12.2. Operational Amplifier Oscillator

The opamp can serve as a square-wave oscillator by adding positive feedback with R₁ and R₃ and replacing R_y with a capacitor C_o, as shown in Fig. 12.2. The frequency of the operation can be computed on the basis of the following derivation. With positive feedback, the circuit is unstable and the output voltage is either V_op or –V_om, that is, the positive or negative limit. The limits are close to V_DD or V_SS. These voltage values are applied to the input through the positive feedback network such that the plus input terminal is either at

Equation 12.11

or

Equation 12.12

where V_om is positive.

However, the negative-feedback circuit consisting of R_f and C_o will cause the capacitor to charge up toward the voltage at the output. For example, suppose that for an interval of time, v_o(t) = V_op ≈ V_DD. Also assume that R₃ = R₁ such that the plus input is at V_op/2. When the capacitor charges up to and just beyond V_op/2, the opamp input V_∊ changes sign and the output will switch over to v_o(t) = –V_om. At this point, the capacitor begins charging toward the opposite polarity, and so forth.

For a given output, the time-dependent capacitor voltage is a simple RC transient. For example, for v_o(t) = V_op,

Equation 12.13

with a steady-state value of V_op. As an oscillator, the capacitor voltage will never reach the steady-state value; the transient terminates at the value given by (12.11).

The output is switching back and forth from V_op to –V_om. Therefore, the initial value of the capacitor voltage in (12.13) is

Equation 12.14

Using this to obtain a complete solution for (12.13) results in

Equation 12.15

We define t = T_plus as the positive time segment of the square-wave period. As noted, this corresponds to the time where the output switches to –V_om; that is,

Equation 12.16

Using (12.16) in the left-hand side of (12.15) gives an equation for T_plus as follows:

Equation 12.17

Then, solving for T_plus results in

Equation 12.18

Following the reasoning that led to (12.15), the falling capacitor-voltage transient is

Equation 12.19

The total time interval (negative segment of the square-wave period) of the falling transient, t = T_minus, corresponds to the capacitor voltage, decreasing to

Equation 12.20

Using (12.20) in (12.19) gives the equation for T_minus, which is

Equation 12.21

This leads to

Equation 12.22

A special case is for R₁ = R₃, where (12.18) and (12.22) become

Equation 12.23

and

Equation 12.24

The oscillator period is T = T_plus + T_minus or, for this special case,

Equation 12.25

A good estimate comes from the approximation V_op ≈ V_om. The result is

Equation 12.26

Note that although T_minus and T_plus can be significantly different [from (12.18) and (12.22)], the sum is well represented by (12.26). Note also that the waveform is strictly a square-wave only if V_op = V_om.

LabVIEW uses (12.25) to calculate the actual value of the capacitor used in the project on the oscillator. In the project, period T is a measured value. Shown in Fig. 12.3 is a project result of the measurement of the output voltage and capacitor voltage for the oscillator. Power-supply voltages were set at plus and minus eight volts and R₁ = R₃.