One of the simplest kinds of oscillators incorporating an operational amplifier can be constructed as shown in Figure 14-42. Here we see that the amplifier is COt” ‘ctl”:;3 in an inverting configuration and drives three cascaded (high-pass) RC sections.The arrungcmcnt is called an RC phase-shift oscillator. The inverting amplifier causes a umo phase shift in the signal passing through it, and the pUi pOSC 0 tile cascaded RC sections is to introduce an additional 180° at some frequency. Recall that the output of a single, high-pass RC network leads its input by a phase angle that depends on the signal frequency. When the signal passes through all three RC sections, there will he some frcqucncy at which the cumulative phase shift is H;o°. When the signal having that frequency is Ied back to the inverting amplifier, as shown in the figure, the total phase shift around the loop will squa 180° -t 180° -180° + 1800= 0°) and oscillation will occur at that frequency,provided the loop gain is 1. The gain necessary to OVt:II..orne lilt :”.,.> ill the RC cascade and bring the loop gain up to 1 IS supplied ‘by the amplifier (vJvm -R,IRt). With considerable algebraic effort, it can be shown (Exercise 14-40) that the !’•..edback ratio determined by the RC cascade (with the feedback connection to Re opened) is

In order for oscillation to occur. the cascade must shift the phase of the signal by Hmo, which means the angle of f3 must be IHO°. When the angle of f3 is umo, f3 is a purely real number. In that case, the imaginary part of the denominator of equation 14-83 is O. Therefore, we can find the oscillation frequency by finding the value of w that makes the imaginary part equal O. Selling it equal to {) and solving for w, we find

Notice that resistor RI in Figure 14-42 is effectively in parallel with the rightmost resistor R in the RC cascade, because the inverting input of the amplifier is at virtual grouno Therefore, when the feedback loop is closed by connecting the cascade to RI.lhe frequency satisfying the phase criterion will be somewhat different than that predicted by equation 14-X5. If RI » R, so that Rill R R, then equation 14-1\5 will closely predict the oscillation frequency.

We can find the gain that the amplifier must supply by finding the reduction in gain caused by the RC cascade. This we find by evaluating the magnitude of f3 at the oscillation frequency: w = 1/(V6RC). At that frequency, the imaginary term in equation 14-XJ IS () and f3 is the real number

The minus sign confirms that the cascade inverts the fccdba ~. at the oscillation frequency. We see that the amplifier must supply a gain of -29 to make the loop gain Af3 = I. Thus, we require

In practice, the feedback resistor is made adjustable to allow for small differences . in component values and Ioi the loading caused by R1

To prevent 1<1 from Im\uing this value of N, We choose R, = 20 kil. (For greater precision, Wl’ could choose the lust R in the cascade and the value Ri s« that R 1\ RI = \300 n.) From equation I4-H7, U, = 29RJ = 29(20 k!l) :-: 580 k.H. The completed circuit is shown”, figure 14-43. Rj is made adjustable so the loop gair can be set precisely to I.

**Ihe Wien-bridge Oscillator**

Figure 14-44 shows a widely used type of oscillator called a Wien bridge. The operational amplifier is used in a noninverting configuration, and the impedance blocks labeled Z, and Z2 form a voltage divider that determines the feedback ratio. Note that a portion of the output voltage is fed back through this impedance divider to the + input of the amplifier. Resistors Rg and Rf determine the amplifier gain and are selected to make the magnitude of the loop gain equal .0 1. If the feedback iml-””lJal1~t::.all! cnosen properly, there will be SOIT!I:’ f”p~l1f~ncvat which there is zero phase shift in the signal fed back to the amplifier input (u+). Since the amplifier is noninverting, it also contributes zero phase shift, so the total phase shift around the loop is 0 at that frequency, :IS required for oscillation. In the most common version of the Wien-bridge oscillator. Z, is a series RC.

In practical Wien-hridge oscillators, Rf is not equal to exactly 2Rr component tolerances prevent R, from being exactly equal to R2 and C, from being exactly equal to C2• Furthermore, the amplifier is not ideal, so o: is not exactly equal to v+. In a circuit constructed for laboratory experimentation, Rf should be made adjustable so that the loop gain can be set as necessary to sustain oscillation. Practical oscillators incorporate a nonlinear device in the R,-Rg feedback loop to provide automatic adjustment of the loop gain, as necessary to sustain oscillation. This arrangement is a form of automatic gain control (AGC), whereby a reduction in signal level changes the resistance of the nonlinear device in a way that restores gain.