11.1. Operational Amplifiers with Resistance Feedback
In this unit, the gain characteristics of the operational amplifiers with resistive feedback are discussed. These are dc amplifiers that are configured for specific gain and input and output resistance characteristics. The operational amplifier without feedback is in the open-loop mode. The dc (bias) configuration is shown in Fig. 11.1. Due to imbalances in the amplifier circuit, which are a result of variations in the parameters of the transistors and circuit components from the values used in the design, the output will probably be latched at either the plus or minus power supply.
Figure 11.1. Open-loop amplifier. Inputs, output, and power-supply pins are connected. Circuit will be bias unstable. Dc VO is likely to be latched at near VDD or VSS.
The circuit can be set into a stable, active mode with the output approximately at zero volts by providing resistive negative feedback as shown in the circuit diagram of Fig. 11.2.
Figure 11.2. Opamp resistor network, including feedback resistor for bias stabilization. Signal output voltage is Vo = avoV∊.
The resistor connected between the output, VO, and the inverting (minus) input effectively applies the output voltage to the opamp input and it is of such a polarity as to drive the output toward zero volts, where it tends to stay. That is, attaching the resistor completes the negative feedback loop from V∊ to VRy. The quantitative aspects of stabilization with the feedback resistor are discussed in Unit 11.6.
This circuit becomes a dc amplifier by installing a signal at either input. These two possibilities are discussed in the following units.
11.1.1. Voltage Gain of the Noninverting Resistor Feedback Amplifier
In the circuit diagram of Fig. 11.3, an input signal voltage Vs is attached to resistor Rx. By definition, the output terminal voltage is positive for a plus input. Hence, this is the noninverting amplifier. Here we obtain the relationship between the amplifier gain, avo = Vo/Vs, and the open-loop gain (opamp gain), avo = Vo/V∊, and the circuit resistors.
Figure 11.3. Non-inverting or voltage amplifier (series-shunt). The input resistance is essentially infinite.
With a signal applied to the plus input terminal, the responding output voltage is fed back to the resistor Ry. The voltage across Ry, Vf, and Vo (signal) are related by
Equation 11.1
This is just the voltage-divider relation, which applies in this case, as negligible current flows into the input terminals of the opamp. Variable Vf is used in lieu of VRy to distinguish it from the dc value of the voltage across Ry. This use is also consistent with the fact that Vf is technically a signal feedback voltage.
The input signal voltage Vs adds up to
Equation 11.2
where (11.1) is used to eliminate Vf. (The voltage drop across Rx is essentially zero.) This leads directly to the relation for amplifier gain, which is
Equation 11.3
with (ideal noninverting gain).
Equation 11.4
AvNI is the limiting form of the noninverting amplifier gain for avo → ∞. This is consistent with the fact that in the limit, V∊ → 0 and Vs = VRy. Thus, the output and input voltages are simply related by the voltage-divider relation. The result, (11.3), indicates that for avo>>AvNI, the voltage gain can simply be expressed in terms of the resistors and therefore is very predictable. If we make, for example, Rf = 10Ry, AvNI = 11, and (11.3) gives Av = 10.998, with avo = 40,000. The value for the opamp gain is typical for our project opamp. Note that due to the high resistance at the opamp input terminals, Rx has no influence on the gain.
The noninverting amplifier has a very high input resistance and a low output resistance, as discussed in Unit 11.4. The amplifier technically falls into the category of a series – shunt feedback configuration or a voltage amplifier.
11.1.2. Voltage Gain of the Inverting Resistor Feedback Amplifier
To obtain the inverting amplifier, the signal is applied to the negative or inverting terminal as shown in Fig. 11.4. A positive signal results in a negative output voltage. The gain expression can be obtained with the loop equation from output to input:
Equation 11.5
Figure 11.4. Inverting or transresistance amplifier (shunt – shunt). The input resistance is equal to the signal-source resistance, Ry.
and the loop equation at the input (voltage drop across Rx is zero)
Equation 11.6
Eliminating Is between (11.5) and (11.6) gives
Equation 11.7
This expression can be manipulated to give the gain as
Equation 11.8
which is
Equation 11.9
where
Equation 11.10
AvI is the gain of the ideal inverting amplifier.
The ideal gain, as in the case of the noninverting amplifier, depends only on resistor values. The approximate form is based on the approximation V∊ = →0, in which case the negative input terminal is at virtual ground. Thus, |vo| and Vs are proportional to Rf and Ry, respectively. The inverting-amplifier gain result includes the fact that the current into the negative terminal is zero. The circuit is a shunt – shunt feedback configuration or a transresistance amplifier.