and it is fed back to the negative input (V-).
The general closed-loop transfer function of the feedback system is expressed as:
The gain G is typically smaller than the open-loop gain and the feedback loop improves the system, making it more stable and less sensitive to A variations.
Considering the case of an ideal amplifier where the open-loop gain A is very large (almost infinite), then the closed-loop gain G(s) is asymptotically 1/B.
As A → infinite: 1 + Aβ ≈ Aβ
Therefore,
Single Pole System
Considering a real amplifier where the open-loop gain Ao is finite, and the system behaves as a single pole circuit, the transfer function is given by:
In a first-order system, the frequency response is well-known:
- The gain starts at Ao at a constant rate until the cutoff frequency ωb. After, it decreases approximately linearly on a logarithmic scale at a rate of -20 dB/decade.
- The ω3dB Bandwidth defines the frequency at which the amplifier’s gain is reduced by 3dB from its DC value.
- The phase shift of approaches -90° (phase lag) as frequency increases.
This single-pole model is commonly used to approximate the behavior of amplifiers with dominant pole characteristics, where one pole determines the frequency behaviour and the bandwidth of the system.
Some Calculations
The transfer function of the first order system can be approximated to A(s) = wt/s, which agrees well for w>wb and values around wt.
Substituting the simplified version of A(s) on the general closed-loop transfer function:
The general closed-loop transfer function is approximated to:
- The DC gain is 1/β, inversely proportional to the feedback factor β.
- The closed-loop Bandwidth is β·ωt
This result shows that adding feedback to a single-pole amplifier increases the bandwidth by a factor of β, while the DC gain is reduced by the feedback factor β.
The reduction in the gain, is exactly equal to the increase in the Bandwidth
The gain is reduced by a factor β, and the pole frequency is increased by the same factor β.
Gain – Bandwidth Tradeoff
The feedback factor β is defining the closed-loop DC gain of 1/β and the closed-loop Bandwidth of β·wt
This relationship is fundamental in feedback amplifier design, where increasing feedback can improve bandwidth at the cost of reducing gain.
Increasing feedback improves the Bandwidth at the cost of reducing Gain
The Gain BandWidth product (GBW) remains constant for negative feedback loop systems, because the division factor β of the gain is canceled out by the pole factor extension β.
The tradeoff can be better understand graphically in the closed-loop Bode plot. If the DC gain is increased (1/β is larger), the Bandwidth (f3dB) is inversely reduced.
Using the exactly values, without the approximation of infinite AoL, the close-loop gain is equal to AoL/(1+βAoL). It can be seen that the reduction in the gain by the factor (1+βAoL) is equal to the bandwidth extension by (1+βAoL).
The Gain-Bandwidth trade-off is the most popular direct relationship trade-off in IC circuit design, but many other trade-offs must be considered.
