As we saw in Experiment No. 22 an opamp is an amplifier with a very high gain (the openloop gain or A_{ol}). This fact limits the opamp usefulness to a few applications (i.e., comparators, to be studied in Experiment 24). In openloop 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 maximum^{1}.
Negative Feedback
To make use of the versatility of opamps 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 180^{o2}. In the case of opamps 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 OpAmps
Part (a) of this figure shows a configuration called noninverting 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 (V_{f}) 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} A_{3} you should remember, a phase of angle of 180^{o} reverts the polarity of a signal.
Effects of Negative Feedback
With negative feedback many features of opamps 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 opamp circuits. In this experiment we will study three basic negative feedback circuits, namely Noninverting amplifiers, Inverting amplifiers, and the Voltagefollower amplifier.
NonInverting Amplifier
An opamp connected in closedloop as a noninverting amplifier is shown in Figure 23.1(a). The input signal is applied to the noninverting 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 V_{f} and is given by Eq. 23.1. Because we apply the input signal to the + (noninverting) 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 closedloop gain we notice that the output voltage is the differential voltage (V_{D}) amplified by the openloop 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
The closedloop voltage gain is defined as
  (23.8)

therefore, using Eq. 23.7, the closedloop voltage gain is given by
  (23.9)

Now, because A_{ol} is very large, the product BA_{ol} is much larger than 1. Therefore, the denominator of Eq. 23.9 can be replaced by BA_{ol}. Then, Eq. 23.9 becomes
Using B from Eq. 23.3, we can express the closedloop gain of the noninverting amplifier by
Thus,
  (23.10)

As you can well see from the Eq. 23.10 the closedloop gain does not depend on the openloop gain of the opamp^{3}. The closedloop gain can be set just by selecting values of R_{f} and R_{1}.
.5ex]
The above derivations shows us that with negative feedback the gain of an opamp 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 opamp is large. When the opamp is used in the noninverting amplifier configuration of Fig. 23.1(a) the input impedance can be increased to any value. If we define Z_{i} the impedance of the opamp without feedback, and A_{ol} the voltage gain without feedback, then the input impedance of the closedloop configuration  denoted by Z_{in}_{(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 feedback^{4}.
We will leave the details of this derivation to the reader.
^{4} Recall that A_{ol} is very large.
Output Impedance
The output impedance of an opamp is relatively small. With negative feedback, however, the output impedance can be reduced to any desired value for the case of the noninverting amplifier.
It can be shown that the output impedance of a noninverting configuration can be practically reduced to a value as small as we want. If we define Z_{o} to be the output impedance of the openloop (no feedback) opamp, then the output impedance (Z_{out(NI)}) for this configuration is given by
  (23.12)

This equation shows that the output impedance of an opamp — if used with negative feedback — is much smaller than its equivalent openloop impedance.
Bandwidth
The bandwidth of a noninverting amplifier is larger than its equivalent openloop amplifier. Without any proof^{5} we state here the relationship between the openloop and closedloop bandwidth. If we define BW_{ol} as the openloop bandwidth, the closedloop bandwidth is given by
  (23.13)

where, as usual, B is the feedback fraction, and A_{ol} is the openloop gain of the opamp at the lowest frequency^{6}.
^{5} As a homework to the reader, we assign the proof of this equation.
^{6} This is called the midfrequency gain and is denoted by A_{ol(mid)}. It is the highest gain of the amplifier.
Inverting Amplifier
An opamp connected in closedloop 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 (R_{f}) and the input resistor (R_{1}). 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 R_{1}.
Voltage Gain
The voltage gain of the inverting amplifier can be calculated by noticing that the differential voltage V_{D} 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 zero^{7}. This fact makes the current through the input resistor R_{1} equal to the current through R_{f}, or
  (23.14)

Figure 23.2 shows this fact.
Figure 23.2: Inverting amplifier virtual ground

Now, because the voltage V_{D} is zero (practically) then the input voltage V_{i} 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 I_{1} = I_{f}. 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 V_{0}/V_{i} by A_{cl}_{(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 opamp parameters, and by choosing the proper resistors R_{1} and R_{f} you can make the gain to acquire any value.
^{7}Remember 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 R_{1}, 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 openloop opamp. In equation form we have
  (23.20)

Bandwidth
The bandwidth of the inverting amplifier is the same as the bandwidth for the noninverting 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 noninverting 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 closedloop 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 noninverting 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)

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 highimpedance sources and lowimpedance loads.
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