12.1: SINUSOIDAL OSCILLATORS

One of the major hazards involved in the application of operational amplifiers is that the user often finds that they oscillate in connections he wishes were stable. An objective of this book is to provide guidance to help circumvent this common pitfall. There are, however, many applications that require a periodic waveform with a controlled frequency, waveshape, and amplitude, and operational amplifiers are frequently used to generate these signals.

If a sinusoidal output is required, the conditions that must be satisfied to generate this waveform can be determined from the linear feedback theory presented in earlier chapters.

Wien-Bridge Oscillator

Figure 12.1 Wien-bridge oscillator

The Wien-bridge corifiguration (Figure 12.1) is one way to implement a sinusoidal oscillator. The transfer function of the network that connects the output of the amplifier to its noninverting input is (in the absence of loading)

\[\dfrac{V_a (s)}{V_o (s)} = \dfrac{RCs}{R^2 C^2 s^2 + 3RCs + 1} \label{eq12.1.1} \]

The operational amplifier is connected for a noninverting gain of 3. Com­bining this gain with Equation \(\ref{eq12.1.1}\) yields for a loop transmission in this positive-feedback system

\[L(s) = \dfrac{3RCs}{R^2 C^2 s^2 + 3RCs + 1} \nonumber \]

The characteristic equation

\[1 - L(s) = 1 - \dfrac{3RCs}{R^2 C^2 s^2 + 3RCs + 1} = \dfrac{R^2 C^2 s^2 + 1}{R^2 C^2 s^2 + 3RCs + 1} \label{eq12.1.3} \]

has imaginary zeros at \(s = \pm (j/RC)\), and thus the system can sustain constant-amplitude sinusoidal oscillations at a frequency \(\omega = 1/RC\).

Quadrature Oscillators

Figure 12.2 Quadrature oscillator.

The quadrature oscillator (Figure 12.2) combines an inverting and a non-inverting integrator to provide two sinusoids time phase shifted by \(90^{\circ}\) with respect to each other. The loop transmission for this connection is

\[L(s) = \left [-\dfrac{1}{R_1C_1s} \right ] \left [\dfrac{R_3C_3 s + 1}{(R_2C_2 s + 1) R_3C_3 s} \right ]\label{eq12.1.4} \]

In this expression, the first bracketed term is the closed-loop transfer function of the left-hand operational amplifier (the inverting integrator), while the second bracketed expression is the closed-loop transfer function of the right-hand operational amplifier. By proper selection of component values, the right-hand amplifier functions as a noninverting integrator. In fact, the discussion of this general connection in Section 11.4.1 shows that only the noninverting input of a differential connection is used as a signal input in this application.

If all three times constants are made equal so that \(R_1 C_1 = R_2 C_2 = R_3C_3 = RC\), Equation \(\ref{eq12.1.4}\) reduces to

\[L(s) = -\dfrac{1}{R^2 C^2 s^2} \nonumber \]

The corresponding characteristic equation for this negative-feedback sys­tem is

\[1 - L(s) = 1 + \dfrac{1}{R^2 C^2 s^2} = \dfrac{R^2 C^2 s^2 + 1}{R^2 C^2 s^2}\label{eq12.1.6} \]

Again, the imaginary zeros of Equation \(\ref{eq12.1.6}\) indicate the potential for constant- amplitude sinusoidal oscillation. Note that, since there is an integration between \(V_a\) and \(V_b\), these two signals will be phase shifted in time by \(90^{\circ}\) with respect to each other.

A similar type of oscillator (without an available quadrature output) can be constructed using a single amplifier configured as a double integrator (Figure 11.12) with its output connected back to its input.

Amplitude Stabilization by Means of Limiting

There is a fundamental paradox that complicates the design of sinusoidal oscillators. A necessary and sufficient condition for the generation of con­stant-amplitude sinusoidal signals is that a pair of closed-loop poles of a feedback system lie on the imaginary axis and that no closed-loop poles are in the right half of the \(s\) plane. However, with this condition exactly satisfied (an impossibility in any but a purely mathematical system), the amplitude of the system output is determined by initial conditions. In any physical system, minor departure from ideal pole location results in an oscillation with an exponentially growing or decaying amplitude.

It is necessary to include some mechanism in the oscillator to stabilize its output amplitude at the desired level. One possibility is to design the oscillator so that its dominant pole pair lies slightly to the right of the imaginary axis for small signal levels, and then use a nonlinearity to limit amplitude to a controlled level. This approach was illustrated in Section 6.3.3 as an example of describing-function analysis and is reviewed briefly here.

Consider the Wien-bridge oscillator shown in Figure 12.1. If the ratio of the resistors connecting the output of the amplifier to its inverting input is changed, it is possible to change the gain of the amplifier from 3 to \(3(1 + \Delta )\). As a result, Equation \(\ref{eq12.1.3}\) becomes

\[1 - L(s) = 1 - \dfrac{3(1 + \Delta )}{R^2 C^2 s^2 + 3RCs + 1} = \dfrac{R^2C^2 s^2 - 3\Delta RCs + 1}{R^2C^2 s^2 + 3 RCs + 1} \nonumber \]

The zeros of the characteristic equation (which are identically the closed-loop pole locations) become second order with \(\omega_n = 1/RC\) and \(\zeta = - (3/2)\Delta\). In practice, \(\Delta\) is chosen to be large enough so that the closed-loop poles remain in the right-half plane for all anticipated parameter variations.

For example, component-value tolerances or dielectric absorption asso­ciated with the capacitors alter the closed-loop pole locations.

Figure 12.3 Wien-bridge oscillator with limiting.

Limiting can then be used to lower the value of \(\Delta\) (in a describing-func­tion sense) so that the output amplitude is controlled. Figure 12.3 shows one possible circuit where a value of \(\Delta = 0.01\) is used. The oscillation fre­quency is \(10^4\) rad/sec or approximately 1.6 kHz. Output amplitude is (allowing for the diode forward voltage) approximately 20 V peak-to-peak. The symmetrical limiting is used since it does not add a d-c component or even harmonics to the output signal if the diodes are matched.

Amplitude Control by Parameter Variation

The use of a limiter to change a loop parameter in a describing-function sense after a signal amplitude has reached a specified value is one way to stabilize the output amplitude of an oscillator. This approach can result in significant harmonic distortion of the output signal, particularly when the oscillator is designed to function in spite of relatively large variations in ele­ment values. An alternative approach, which often results in significantly lower harmonic distortion, is to use an auxillary feedback loop to adjust some parameter value in such a way as to place the closed-loop poles pre­cisely on the imaginary axis, precluding further changes in the amplitude of the oscillation, once the desired level has been reached. This technique is frequently referred to as automatic gain control, although in practice some quantity other than gain may be varied.

As an example of this type of amplitude stabilization, let us consider the effect on performance of varying resistor \(R_3\) in the quadrature oscillator (Figure 12.2). We assume that \(C_1 = C_2 = C_3\), and that \(R_1 = R_2 = R\), while \(R_3 = (1 + \Delta ) R\). In this case the loop transmission of the system (see Equation \(\ref{eq12.1.4}\)) is with a corresponding characteristic equation

\[1 - L(s) = \dfrac{R^3 C^3 (1 + \Delta) s^3 + R^2 C^2 (1 + \Delta) s^2 + RC(1 + \Delta) s + 1}{R^2 C^2 s^2 (1 + \Delta )(RCs + 1)} \nonumber \]

If we assume a small value for \(\Delta\), the zeros of the characteristic equation can be readily determined, since

\[\begin{array} {l} {R^3 C^3 (1 + \Delta) s^3 + R^2 C^2 (1 + \Delta) s^2 + RC (1 + \Delta ) s + 1} \\ {\left [RC \left (1 + \dfrac{\Delta}{2} \right ) s + 1 \right ] \left [R^2C^2 \left (1 + \dfrac{\Delta}{2} \right ) s^2 + RC \dfrac{\Delta}{2} s + 1 \right ] |\Delta | \ll 1} \end{array}\label{eq12.1.10} \]

The performance of the oscillator is, of course, dominated by the complex-conjugate root pair indicated in Equation \(\ref{eq12.1.10}\), and this pair has a natural frequency \(\omega_n \simeq 1/RC\) and a damping ratio \(\zeta \simeq A/4\). The important feature is that the closed-loop poles can be made to lie in either the left half or the right half of the s plane according to the sign of \(\Delta\).

The design of the amplitude-control loop for a quadrature oscillator provides an interesting and instructive example of the way that the feedback techniques developed in Chapters 2 to 6 can be applied to a moderately complex circuit, and for this reason we shall investigate the problem in some detail. The difficulties are concentrated primarily in the modeling phase of the analytical effort.

Our intent is to focus on amplitude control, and this control is to be accomplished by moving the closed-loop poles of the oscillator to the left- or the right-half plane according to whether the actual output amplitude is too large or too small, respectively. We assume that the signal \(v_A(t)\) (see Figure 12.2) has the form

\[v_A (t) = e_A (t) \sin \omega t \nonumber \]

This representation, which models the signal as a constant-frequency sinusoid with a variable envelope \(e_A(t)\), is not exact, because the instan­taneous frequency of the sinusoidal component of \(v_A\) is a function of \(\Delta\). However, if the amplitude-control loop has a very low crossover frequency compared to the frequency of oscillation so that magnitude changes are relatively slow, we can consider the amplitude \(e_A\) alone and ignore the sinusoidal portion of the expression. In this case the exact frequency of the sinusoid is unimportant.

In order to find the dependence of \(v_A\) on the control parameter \(\Delta\), assume that the circuit is oscillating with \(\Delta = 0\) so that the closed-loop poles of the oscillator are precisely on the imaginary axis. With this constraint the envelope is constant with some operating point value \(E_A\) so that

\[v_A (t) = E_A \sin \omega t \nonumber \]

where \(\omega = 1/RC\). If \(\Delta\) undergoes an incremental step change to a new value \(\Delta_1\) at time \(t = 0\), the oscillator poles move into the left-half plane (for positive \(\Delta_1\)), and

\[v_A (t) \simeq E_A e^{-\zeta \omega_n t} \sin \omega_n t \label{eq12.1.13} \]

Inserting values for \(\zeta\) and \(\omega_n\) from Equation \(\ref{eq12.1.10}\) into Equation \(\ref{eq12.1.13}\) yields

\[v_A (t) \simeq E_A e^{-(\Delta_1 t/4RC)} \sin \dfrac{t}{RC} \nonumber \]

The envelope for this signal is

\[e_A (t) = E_A e^{-(\Delta_1 t/4RC)} = E_A \left [ 1 - \dfrac{\Delta_1 t}{4RC} + \dfrac{1}{2} \left (\dfrac{\Delta_1 t}{4RC} \right )^2 - \cdots + \right ] \nonumber \]

If \(\Delta_1 t/4RC\) is small (a condition insured by a sufficiently small value of \(\Delta_1\)), we can separate \(e_A(t)\) into operating-point and incremental components as

\[e_A (t) = E_A + e_a (t) \simeq E_A - \dfrac{E_A \Delta_1}{4RC} t \nonumber \]

Thus a positive incremental step change in \(\Delta\) leads to an incremental envelope change that is a linearly decreasing function of time. This condi­tion implies that the linearized transfer function that relates envelope amplitude to \(\Delta\) is

\[\dfrac{E_a (s)}{\Delta (s)} = - \dfrac{E_A}{4RCs} \label{eq12.1.17} \]

This linearized analysis confirms the feeling that control of the value of \(\Delta\) is in fact a reasonable way to stabilize the amplitude of the oscillation, since the incremental change in the envelope of the oscillation is proportional to the time integral of \(\Delta\).

Figure 12.4 Quadrature oscillator with amplitude stabilization.

Further design of the amplitude-control loop depends on the actual topology of the system. Figure 12.4 shows one possible implementation in mixed circuit and functional block-diagram form. The envelope of the signal to be controlled is determined by an amplitude-measuring circuit. This circuit may be a simple diode-resistor-capacitor peak detector in cases where high precision is not required, or it may be an active "super­diode" type of connection (an example is given in Section 12.5.1) in more

demanding applications. In either case, the design of this circuit is not particularly difficult and will not be discussed here. The envelope of the signal is compared with a reference value, and the resulting error signal passes through a linear controller with a transfer function \(a(s)\). The output of the controller is used to drive a field-effect transistor that functions as a variable resistor whose value determines \(\Delta\).

The FET connection incorporates local compensation to linearize its characteristics as shown in the following development. If a junction FET is biased into conduction with a small voltage applied across its channel, and its gate reverse biased with respect to its channel, the drain current is approx­imately related to terminal voltages as

\[i_D = K \left [(v_{GS} + V_P) v_{DS} - \dfrac{v_{DS}^2}{2} \right ] \label{eq12.1.18} \]

where \(K\) is a constant dependent on transistor construction, and \(V_P\) is the magnitude of the gate-to-source pinch-off voltage.

The dependence of \(i_D\) on the square of the drain-to-source voltage is undesirable, since this term represents a nonlinearity in the channel resist­ance of the device, and this nonlinearity will introduce harmonic distortion into the oscillator output. The nonlinearity can be eliminated by adding half of the drain-to-source voltage to the gate-to-source voltage via resistors as shown in Figure 12.4. The resistors are large enough so that they do not significantly shunt the drain-to-source resistance of the FET under normal operating conditions. With the topology shown,

\[v_{GS} = \dfrac{1}{2} (v_C + v_{DS}) \label{eq12.1.19} \]

Substituting Equation \(\ref{eq12.1.19}\) into Equation \(\ref{eq12.1.18}\) shows that

\[i_D = K\left [\left (\dfrac{v_C}{2} + \dfrac{v_{DS}}{2} + V_P \right ) v_{DS} - \dfrac{v_{DS}^2}{2} \right ] = K \left (\dfrac{v_C}{2} + V_P \right ) v_{DS} \nonumber \]

or

\[R_{DS} = \dfrac{\partial v_{DS}}{\partial i_D} = \dfrac{1}{K[(v_C/2) + V_P]}\label{eq12.1.21} \]

This equation indicates that the incremental resistance of the FET is inde­pendent of drain-to-source voltage when the network is included.

For purposes of design, we assume that the FET is characterized by \(V_P = 4\) volts and \(K = 10^{-3}\) mho per volt. Recall that stable-amplitude oscillations require that all three \(R-C\) time constants be identical; thus the operating point value of \(R_{DS}\) is 500 ohms. Equation \(\ref{eq12.1.21}\) combined with FET parameters indicates that this value results with an operating-point value for the control voltage of -4 volts. The incremental change in \(R_{DS}\) as a function of the control voltage at this operating point, obtained by differentiating \(\ref{eq12.1.21}\) with respect to \(v_C\),

\[\dfrac{\partial R_{DS}}{\partial v_C}|_{v_C = -4V} = -125 \Omega /V \label{eq12.1.22} \]

Earlier modeling was done in terms of \(\Delta\), the fractional deviation of the resistance \(R_3\) in Figure 12.2 from its nominal value. This resistor consists of the FET plus a \(9.5\ k\Omega\) resistor in the actual implementation. The incremental dependence of \(\Delta\) on the control voltage is determined by dividing Equation \(\ref{eq12.1.22}\) by the anticipated operating-point value of the total resistance, \(10\ k\Omega\). Thus

\[\dfrac{\partial \Delta}{\partial v_C}|_{v_C = -4V} = -0.0125 V^{-1} \label{eq12.1.23} \]

Figure 12.5 Linearized block diagram for amplitude-control loop.

The relationships summarized in Equations \(\ref{eq12.1.17}\) and \(\ref{eq12.1.23}\) combined with the system topology and an assumed operating-point value for the en­velope \(E_A = 10\) volts lead to the linearized block diagram for the amplitude­-contrrol loop shown in Figure 12.5. The negative of the loop transmission for this system is

\[\dfrac{E_a (s)}{E_e (s)} = a(s) \times \dfrac{312.5}{s} \nonumber \]

A number of factors govern the choice of \(a(s)\) for this application including:

(a) The actual FET gate-to-source voltage required under quiescent con­ditions is strongly dependent on FET parameters and the exact values of the other components used in the circuit. The easiest way to insure that the difference between the envelope and the reference is constant in spite of these variable parameters is to include an integration in \(a(s)\) since this integration forces the operating-point value of the error to zero.

(b) The analysis is predicated on a much lower crossover frequency for the amplitude-control loop than the frequency of oscillation, \(10^4\) radians per second. However, a very low frequency control loop accentuates the effect on amplitude of rapid changes in quantities like the supply voltages. A somewhat arbitrary compromise is to choose a crossover frequency of 100 radians per second.

(c) Since the analysis is based on a hierarchy of approximations, the system should be designed to have a very conservative phase margin.

(d) The controller transfer function should include low-pass filtering. The detector signal that indicates the envelope amplitude invariably in­cludes components at the oscillation frequency or its harmonics. If these components are not filtered so that they are at an insignificant level when applied to the FET gate, the resultant channel-resistance modulation intro­duces distortion into the oscillator output signal.

A controller transfer function that incorporates these features is

\[a(s) = \dfrac{3.2 (0.1s + 1)}{s(10^{-3} s + 1)^2} \nonumber \]

The negative of the loop transmission with this value for \(a(s)\) is

\[\dfrac{E_a (s)}{E_e (s)} = \dfrac{}{} \nonumber \]

The system crossover frequency is 100 radians per second, and phase margin exceeds \(70^{\circ}\) with this value for \(a(s)\).

Figure 12.6 Controller circuit.

A possible circuit that provides the negative of the desired \(a(s)\) is shown in Figure 12.6. In many cases of practical interest, this inversion can be can­ celled by some rearrangement of the amplitude-measuring circuit. The second required filter pole is obtained with a passive network. The filter network impedance level is low enough so that the network is not disturbed by the \(2-M\Omega\) load connected to it.

The reference level required to establish oscillator amplitude can be applied to the controller by adding another input resistor to the operational amplifier. It may also be possible to realize part of the amplitude-measuring circuitry with this amplifier. An example of this type of function combination is given in Section 12.5.1.

Two practical considerations involved in the design of this oscillator deserve special mention. First, the signal \(v_B\) normally has lower harmonic distortion than does \(v_A\) since the integration of the first amplifier filters any harmonics that may be introduced by the FET. Second, it is possible to vary the reference amplitude for this circuit and thus modulate the amplitude of the oscillator output. However, the control bandwidth in this mode will be relatively small, and performance will change as a function of quiescent envelope amplitude since the loop-transmission magnitude is dependent on operating levels.

The performance of an oscillator of this type can be quite impressive. Amplitude control to within \(1\ mV\) peak-to-peak is possible if "superdiodes" are used in the envelope detector. Harmonic distortion of the output signal can be kept a factor of \(10^4\) or more below the fundamental component.