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Electronics I Laboratory Manual

Basic Feedback Op-Amp Amplifiers

As we saw in Experiment No. 22 an op-amp is an amplifier with a very high gain (the open-loop gain or Aol).  This fact limits the op-amp usefulness to a few applications (i.e., comparators, to be studied in Experiment 24).  In open-loop applications almost any differential input voltage will drive the output of the amplifier to its output saturation value.  If the differential voltage is positive, the output voltage will be the maximum positive value. If the differential input is negative the output voltage remains in its negative maximum1.

Negative Feedback

To make use of the versatility of op-amps we normally use them connected in negative feedback.  When an amplifier is connected in negative feedback a fraction of the output voltage is returned back to the input with a phase angle of 180o2.  In the case of op-amps if we take a fraction of the output and connect it to the inverting input (the input that normally is represented with a negative sign) the signal which is fed back is effectively negative with respect to the output voltage.  This is illustrated in Figure 23.1.

Figure 23.1: Negative Feedback in Op-Amps

Part (a) of this figure shows a configuration called non-inverting whereas part (b) represents the inverting amplifier.  In the next section we will explore in more detail these configurations.  In both cases, as you can see, the output voltage is fed back to the input (inverting) terminal through a network of two resistors.  These resistors form a voltage divider circuit.  By applying the voltage divider rule we can determine the feedback voltage (Vf) that is shown in these circuits.  This voltage is given by

 (23.1)

Or,

 (23.2)

where

 (23.3)

is the fraction of the output voltage fed back to the input.

1 The reader should review these concepts that were presented in the previous chapter.

2 A3 you should remember, a phase of angle of 180o reverts the polarity of a signal.

Effects of Negative Feedback

With negative feedback many features of op-amps are improved, or are made suitable for certain applications that otherwise could not be developed.  Some of these features are: voltage gain, input impedance, output impedance and bandwidth, among others.  The value of these parameters are dependent on the type of op-amp circuits.  In this experiment we will study three basic negative feedback circuits, namely Non-inverting amplifiers, Inverting amplifiers, and the Voltage-follower amplifier.

Non-Inverting Amplifier

An op-amp connected in closed-loop as a non-inverting amplifier is shown in Figure 23.1(a).  The input signal is applied to the non-inverting input (the terminal labeled +) and certain fraction (B) of the output voltage is fed back to the negative or inverting input.  This fraction, as we saw, is Vf and is given by Eq. 23.1.  Because we apply the input signal to the + (non-inverting) terminal the output voltage will have the same polarity as the input signal.  However, the fraction of the output voltage fed back to the inverting terminal will be inverted at the input, so the output portion due to this voltage will not be inverted.  The total output voltage will be the sum of the output voltage due to the input signal plus the contribution of the feedback signal.  Several parameters will be studied here.

Voltage Gain

To determine the closed-loop gain we notice that the output voltage is the differential voltage (VD) amplified by the open-loop voltage gain.  The differential voltage is given by

 (23.4)

The output voltage is expressed by

 (23.5)
 (23.6)

but

Substitute this value into Eq. ?? and get

Expanding the parenthesis and applying simple algebraic rules we get

 (23.7)

The closed-loop voltage gain is defined as

 (23.8)

therefore, using Eq. 23.7, the closed-loop voltage gain is given by

 (23.9)

Now, because Aol is very large, the product BAol is much larger than 1.  Therefore, the denominator of Eq. 23.9 can be replaced by BAol.  Then, Eq. 23.9 becomes

Using B from Eq. 23.3, we can express the closed-loop gain of the non-inverting amplifier by

Thus,

 (23.10)

As you can well see from the Eq. 23.10 the closed-loop gain does not depend on the open-loop gain of the op-amp3.  The closed-loop gain can be set just by selecting values of Rf and R1.
.5ex]
The above derivations shows us that with negative feedback the gain of an op-amp can be lowered to practically any value.  This makes the amplifier more stable so we can use it in more applications.

3 Don’t you think that this is extraordinary?  Why is this so?

Input Impedance

The input impedance of an op-amp is large.  When the op-amp is used in the non-inverting amplifier configuration of Fig. 23.1(a) the input impedance can be increased to any value.  If we define Zi the impedance of the op-amp without feedback, and Aol the voltage gain without feedback, then the input impedance of the closed-loop configuration - denoted by Zin(NI) - can be determined by mean of the following equation

 (23.11)

where B is the feedback fraction.  Note that this equation shows that the input impedance for this configuration is much larger than the input impedance without feedback4.

We will leave the details of this derivation to the reader.

4 Recall that Aol is very large.

Output Impedance

The output impedance of an op-amp is relatively small.  With negative feedback, however, the output impedance can be reduced to any desired value for the case of the non-inverting amplifier.

It can be shown that the output impedance of a non-inverting configuration can be practically reduced to a value as small as we want.  If we define Zo to be the output impedance of the open-loop (no feedback) op-amp, then the output impedance (Zout(NI)) for this configuration is given by

 (23.12)

This equation shows that the output impedance of an op-amp — if used with negative feedback — is much smaller than its equivalent open-loop impedance.

Bandwidth

The bandwidth of a non-inverting amplifier is larger than its equivalent open-loop amplifier.  Without any proof5 we state here the relationship between the open-loop and closed-loop bandwidth.  If we define BWol as the open-loop bandwidth, the closed-loop bandwidth is given by

 (23.13)

where, as usual, B is the feedback fraction, and Aol is the open-loop gain of the op-amp at the lowest frequency6.

5 As a homework to the reader, we assign the proof of this equation.

6 This is called the mid-frequency gain and is denoted by Aol(mid).  It is the highest gain of the amplifier.

Inverting Amplifier

An op-amp connected in closed-loop as an inverting amplifier is shown in Figure 23.1(b).  A fraction of the output voltage is applied to the inverting input to provide negative feedback.  This fraction is applied through two resistors, the feedback resistor (Rf) and the input resistor (R1).  These two resistors form the divider circuit needed to generate the fraction, B, of the output voltage fed back to the input.  (See Eq. 23.3 on page 233.)  The input signal is applied to the inverting input through the input resistor R1.

Voltage Gain

The voltage gain of the inverting amplifier can be calculated by noticing that the differential voltage VD is very small.  This condition is known as virtual ground because the voltage at the negative input is the same as the voltage at the positive terminal.  Also the input current to the amplifier (the current through the input impedance) is practically zero7.  This fact makes the current through the input resistor R1 equal to the current through Rf, or

 (23.14)

Figure 23.2 shows this fact.

 Figure 23.2: Inverting amplifier virtual ground

Now, because the voltage VD is zero (practically) then the input voltage Vi is given by

 (23.15)

The output voltage, on the other hand, is equal to

 (23.16)

The expression is negative because in the diagram we assumed that the current flows as indicated.

But I1 = If.  Therefore we can write the voltage gain of this amplifier as follows

 (23.17)

The two currents cancel out in the above equation.  If we denote V0/Vi by Acl(I), then the voltage gain of the inverting amplifier is expressed as

 (23.18)

The above equation shows that the voltage gain of the inverting amplifier is out of phase by 180° with respect to the input signal.  Also, the gain is independent of the op-amp parameters, and by choosing the proper resistors R1 and Rf you can make the gain to acquire any value.

7Remember that for a good amplifier the input impedance is extremely large.  Thus, the current through it is extremely low.

Input Impedance

It can be shown that the input impedance of the inverting amplifier (see Fig. 23.1(b)) is approximately equal to R1, the resistance connected directly to the inverting terminal; or, in equation form

 (23.19)

Output Impedance

The output impedance of an inverting amplifier is approximately equal to the output impedance of the open-loop op-amp.  In equation form we have

 (23.20)

Bandwidth

The bandwidth of the inverting amplifier is the same as the bandwidth for the non-inverting amplifier.  Its value is calculated with Eq. 23.21 which is the same as Eq.13.  For completeness we repeat this equation here:

 (23.21)

 Figure 23.3: Voltage Follower Circuit

Voltage Follower

The voltage follower is a special case of the non-inverting amplifier.  Figure 23.3 shows a typical connection to achieve this configuration.  As you can see the two resistors are replaced with simple wires.  This means that 100% of the output voltage is returned via negative feedback to the input.

Voltage Gain

The voltage gain of the voltage follower is, according to Eq. 23.3, 1/B.  Since in this case B = 1, its closed-loop gain of the votage follower is unity, or, in equation form

 (23.22)

Input Impedance

Because the voltage follower is the special case of the non-inverting amplifier with B = 1, its input impedance can be determined by using Eq. 23.11 with B = 1, or

 (23.23)

Output Impedance

The output impedance of the voltage follower can be found by using Eq. 23.12 with B = 1, as in the last case.

 (23.24)

Bandwidth

The bandwidth of the voltage follower amplifier can be found using Eq. 23.13 with B = 1.  In equation form we get

 (23.25)

 Note:

The voltage follower is used as a buffer amplifier because ofits very high input impedance and very low output impedance.  These features make it ideal for interfacing high-impedance sources and low-impedance loads.

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