Closed-Loop Gain
Understanding Closed-Loop Gain
The concept of negative feedback, formalized by Harold Black in 1927, transformed amplifier design by making gain accuracy dependent on passive feedback components rather than active device parameters. An amplifier with 40 dB of open-loop gain (A = 10,000) and a feedback factor β = 0.1 (sampling 10% of the output) has a loop gain Aβ = 1,000 (60 dB). The closed-loop gain is 10,000/1,001 = 9.99, or essentially 1/β = 10 (20 dB). Even if the open-loop gain drifts by ±50% due to temperature or device variation, the closed-loop gain changes by only ±0.05%, because the high loop gain suppresses the effect of open-loop gain variations by the factor (1 + Aβ).
In RF amplifier design, feedback techniques include resistive shunt feedback (a resistor from output to input), series feedback (emitter/source degeneration), and transformer-coupled feedback. Resistive shunt feedback is the most common in broadband designs, simultaneously reducing gain, flattening frequency response, and improving input/output impedance matching. A 50 Ω broadband amplifier using a GaAs PHEMT with 15 dB open-loop gain can achieve flat 10 dB gain from DC to 20 GHz with a properly designed feedback network. The 5 dB of feedback improves input return loss from 6 dB (open-loop) to better than 15 dB and reduces harmonic distortion by approximately 5 dB. The tradeoff is noise figure: the feedback resistor adds thermal noise, typically degrading NF by 0.5 to 1.5 dB compared to the open-loop value.
Feedback Amplifier Equations
ACL = A / (1 + Aβ) ≈ 1/β when Aβ >> 1
Gain-Bandwidth Product:
GBW = AOL × f-3dB(OL) = ACL × f-3dB(CL) (constant)
Distortion Reduction:
THDCL = THDOL / (1 + Aβ)
Where A = open-loop voltage gain, β = feedback fraction (dimensionless, 0 to 1), Aβ = loop gain. Example: A = 40 dB (100), β = 0.1, Aβ = 10, ACL = 100/11 = 9.09 (19.2 dB). THD reduced by factor of 11.
Feedback Type Comparison
| Feedback Type | Topology | Input Z Effect | Output Z Effect | RF Application |
|---|---|---|---|---|
| Shunt-shunt (transresistance) | Rf from output to input | Decreases | Decreases | Broadband LNA, TIA |
| Series-series (transconductance) | Source/emitter degeneration | Increases | Increases | PA linearization |
| Shunt-series (current) | Current sensing, voltage feedback | Decreases | Increases | Current-mode amplifiers |
| Series-shunt (voltage) | Voltage divider feedback | Increases | Decreases | Op-amp, baseband |
| Transformer-coupled | Coupled winding feedback | Controlled by turns ratio | Controlled by turns ratio | Wideband PA, CATV |
Frequently Asked Questions
How does negative feedback improve linearity?
Distortion is reduced by the loop gain factor (1 + Aβ). With 1% open-loop THD and Aβ = 10 (20 dB loop gain), closed-loop THD drops to 0.1%. IP3 improves by 20log(1 + Aβ) dB. The practical limit is that the feedback network must maintain greater than 45 degrees phase margin at the unity loop gain frequency to prevent oscillation.
What is gain-bandwidth product?
GBW is the product of gain and bandwidth, approximately constant for a single-pole amplifier. With 40 dB gain and 10 MHz bandwidth, GBW = 1 GHz. At 20 dB closed-loop gain, bandwidth extends to 100 MHz. At 10 dB, bandwidth reaches 316 MHz. Wideband RF amplifiers use heavy feedback to achieve flat gain across DC to 40 GHz, accepting 8 to 12 dB per stage.
Why is loop gain stability analysis critical?
If loop gain reaches 0 dB where phase shift equals -180 degrees, the amplifier oscillates (Barkhausen criterion). Phase margin should exceed 45 degrees (preferably 60 degrees) and gain margin should exceed 10 dB. Parasitic capacitances and inductances in the feedback network create additional poles that erode phase margin. Stabilization uses dominant pole compensation, resistive loading, and careful layout.