Feedback and loop gain

Wrap an amplifier in negative feedback, measure its loop gain, and predict the closed-loop gain error, bandwidth and output impedance from it.

Lesson: E6.2
Time: 30 min
Runs on: ngspice
Check: 5 questions

A buffer feeds a signal between 0.8 and 1.0 V to a 12-bit converter, whose step is 1.8 V/4096 = 0.44 mV. Built on the five-transistor OTA of E5.3, its error drifted from −1.28 mV to −3.11 mV across that range. The same buffer built on the two-stage op-amp stayed within 0.18 mV.

DC sweepBuffer error across the input rangev(out) − v(in) of two unity-gain buffers on the same input: the one-stage OTA-A and the two-stage OPA-B.

OTA-A buffer
OPA-B buffer

Input 800 mVOTA-A buffer = -1.279 mV · OPA-B buffer = -176.9 µV
At 0.8 V the OTA-A buffer is 1.28 mV low, the OPA-B buffer 0.18 mV low.
Input 1 VOTA-A buffer = -3.114 mV · OPA-B buffer = -142.5 µV
At 1.0 V the OTA-A buffer is 3.11 mV low. The error grew by 1.83 mV over 200 mV: the output moves only 0.991 of each input change. OPA-B is 0.14 mV low.

Read the plot as a table

Input	OTA-A buffer	OPA-B buffer
800 mV	-1.279 mV	-176.9 µV
850 mV	-1.724 mV	-141.6 µV
900 mV	-2.166 mV	-131.8 µV
950 mV	-2.624 mV	-133.3 µV
1 V	-3.114 mV	-142.5 µV

Measured on ngspice 42, a DC sweep in 1 mV steps.

Both are the same loop, with the output wired back to the − input; what differs is how much gain goes around it.

The loop and its gain

A negative-feedback loopArrows show the signal. The amplifier multiplies the difference between the input and the returned fraction of the output.

The input arrives
v_in reaches the input of the amplifier. In an op-amp the difference is taken by the input pair itself: + minus −.
The amplifier multiplies
The amplifier makes v_out = A times the difference. For OTA-A, A = 136.7.
A fraction comes back
The feedback network returns β·v_out to the − input. A signal that goes once around the loop is multiplied by A and by β: the loop gain T = Aβ.
The loop settles
v_out = A(v_in − βv_out), so v_out/v_in = A/(1 + T). A buffer has β = 1 and T = A.

Read the diagram as text

Input: the signal to copy
Difference: input minus β × output
Amplifier A: output = A × difference
Output: what the load sees
Feedback β: returns β × output

Input → Difference: input
Difference → Amplifier A: difference
Amplifier A → Output: output
Output → Feedback β: sample
Feedback β → Difference: β × output

In the buffer of the hook, the feedback network is a wire: β = 1.

Solve the loop and the closed-loop gain is A/(1 + T). Write it as (1/β)·T/(1 + T). When T is large, the gain is 1/β, set by the feedback network alone. What is left over is the gain error, 1/(1 + T): it depends on the loop gain, not on A alone.

The feedback is negative: the returned fraction is subtracted, so an output that rises too far shrinks the difference and pulls itself back down. Positive feedback would add instead and drive the output to a rail, which the comparator of E8 does on purpose.

A buffer has β = 1, so T = A. With OTA-A, T = 136.7 and the error should be 1/137.7 = 0.73 %. With OPA-B, T = 11019 and the error falls to 0.009 %, which is why op-amp designers chase DC gain.

Gain error is 1/(1 + T)

Measure it. Each open-loop gain comes from the inductor bench of E5. A 1 TH inductor from the output to the − input and 1 F from there to ground close the loop for DC, so the bias is right. The AC signal sees an open loop. The gain-of-two loop is opened the same way at M1's gate, and v(tk) at the divider is T itself.

Circuit	Loop gain T	A/(1 + T) · 1/β	Measured gain	Error
OTA-A buffer	136.7 (42.7 dB)	0.99274	`cl_ota = 9.910593e-01`	0.89 %
OPA-B buffer	11019 (80.8 dB)	0.99991	`cl_buf = 1.000058e+00`	+0.006 %
OPA-B, gain of two, 20 kΩ + 20 kΩ	768.9	1.99740	`cl_two = 1.997700e+00`	0.115 %

Measured at 10 Hz on ngspice 42. Error is 1 − measured/(1/β).

The OTA-A buffer misses its prediction by 0.17 %. In a buffer both inputs move with the signal, so the OTA also sees a common-mode input. Its common-mode gain A_cm = −0.233 (E5.2) adds to the output: v_out/v_in = (A + A_cm)/(1 + A) = 0.99104, against 0.99106 measured. The same term pushes OPA-B's buffer a hair above 1.

The gain-of-two amplifier shows a second effect. Its 40 kΩ of feedback resistors hang on the output, and OPA-B's second stage has a high output resistance. The open-loop gain falls from 11019 to 1538 (63.7 dB), so T = 1538 × 0.5 = 769. A feedback network that loads the output cuts A, and with it T.

What the loop buys: bandwidth and impedance

AC analysisOTA-A open loop and as a bufferGain of OTA-A on the open-loop bench, and of the same OTA as a unity-gain buffer, both into 1 pF.

open loop A
buffer

Frequency 1 kHzopen loop A = 42.71 dB · buffer = -0.07801 dB
At 1 kHz the open-loop gain is 42.7 dB (136.7). The buffer gives −0.08 dB: 0.991.
Frequency 188 kHzopen loop A = 39.7 dB · buffer = -0.07821 dB
The open-loop gain is 3 dB down at 188 kHz. The buffer has not moved: its loop gain is still 97.
Frequency 25.54 MHzopen loop A = -0.004288 dB · buffer = -2.792 dB
The open-loop gain reaches 1 at 25.5 MHz. Here T = 1 and the loop runs out of gain.
Frequency 27.38 MHzopen loop A = -0.617 dB · buffer = -3.089 dB
The buffer is 3 dB down at 27.4 MHz, 146 times the open-loop bandwidth. It traded 42.7 dB of gain for that bandwidth.

Read the plot as a table

Frequency	open loop A	buffer
10 Hz	42.71 dB	-0.07801 dB
100 Hz	42.71 dB	-0.07801 dB
1 kHz	42.71 dB	-0.07801 dB
10 kHz	42.7 dB	-0.07801 dB
100 kHz	41.63 dB	-0.07806 dB
188 kHz	39.7 dB	-0.07821 dB
1 MHz	28.04 dB	-0.08367 dB
10 MHz	8.185 dB	-0.6122 dB
25.54 MHz	-0.004288 dB	-2.792 dB
27.38 MHz	-0.617 dB	-3.089 dB
100 MHz	-12.57 dB	-12.61 dB
1 GHz	-38.63 dB	-40.2 dB

Measured on ngspice 42, 50 points per decade.

Bandwidth. The loop holds the gain at 1/β for as long as T stays large. Above the open-loop pole A falls as 1/f, and the closed loop ends near the frequency where T = 1. For a buffer that is f_u; for a loop with β < 1 it is about β·f_u. Gain is traded for bandwidth, with the gain-bandwidth product as the exchange rate.

Impedance. Push 1 A of AC into the output and read the voltage: at 1 kHz OTA-A alone shows 841.7 kΩ, the r_o2 ‖ r_o4 of its output. As a buffer it shows 6.10 kΩ, and the ratio of the two is 137.9, which is 1 + T: the loop sees any change at the output and corrects it.

Desensitivity. The loop also holds the gain when A moves. Heat the lab's gain-of-two amplifier to 125 °C: T falls by 15.9 %, and the closed-loop gain moves by 0.020 %. The prediction, 15.9 %/(1 + T), is 0.021 %.

27 °C

.lib sky130 tt
* (default temperature, 27 C)
XH in hn oh vdd opa
RFH oh hn 20k
RGH hn ref 20k

t_two = 7.689262e+02, cl_two = 1.997700e+00

125 °C

.lib sky130 tt
.temp 125
XH in hn oh vdd opa
RFH oh hn 20k
RGH hn ref 20k

t_two = 6.465233e+02, cl_two = 1.997303e+00: T −15.9 %, gain −0.020 %.

The lab netlist at two temperatures, measured on ngspice 42.

Feedback costs one thing. A has phase shift, and near f_u the returning signal arrives late. If it arrives a full half-cycle late, the loop rings or oscillates: E6.3 measures how late is too late.

Asked in interviews

What is loop gain?
The gain a signal sees going once around the loop, T = Aβ. Break the loop where it does not change the bias or the loading, inject a signal, and see what comes back.
What is the gain error of a buffer with 60 dB of open-loop gain?
T = 1000, so the gain is 1000/1001 and the error 1/(1 + T) ≈ 0.1 %. A real buffer adds the common-mode gain: OTA-A's 0.73 % became 0.89 %.
What does feedback do to output impedance?
Voltage feedback divides it by 1 + T: OTA-A's 841.7 kΩ fell to 6.10 kΩ as a buffer. The effect fades with frequency as T falls.

Try it

Run the netlist and predict cl_two from t_two before you read it: 2·T/(1 + T). Open Plots to see all five gains against frequency.

Now add .temp 125 under the .lib line and run again: a_opa and t_two fall, cl_two barely moves. Then change both 20 k resistors to 200k, in both copies: the load on the output shrinks, a_loaded rises toward a_opa, and the error falls.

SPICEngspice 42 · simulator service

Edit the netlist, then run it

Expected measurements: a_opa = 1.101886e+04 cl_buf = 1.000058e+00 a_loaded = 1.537852e+03 t_two = 7.689262e+02 cl_two = 1.997700e+00

Check yourself

Predict the outputAn ideal amplifier with a gain of 100 is wired as a buffer: its output drives its own − input. The run prints the real part of v(out) for 1 V in. Type the line meas prints (format: g = 6.319360e-01).

* an ideal amplifier of gain 100 wired as a buffer
VIN in 0 dc 0 ac 1
E1 out 0 in out 100
.control
ac lin 3 1k 3k
meas ac g FIND vr(out) AT=1k
.endc
.end

ConceptAn op-amp with open-loop gain A sits in a loop with feedback factor β. Which is its gain error, the fraction by which the gain falls short of 1/β?

1/(1 + Aβ)1/A1/(1 + A)1/(1 + A/β)

ConceptThe same op-amp builds a buffer and a gain-of-ten amplifier. Which has the larger gain error, and why?

The buffer: its output swings as far as its input, so A fallsThe gain-of-ten: β = 0.1 makes T ten times smallerBoth the same: the error depends only on AThe buffer: it has no resistors to fix its gain

ConceptRead the open- and closed-loop Bode figure. Where does the buffer's gain fall 3 dB?

Near 1 kHz, where the loop gain is largestNear 1 MHz, where the open-loop gain is 100Near 27 MHz, close to where the open-loop gain reaches 1Near 188 kHz, where the open-loop gain is 3 dB down

ConceptOTA-A alone has an output resistance of 841.7 kΩ. As a buffer, T = 136.7. About what output resistance does the buffer show at 1 kHz?

115 MΩ841.7 kΩ: feedback does not change the transistors0 Ω: an ideal buffer6.1 kΩ

Remember

The loop gain T = Aβ is the gain once around the loop; the closed-loop gain is (1/β)·T/(1 + T).
The gain error is 1/(1 + T): 0.73 % for OTA-A as a buffer (0.89 % measured, with its common-mode gain), 0.009 % for OPA-B.
A feedback network that loads the output lowers A and T: 40 kΩ cut OPA-B's gain from 11019 to 1538.
Feedback divides output impedance by 1 + T (841.7 kΩ to 6.10 kΩ) and changes in A by 1 + T (15.9 % to 0.020 %).
The closed loop ends near the frequency where T = 1: the buffer reached 27.4 MHz from a 188 kHz open-loop pole.