In this chapter, we will explore some useful an: aplications of operational amplifiers. In connection with several of these. ve will introduce some important conccnt« i:l lnding oscillation theory. filtering, and wavcshuping, that have broad yond just their relevance 10 upcrutionul-amplilicr circuits. Unless otherwise noted ..we will assume that the operational umpliflcrs arc ideal. ‘enough to ideal that we can ignore small deviations from theory caused ie gain. finite input impedance. and nonzero output impedance.

**VOLTAGE SUMMATION, SUBTRACTION, AND SCALING**

**Voltage Summation**

Wl! have seen tl at it is possible 10 scule a signal voltage. thai is. to multiply it b) ” Iixed constant, through an appropriate choice of external resistors that determine the closed-loop gain of an amplifier circuit. This operation can be accomplished in either an inverting or noninverting configuration. t is also possible to sum several signal voltages in 01 e operationa -amplificr circuit and at the same time scale each by a different factor. For example. given inputs Vio V~. and VI, we might wish to generate an output equal to 2vI -:- O.5V2 + 4u.l. The latter SUIll is called – ‘inear combination of VI. V2, and v,, and the circuit that produces it is often calk rl a.earcombination circuit.

An inverting amplifier circuit that can be used to sum and scale three input signals. Note that input signals U” V2, and v.: are applied through separate resistors HI, R:, and RJ to the summing junction of the amplifier and that there is a single feedback resistor Rro Resistor R,. is the offset compensation resistor discussed in Chapter 13.

Following the same procedure we used in Chapter \J to derive the output of an inverting

An operational-amplifier circuit that produces an Oil/put equal to the (inverted) SIIIII of three separately scaled input signals

Equation 14-3 hows that the output is the inverted sum of the separately scaled inputs, i.e., a weighted sum, or linear combination of the inputs. By appropriate choice of values for R” R2, and RJ, we can make the scale factors equal to whatever . constants we wish, within practical limits. If we choose R, = R2 = RJ = R, then we obtain

The theory can be extended in an obvious way to two, luur, or allY reasonable number of inputs. ‘lhc feedback ratio for the circuit is

where R,. = R, II R~ 1\ R.I’ Using this value of fJ, we can apply the theory developed in Chapter 13 to determine all the performance characteristics that depend Oil fJ. including closed-loop bandwidth and output offset Vase Vi”)’ The optimum value of the bias-current compensation resistor is

l. Design an operational-amplifier circuit that will produce an output equal to DESIGN – (4vl + Vz +. 0.1 V3)’

2. Write an expression for the output and sketch its waveform when VI 2 sin wt V, Vz :::: +5 V de, and V3 = -100 V de.

**Solution**

1. We arbitrarily choose R, ::::60 kil. Then

2. v” = -[4(2 sin wt) + 1(5) + O.I(-100)] = -8 sin wI – 5 + 10 = 5 – 8 sin wt. This output is sinusoidal with a 5-V offset and varies between 5 – H = -3 V and 5 + 8 == 13 V. It is sketched if’ Figure 14-3.

A noninverting version of the linear-combination circuit. In this example, only two inputs arc connected and it can be shown (Exercise 14-4) that

Although this circuit does not invert the scaled sum, it is somewhat more cumbersome than the inverting circuit in terms of selecting resistor values to provide precise scale factors. Also, it is limited to producing outputs of the form K [avt + (1 – a )vJ where K and a are positive constants. Phase inversion is often of no consequence, but

in those applications where a noninverted slim is required, it can also be obtained using the inverting circuit of Figure 14-1, followed by a unity-gain inverter.