Op-Amps with negative feedback

– There are several basic ways in which an op-amp can be connected using negative feedback to stabilize the gain and increase frequency response.

– The large open-loop gain of an op-amp creates instability because a small noise voltage on the input can be amplified to a point where the amplifier is driven out of the linear region.

– Open-loop gain varies between devices.

– Closed-loop gain is independent of the open-loop gain.

Closed-Loop voltage gain, A_cl

– It is the voltage gain of an op-amp with external feedback.

– Gain is controlled by external components.

Noninverting Amplifier

– The op-amp circuit shown below is a non-inverting amplifier in a closed-loop configuration.

– Input signal is applied to the non-inverting input.

– The output is applied back to the inverting input through feedback (closed loop) circuit formed by the input resistor R_i and the feedback resistor R_f.

– This creates a negative feedback.

– The two resistors create a voltage divider, which reduces V_out and connects the reduced voltage V_f to the inverting input.

– The feedback voltage is:

V_f = R_i/(R_i + R_f)V_out

– The difference between the input voltage and the feedback voltage is the differential input to the op-amp.

– This differential voltage is amplified by the open loop gain, A_ol, to get V_out­.

V_out­ = A_ol(V_in – V_f)

– Let B = R_i/(R_i + R_f). Thus V_f = BV_out and

V_out = A_ol(V_in – BV_out)

– Manipulate the expression to get:

V_out = A_olV_in - A_olBV_out

V_out + A_olBV_out = A_olV_in

V_out(1 + A_olB) = A_olV_in

Overall Gain = V_out/V_in = A_ol/(1 + A_olB)

– Since A_olB >> 1, the equation above becomes:

V_out/V_in = A_ol/(A_olB) = 1/B

– Thus the closed loop gain of the noninverting (NI) amplifier is the reciprocal of the attenuation (B) of the feedback circuit (voltage-divider).

A_cl(NI) = V_out/V_in = 1/B = (R_i + R_f)/R_i

– Finally:

A_cl(NI) = 1 + R_f/R_i

– Notice that the closed loop gain is independent of the open-loop gain.

Example

Determine the gain of the amplifier circuit shown below. The open loop gain of the op-amp is 150000.

Solution

This is a noninverting amplifier op-amp configuration. Therefore, the closed-loop voltage gain is

A_cl(NI) = 1 + R_f/R_i = 1 + 100 k?/4.7 k? = 22.3

Voltage-Follower (VF)

– Output voltage of a noninverting amplifier is fed back to the inverting input by a straight connection.

– The straight feedback has a gain of 1 (i.e. there is no gain).

– The closed-loop voltage gain is 1/B, but B = 1. Thus, the A_cl(VF) = 1.

– It has very high input impedance and low output impedance.

Inverting Amplifier (I)

– The input signal is applied through a series input resistor R_i to the inverting input.

– The output is fed back through R_f to the same input.

– The noninverting input is grounded.

– For finding the gain, let’s assume there is infinite impedance at the input (i.e. between the inverting and non-inverting inputs).

– Infinite input impedance implies zero current at the inverting input.

– If there is zero current through the input impedance, there is NO voltage drop between the inverting and noninverting inputs.

– Thus, the voltage at the inverting input is zero!

- The zero at the inverting input is referred to as virtual ground.

– Since there is no current at the inverting input, the current through R_i and the current through R_f are equal:

I_in = I_f.

– The voltage across R_i equals V_in because of virtual ground on the other side of the resistor. Therefore we have that

I_in = V_in/R_i.

– Also, the voltage across R_f equals –V_out, because of virtual ground. Therefore:

I_f = -V_out/R_f

– Since I_f = I_in, we get that:

-V_out/R_f = V_in/R_i

– Or, rearranging,

V_out/V_in = -R_f/R_i

– So,

A_cl(I) = -R_f/R_i

– Thus, the closed loop gain is independent of the op-amp’s internal open-loop gain.

– The negative feedback stabilizes the voltage gain.

– The negative sign indicates inversion.