**Oscillator Amplitude Transfer Function**

Oscillator amplitude control loop (ALC) design is complicated by the inherent nonlinearity of the loop. A systematic analysis of the problem begins with the dependencies of the amplitude, which ideally is *V _{g} * . When the oscillator pole-pair is not exactly on the

*jω*-axis, the amplitude is

where *α * = *ζ* x*ω _{n} * is the real component of the complex pole-pair,

*ζ*is the

*damping*(= 1/2x

*Q*) and

*ω*is the resonant or undamped frequency of oscillation. On the

_{n}*jω*-axis,

*ζ*= 0 and the oscillator circuit response is undamped. When the pole-pair wanders off the imaginary axis to the right,

*ζ*> 0, and the amplitude increases exponentially in time. To the left, it decreases exponentially and

*ζ*< 0. The goal of ACL design is to keep the pole-pair on the axis.

What the ACL controls is *ζ* , and because we are interested in how this control behaves in time, *ζ* is not a constant as it usually is in circuits but is *ζ* (*t* ). This makes the amplitude function nonlinear in that

is now a function of time for both constant *ζ* and for *ζ* (t):

. How *ζ* (*t* ) varies over time depends on the ACL dynamics. The problem can be approached generally, without regard to oscillator type, by taking the derivative of the amplitude;

(An alternative is to consider the amplitude a function of time only and differentiate. This results in the same expression, though our interest in *ζ* as the control parameter is emphasized in taking partial derivatives.) This derivative is a total-variable quantity and requires an operating-point at *t* = *t* _{0} and *ζ* = *ζ* _{0} at which the expression is evaluated. It assumes that *ω _{n} * is constant, and though

*ω*varies slightly with

_{n}*ζ*in some oscillators, it is not the parameter controlled by the ACL and is not a consideration in ACL design.

The derivative is evaluated by expressing total-variable quantities as the sum of the constant (static) op-pt values and a small variation from it, expressed as a differential. Thus the total variable is the op-pt value plus a differential value;

The resulting differential quantities are the *incremental* quantities of the linearized circuit. Introducing the expanded total-variable expressions into the derivative,

Because the ACL is intended to keep

it is desired that any deviation from *V _{g} * be immediately corrected. Consequently, little time will have elapsed before the correction is made and the operating-point in time is near

*t*= 0 s from the instant of the deviation;

*t*

_{0}= 0 s. At

*t*

_{0}, the pole-pair is on the

*jω*-axis and

*ζ*

_{0}= 0. Both

*dζ*and

*dt*are small changes from their static values and their product can be set to zero. Then substituting for

*ζ*

_{0}and

*t*

_{0}, the exponential reduces to

The op-pt values are constants and the derivative reduces to

Now all of the quantities are differentials and thus are incremental. Consequently, the system has been linearized and the Laplace transform can be taken so that the ACL can be analyzed in the complex-frequency domain:

where it is understood that *ζ* (*s* ) refers to the incremental *ζ* . Then the desired transfer function is

The incremental amplitude is related to *ζ* by a quasistatic value of –*V _{g} * and a pole at the origin that crosses a gain of one at the oscillator frequency.

**Forward-Path Transfer Function**

Now that the oscillator block of the ACL is found, *ζ* must still be related to a controlled circuit variable. The circuitry in the ACL between the error summing function – the input circuit of an op-amp – and the oscillator is the ACL *forward path* , *G* , of the loop. Then the oscillator and amplitude extractor are in the feedback path, *H* .

In the state-variable oscillator, a single resistor between the summing stage and first integrator following it sets *ζ* . This resistor can be replaced by a multiplier to adjust *ζ* with a controlling voltage. In the Wien-bridge oscillator, the pole polynomial is the feedback factor,

For oscillation, 1 + *G* x*H* = 0 and the numerator of *D* (*s* ) = 0. It is the quadratic form of the pole-pair which in general form is

where *ζ* is damping and the resonant time constant, *τ _{n} * = 1/

*ω*. The goal is to vary

_{n}*G*

_{0}, the gain of the noninverting side of the Wien-bridge op-amp, to be exactly 3 so that

*ζ*= 0. The noninverting gain, from the familiar op-amp formula, is

Thus *G* _{0} can be varied by varying *R _{i} * , and the JFET rDS (or including the shunting

*R*,

_{p}*r*) is placed in series with an external resistor,

_{out}*R*’ so that

_{i}Ri = Ri’ + rout

The fractional range of *r _{out} * ,

*r*/

_{out}*R*, is chosen according to the required range on

_{i}*R*and this is intended to be small. Taking into account resistor tolerances and small transient disturbances of the oscillator, typically a 10 % range should be adequate. Then the maximum

_{i}*r*≈ 0.1x

_{out}*R*.

_{i}The previous coverage of the JFET circuit produced the transmittance from the output of the forward-path error amplifier, *v _{i} * , to

*r*. The incremental

_{out}*g*

_{0}, or

*G*

_{0}, is the change in

*G*

_{0}with

*r*, or

_{out}This transmittance can be linked to the previous block of the oscillator transfer function through the simple relationship of the total-variable *ζ* to *G* _{0} for the Wien-bridge oscillator;

Then with all transfer function variables incremental, the connected cascade blocks of the ACL have transfer function

For oscillators other than the Wien-bridge, such as the quadrature, no *G* _{0} appears in the expression for 1 + *G* x*H* . In this case, one of the timing resistances of an RC time constant is varied. This causes a pole at the origin to shift its unity-gain frequency. Such a pole is of the form

where *ω _{T} * is the frequency at a gain of one. By varying

*R*, the frequency-response magnitude plot of the pole (of –1 slope) slides up and down; increasing

*R*decreases gain by reducing

*ω*so that for a given frequency, the gain is lower than before

_{T}*R*was reduced. Similar reasoning applies to state-variable and other types of oscillators.

**Amplitude Extractor and Error Amplifier**

What remains in the ACL is the error amplifier (input block of *G* ) and amplitude extractor (output block of *H* ). The error amplifier is typically a one-op-amp differential-input amplifier with a fixed gain, driven by a voltage reference that sets

.

Its quasistatic gain is set to provide adequate amplitude accuracy, and additional poles and zeros for loop compensation can be added to it.

The amplitude extractor inputs the oscillator sine-wave and outputs ideally a constant voltage equal to its amplitude. Various circuits can be applied for this function. All of them are in the category of sampling circuits because the amplitude information occurs only once per oscillator half-cycle. Whether the amplitude is extracted by a half-wave or full-wave peak detector followed by an RC integrator or by a synchronous rectifier that integrates over one or both half-cycles and scales the resultant average for the peak value, all have transfer functions similar to zero-order hold (ZOH) circuits.

A simple half-wave op-amp peak detector with an RC filter illustrates.

The diode conducts around the input peaks, from – *θ _{c} * to +

*θ*degrees of the cycle. The average voltage during this conduction time is

_{c}The ZOH sin*θ* /*θ* = sinc*θ* function appears. If the conduction angle is expressed as a duty ratio instead, where

then

Some values are tabulated below. A peak detector that conducts over the full half-cycle (*D* = 0.5) is equivalent to a synchronous detector.

For smaller *D* , the sampling time is less and the ZOH gain closer to one. Even for sloppy peak detection, the loss in gain is not severe – less than 2 % for a 10 % conduction time over the cycle. Consequently, the amplitude extractor does not significantly impact loop gain though its varying output voltage can modulate the oscillator sine-wave and increase distortion. To minimize output ripple, the RC integrator time constant can be made large, but if its pole is too close to that of the oscillator transfer function, ACL instability might occur. The dynamic effect of the amplitude extractor on the ACL is that of a slight gain reduction which can be ignored, and the addition of a finite real pole from the RC integrator, at τ * _{A} * =

*R*x

_{A}*C*.

_{A}Finally, the error amplifier, in its simplest (and often adequate form) contributes nothing more than quasistatic gain, *K* _{0} . Then the total ACL gain is

and is negative (inverting) because *V _{P} * < 0 V. The loop gain can be subjected to Bode-plot stability analysis. The quasistatic gain constitutes most of the expression, with a pole at the origin and another at –1/ τ

*.*

_{A}This kind of loop is not difficult to stabilize by adjusting the quasistatic gain. If the RC integrator pole is well below the oscillator frequency, then *ω _{A} * = 1/ τ

*<<*

_{A}*ω*and at

_{n}*ω*, the G·H magnitude plot steepens from –1 to –2 in slope, to cross unity gain at

_{A}*ω*= √(

_{T}*G*x

*H*)

_{0}x

*ω*x

_{A}*ω*. The phase at this frequency determines the phase margin and the dynamic response of the ACL.

_{n}
Really enjoyed this article. This topic is very difficult to find meaningful references on. It would also be interesting to take it farther and carry through a design, along with testing the disturbance and startup response.

Thanks for the writeup.

Dave