8 Harmonic oscillators (low Q)

In stable systems, the loop gain $A(\mathit{𝑗𝜔})\beta (\mathit{𝑗𝜔})$ does not encircle the point “-1” in the complex plane in the clockwise These stable systems are usually linear, except when explicitly using non-linear components or when running into some limit like clipping to the voltage supply. Because these systems are linear(-ish),:

• the linear transformation between time domain and frequency domain can be applied. Consequently these linear systems can (also) be analysed in the frequency domain.
• an excitation at a sinusoidal input signal at radian frequency $\omega$ results in signals throughout that system also to be at this $\omega$

If the “-1” point in the complex plane is encircled in the clockwise direction by the loop gain $A(\mathit{𝑗𝜔})\beta (\mathit{𝑗𝜔})$, then that system is unstable. Such an unstable system may be in a meta-stable state or it goes to or stays in either of the extreme states — possibly periodically changing its state. Usually these extreme states correspond to an output voltage equal to the positive or negative supply voltage. Theoretically a system can stay in a meta stable point indefinitely⁴⁸ ; in actual circuits any disturbance (noise, interference, birds flying by) will make the system leave the meta-stable point and go to either of the stable extreme states. These fundamentally unstable systems behave non-linearly and hence cannot be analyzed in frequency domain: they can only be analyzed in time domain. As interesting and as useful as they may be, we’re not going to analyze, introduce or show these unstable circuits⁴⁹ .

Exactly between the stable systems and the unstable systems are systems for which the loop gain $A(\mathit{𝑗𝜔})\beta (\mathit{𝑗𝜔})$ crosses the –1 point in the Nyquist plot. Systems for which the loop gain curve crosses the “-1” point will be shown to oscillate harmonically (sinusoidal) at the frequency corresponding at which the Nyquist plot crosses that – 1 point.

8.1 Introduction into harmonic oscillators

Harmonic oscillators are systems for which the Nyquist plot crosses the “$-1$” point for one specific (non-zero, finite) frequency. The hows and whys for this are briefly introduced below. After that, a variety of harmonic oscillators are discussed. Although the basics are identical, the actual circuit implementation of oscillators can be very different, all with their specific advantages and disadvantages.

As starting point, let’s assumed the negative feedback configuration shown in Figure 8.1a, having a forward gain $A(\mathit{𝑗𝜔})$ and feedback gain $\beta (\mathit{𝑗𝜔}$. Using this configuration, it will be shown that harmonic oscillators can be constructed by ensuring that the Nyquist factor $A(\mathit{𝑗𝜔})\beta (\mathit{𝑗𝜔})=-1$. for one specific non-zero and finite frequency $\omega ={\omega }_{\mathit{𝑜𝑠𝑐}}$. Originally the Nyquist factor is defined as $A(\mathit{𝑗𝜔})\beta (\mathit{𝑗𝜔})$, for a negative feedback to (hence) the inverting input of the amplifier as shown in Figure 8.1a. This negative feedback arrangement is common practice for amplifier circuits and other stable systems, but it is not necessary for oscillator circuits.

Harmonic oscillators from systems with negative feedback

The signal transfer – or closed loop gain – of the system with negative feedback in Figure 8.1a, is

Assuming that the Nyquist plot crosses the –1-point, for the specific (radian) frequency ${\omega }_0$ where the Nyquist curve actually crosses the –1-point, the denominator of the closed loop transfer function of the inverting configuration in Figure 8.1a is zero. The transfer for that frequency then goes to infinity:

which implies that the circuit can create an output signal with a finite and non-zero magnitude at ${\omega }_0$ from nothing. An output signal for one non-zero finite frequency is a harmonic signal (a sine): we have a harmonic oscillator and the circuit consequently behaves as a sine generator. Note that this happens under two very specific conditions that both should be satisfied for the same ${\omega }_0$:

• the magnitude of $A(j{\omega }_0)\cdot {\beta }_1(j{\omega }_0)$ is unity and
• the argument of $A(j{\omega }_0)\cdot {\beta }_1(j{\omega }_0)$ must be $\pm \pi$ for only one ${\omega }_0$.

Harmonic oscillators from systems with positive feedback

Stable circuits rarely use positive feedback; for oscillators it can however be useful. Above, is was stated that for an infinite closed loop gain at a single frequency, that system can oscillate harmonically. Assuming the configuration with positive feedback shown in Figure 8.1b, it can readily be derived that the closed loop gain is

This closed loop gain goes to infinity for $A(j{\omega }_0){\beta }_2(j{\omega }_0)=1$, for which case harmonic oscillation is achieved at $\omega ={\omega }_0$. Compared to the case with feedback to the inverting input of the amplifier, this oscillation condition is different by a factor $-1$:

Harmonic oscillators: general formulation

Oscillation requires feedback. As shown above, both systems with feedback to the negative input of an amplifier and systems with feedback to the non-inverting input of an amplifier can be used for harmonic oscillators. In both cases, the total loop gain including any (inversion) sign must equal $1$ for one single (non-zero and finite) frequency, to get harmonic oscillation. This condition is usually referred to as the oscillation condition or Barkhausen’s stability criterion.

Harmonic oscillation requires a total loop gain equal to 1, meaning that the modulus of the loop gain is 1, and the loop phase is 0 $\pm k\cdot 2\pi$ rad. Under this condition, after passing the loop any integer number of times, an initially present sine is unchanged in magnitude and in phase. This implies that the sine is sustained indefinitely for loop gain=1.

Harmonic oscillation: total loop gain $\equiv$1, as a result

an initial sine remains unchanged after any N times looping through the system.

The Nyquist plot nicely shows (the degree of) stability for systems where the feedback is explicitly to the inverting input of the amplifier. The condition for stability is then that the –1-point is not net encircled in a clockwise direction by the Nyquist contour. This Nyquist contour starts at $\omega =0$, goes to $\omega \to \infty$ and thereafter goes via $\omega \to -\infty$ back to $\omega =-0$.

Oscillators must oscillate in a stable manner; according to feedback theory that does not corresponds to a stable circuit. Moreover, whereas the Nyquist contour and the Nyquist criterion for stability assumes feedback to the inverting input of an amplifier, for oscillators there is no need to assume negative or positive feedback: only the Barkhausen-criterion must be satisfied. This Barkhausen-criterion states that $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})=1$ for one specific frequency. For that reason, in this chapter, typically the loop gain $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ is plotted across all positive frequencies in a polar plot. This typically looks like a $180^{\circ }$-rotated version of (half) the Nyquist plot, where that $180^{\circ }$ is due to explicitly including the “-1” of an inverting amplifier input in the loop gain.

8.2 Harmonic oscillators and quality factor Q

To make the loop gain curve cross the “$1$” point in a polar plot, we need an amplifier and a suitable feedback circuit that combined yield a suitable $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$. Many variations on this are possible that each neatly oscillates at a specific frequency. Within this book, we categorize these oscillators based on the so-called quality factor $Q$. The quality factor $Q$ of a system is a measure for the loss of energy during one oscillation period. The definition of the quality factor $Q$ is

where the $E_{\mathit{𝑡𝑜𝑡𝑎𝑙}}$ term is the total energy within the oscillator and $P_{\mathit{𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑖𝑜𝑛}-\mathit{𝑝𝑒𝑟𝑖𝑜𝑑}}$ the dissipated power in one period; the $\omega$ is the oscillation (angular) frequency. This relation can be rewritten into an equation relating the total energy in the oscillator and the decrease in energy per oscillation period:

A higher value of $Q$ corresponds to less energy loss per oscillation cycle which usually translates into relatively power efficient oscillators. These are also (for the same reason) the preferred harmonic oscillators for generating sine waves at high frequencies. Alternatively, low-Q harmonic oscillators are relatively power hungry but typically have advantages in terms of tunability. Low-Q oscillators are covered in this chapter while high-Q oscillators are covered in chapter 9.

8.3 Harmonic oscillators with a low Q

Harmonic oscillators with a low quality factor dissipate a relatively large amount of energy per oscillation period. Because of this dissipation, the circuit must, by definition, contain dissipating components: resistors. A number of implementations of these harmonic oscillators, on a high level and assuming one amplifier with frequency independent gain $A$, is given in Figure 8.2.

Assuming the amplifier to be ideal (for simplicity reasons), with a frequency independent gain $A$, infinite input impedance and zero output impedance, then the feedback circuit must ensure there is only one non-zero finite frequency for which the loop gain is exactly 1. Hence, the $\beta$ circuit must have a frequency-dependent transfer function. For the three oscillator concepts in Figure 8.2:

• feedback to the $-$-input as shown in Figure 8.2a:

  for harmonic oscillation, the total loop gain is $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})=-A_1{\beta }_1(\mathit{𝑗𝜔})=1$ at only the oscillation frequency. Assuming a constant real and positive $A_1$ then at oscillation frequency ${\omega }_0$ the phase shift of the feedback network ${\beta }_1$ equals $180^{\circ }\pm N\cdot 360^{\circ }$. To accomplish this kind of phase shift at least a third order feedback circuit⁵⁰ should be used.

• feedback to the $+$-input as shown in Figure 8.2b:

  the total loop gain is $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})=+A_2{\beta }_2(\mathit{𝑗𝜔})$, which equals 1 for one frequency to achieve harmonic oscillation. The phase shift of ${\beta }_2$ for this frequency is $0^{\circ }\pm N\cdot 360^{\circ }$. To create a frequency-dependent circuit with a phase shift of $0^{\circ }$ at only one (angular) frequency, we need a second or higher-order circuit.

• feedback to both inputs as shown in Figure 8.2c:

  The total loop gain is now just somewhat harder to get due to having 2 feedback paths: it is $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})=(+{\beta }_p(\mathit{𝑗𝜔})-{\beta }_n(\mathit{𝑗𝜔}))\cdot A_3$. For harmonic oscillation this total loop transfer equals 1 for only the oscillation frequency. Again, this requires at least a second-order circuit. This at-least-second-order signal transfer may of course be distributed over ${\beta }_p$ and ${\beta }_n$.

All these principles, concerning harmonic oscillators with a low Q, can be found in many examples in literature or in any old stow away box with electronic junk. A few interesting harmonic oscillators with low Q will now be discussed.

8.3.1 Wien bridge oscillator

A well known implementation of a harmonic oscillator which uses the principle of Figure 8.2b — feedback to the non-inverting input node of an amplifier — is the so-called Wien bridge oscillator. The original circuit is given in Figure 8.3 and originates from bridge measurement equipment.

This oscillator circuit schematic may be viewed as an amplifier (with gain $A$) and a feedback network implemented by the resistors and capacitors. For oscillators, only the loop gain is important and splitting up this loop gain in forward and feedback gain is not required although it may be done intuitively.

In this specific case it may seem natural to split the loop gain $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ into an explicit feedback path formed by the circuitry in the gray rectangle and into an amplification part. Then the input of the $\beta$ network is connected to the output of the amplifier, while the output of the $\beta$ network is connected to the non-inverting input of the amplifier. The signal transfer of the hereby defined $\beta$-network in the Wien bridge oscillator in Figure 8.3 is then:

The oscillation condition states that harmonic oscillation occurs if $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})=\mathit{𝐴𝛽}(\mathit{𝑗𝜔})=1$ for one specific finite non-zero $\omega ={\omega }_{\mathit{𝑜𝑠𝑐}}$:

Equating the loop transfer to 1 yields two expression/conditions because the loop gain is complex (with a real part and a complex part that both yield one equation). The oscillation frequency ${\omega }_{\mathit{𝑜𝑠𝑐}}$ follows from equating the total loop transfer to be real (with a zero complex part). The required voltage amplifier gain $A$ follows from equating the loop gain magnitude to 1 at $\omega ={\omega }_{\mathit{𝑜𝑠𝑐}}$. For the Wien bridge oscillator:

Assuming a real valued amplifier gain $A$, the numerator of the loop gain for this specific circuit is real valued. To get a real valued loop gain, then also the denominator of the loop gain must be real valued. Being a complex number for this specific circuit, the upper condition can then be rewritten into

after which the $\mathrm{Re}<!--\; FUNCTION\; APPLICATION\; -->(...)=1$ condition — at $\omega ={\omega }_{\mathit{𝑜𝑠𝑐}}$ — can be solved, yielding

This Wien bridge oscillator circuit will hence oscillate harmonically at the (radian) frequency given in (8.3) with amplifier gain as given in (8.4).

The Bode plot of this loop gain $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ is shown in Figure 8.4, where ${\omega }_0=1∕\mathit{𝑅𝐶}$.

The corresponding polar plot of the loop gain $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ is shown below. In this polar plot, both the direction along the curve, for increasing $\omega$ and a few relevant (radian) frequencies are indicated.

8.3.2 Alternative Wien bridge oscillator configurations

Figure 8.6 shows a few alternative oscillator circuits using a variant of the “Wien bridge circuit”. All of these are combinations of a high-pass and a low-pass configuration to get zero phase shift at the oscillation frequency.

More variants can be constructed, especially if also inductors are included. Similar oscillators with feedback to both the inverting and non-inverting input of the amplifier can also be constructed. The approach to get any circuit to oscillate is however the same: derive the loop gain, equate that to unity for one single finite and non-zero frequency. Equating yields 2 equation, where setting the loop gain to a real value yields/sets the oscillation frequency and thereafter setting the magnitude of the loop gain at the oscillation frequency to unity sets the required gain factor for the amplifier.

8.3.3 Three-stage (low pass) phase-shift oscillator

Harmonic oscillators that use the inverting amplifier configuration shown in Figure 8.2a are usually so-called phase-shift oscillators. For these oscillators the total loop transfer $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ equals unity at one specific finite and non-zero oscillation frequency. Splitting the oscillator into a forward and feedback gain (again: not necessary as only the total loop gain is important!):

• the $\beta$ circuit must have a phase shift of $-180^{\circ }\pm N\cdot 360^{\circ }$ at oscillation. This requires at least a third order $\beta$-network when assuming a real voltage amplification factor.
• the gain $A$ at the oscillation frequency must be such that the magnitude of the total loop transfer equals unity, $A=1∕\beta (j{\omega }_0)$

Three stage low pass phase shift oscillator

The simplest form of an oscillator circuit conform the setup in Figure 8.2a is given in Figure 8.7. This circuit schematic includes an inverting amplifier and a frequency dependent part consisting of 3 first order low pass RC sections. For this frequency dependent part, the phase shift decreases monotonically with increasing frequency, from 0${}^{\circ }$ at DC to $-270^{\circ }$ at infinitely high frequencies. Using an inverting amplifier, the phase shift of the loop gain varies monotonically between 180${}^{\circ }$ at DC to $-90^{\circ }$ across frequencies and hence there is one frequency for which the phase shift is $0^{\circ }$.

The general transfer function of the circuit in Figure 8.7 cannot be easily derived, since all components influence each other. The calculations are less cumbersome if we assume the RC-members not to influence each other (which is not true in general!). A requirement for this assumption is $R_3>>R_2>>R_1$ and $C_3<<C_2<<C_1$. If we then choose $R_3{C}_3=R_2{C}_2=R_1{C}_1=\mathit{𝑅𝐶}$, then the loop gain is:

To operate as harmonic oscillator, this loop gain must equal (the complex number) 1 for only one frequency. Setting $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ to a real value yields the oscillation frequency. Noting that the numerator of $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ is real, in (8.6) this implies that the denominator at harmonic oscillation also must be real, which is the case for ${\omega }_{\mathit{𝑜𝑠𝑐}}$ as given by:

At $\omega ={\omega }_0$, $A_{\mathit{𝑙𝑜𝑜𝑝}}=A∕8$ from which it follows that for stable oscillation the amplifier must have gain $A=8$. Note that the amplifier is used in an inverting configuration.

The Bode plot of this loop gain is shown in Figure 8.8, where ${\omega }_0=1∕\mathit{𝑅𝐶}$.

The polar plot of the loop gain for this circuit is given in Figure 8.9. Note that this polar plot can (relatively) easily be constructed from the Bode plot of $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ by picking a few sets of $|A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})|$ and $\angle A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ and plotting these along with a smooth connecting curve in a polar plane.

8.3.4 Alternative implementation of the three stage low pass phase shift oscillator

The oscillator described in the previous section was easy to analyze under assumption that the RC-sections do not (significantly) load each other. That condition was satisfied for $R_3>>R_2>>R_1$ and $C_3<<C_2<<C_1$. One way to ensure that RC-sections actually do not load each other is using (ideal) voltage buffers between each RC-section:

Assuming $R_3{C}_3=R_2{C}_2=R_1{C}_1=\mathit{𝑅𝐶}$, the loop gain of this circuit is

The analysis or dimensioning to get harmonic oscillation can be executed the exact same way as for the previously analyzed 3-stage phase shift oscillator.: writing out $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ and equating that to (a complex) 1. Another way is to explicitly leverage the symmetry in the circuit.

The circuit schematic in Figure 8.10 consists of 3 identical parts, that individually consist of an amplifier and an RC-low pass section. For harmonic oscillation,

This translates into the following condition, for harmonic oscillation, for each individual section:

Using an arbitrary amplifier with a positive or negative real voltage gain, and noting that the phase shift of a single first order low pass section is $0\le \angle A_{\mathit{𝑠𝑒𝑐𝑡𝑖𝑜𝑛}}(\mathit{𝑗𝜔})\le -90^{\circ }$, possible solutions for (8.7) and (8.8) are

• assuming $A>0$:

  $$\begin{array}{rcll}& & \angle A_{\mathit{𝑠𝑒𝑐𝑡𝑖𝑜𝑛}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})=0^{\circ }& \\ & & \mathit{\angle 𝑅𝐶}=0^{\circ }\Rightarrow {\omega }_{\mathit{𝑜𝑠𝑐}}=0& \\ & & |A_{\mathit{𝑠𝑒𝑐𝑡𝑖𝑜𝑛}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})|=1& \end{array}$$

• for $A<0$:

  $$\begin{array}{rcll}& & \angle A_{\mathit{𝑠𝑒𝑐𝑡𝑖𝑜𝑛}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})=120^{\circ }& \\ & & \mathit{\angle 𝑅𝐶}=-60^{\circ }\Rightarrow {\omega }_{\mathit{𝑜𝑠𝑐}}=\frac{\sqrt{3}}{\mathit{𝑅𝐶}}& \\ & & |A_{\mathit{𝑠𝑒𝑐𝑡𝑖𝑜𝑛}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})|=1& \\ & & A=-2& \end{array}$$

Note that the “harmonic oscillator” with 3 low pass RC sections and positive amplifier gain “oscillates” at $\omega =0$: it yields a DC value. This is usually not regarded as harmonic oscillation. The variant with a negative voltage gain $A$ can oscillate at a finite non-zero frequency, for which the (per section) voltage gain is $A=-\sqrt{1+3}=-2$. Being essentially the same harmonic oscillator as in the previous section, the Bode graph and polar plot for the circuit in Fig. 8.10 are also given in Figure 8.8 respectively in Figure 8.9

8.3.5 Three stage phase shift oscillator with high pass sections

The 3-stage relaxation oscillators above were implemented with low pass RC-sections. Implementing them with high-pass sections yields the following circuit:

Assuming A$>$0
Assuming a real valued positive voltage gain $A$, and assuming that the RC-sections do not load each other, the loop gain is

Defining ${\omega }_0=1∕\mathit{𝑅𝐶}$, the Bode plot of this loop gain is shown in Figure 8.12. It can be seen that the phase condition for harmonic oscillation — $\angle A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})=N\cdot 360^{\circ }$ — can only be obtained for $\omega \to \infty$ which is not regarded as an oscillation frequency. Therefore this specific circuit does not implement a harmonic oscillator.

Assuming A$<$0
Assuming a real value negative amplifier gain $A$ and ignoring loading effects of the three RC section (for simplicity reasons), the corresponding Bode plot is shown in Figure 8.13. The phase condition for the loop gain yields at the oscillation frequency

For a single first order high pass RC section, the phase shift is between $+90^{\circ }$ and $0^{\circ }$. From this is can be concluded that the only possible solution for harmonic oscillation is for a phase shift per RC section equal to $+60^{\circ }$. This phase shift occurs for ${\omega }_{\mathit{𝑜𝑠𝑐}}=1∕\sqrt{3}\mathit{𝑅𝐶}$. To get $|A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})|=1$ the gain of the amplifier then needs to be $A=-8$.

The polar plot of the loop gain for this circuit is given in Figure 8.14.

8.3.6 More-than-three stage low pass phase shift oscillator (negative gain)

RC phase shift oscillators with more than 3 stages can be realized. In this section a number of designs are treated briefly. Let’s first assume that $N$ low pass sections are used that do not load each other significantly. This can be accomplished by e.g. including voltage buffers between RC-sections or by significantly scaling up the impedance of consecutive RC-sections. Assuming an amplifier with a real negative gain and leveraging the symmetry in the circuit it can readily be derived that:

• for $N=4$ the phase shift of the loop gain lies between $180^{\circ }$ and $180^{\circ }-4\cdot 90^{\circ }=-180^{\circ }$, where the $180^{\circ }$ originate from the negative real amplifier gain. Denoting the phase shift of the RC-section as $\angle \mathrm{\text{RC-section}}$, for this situation, oscillation occurs for

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=-45^{\circ }& \\ & & {\omega }_{\mathit{𝑜𝑠𝑐}}=1∕\mathit{𝑅𝐶}& \\ & & A=-{\left(\sqrt{1+\tan <!--\; FUNCTION\; APPLICATION\; -->{(45^{\circ })}^2}\right)}^4=-4& \end{array}$$

• for $N=5$ the phase shift of the loop gain lies between $180^{\circ }$ and $180^{\circ }-5\cdot 90^{\circ }=-270^{\circ }$, again with the $180^{\circ }$ due to the negative real amplifier gain. Then harmonic oscillation occurs for

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=-36^{\circ }& \\ & & {\omega }_{\mathit{𝑜𝑠𝑐}}=\mathit{𝑡𝑎𝑛}(36^{\circ })∕\mathit{𝑅𝐶}& \\ & & A=-{\left(\sqrt{1+\tan <!--\; FUNCTION\; APPLICATION\; -->{(36^{\circ })}^2}\right)}^5\approx -2.88& \end{array}$$

• for $N=6$, the loop phase can be $N\cdot 360^{\circ }$ for $\angle \mathrm{\text{RC-section}}=-30^{\circ }$ and for $\angle \mathrm{\text{RC-section}}=-90^{\circ }$. This latter phase shift corresponds to $\omega \to \infty$ which is not regarded as an oscillation frequency.

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=-30^{\circ }& \\ & & {\omega }_{\mathit{𝑜𝑠𝑐}}=\mathit{𝑡𝑎𝑛}(30^{\circ })∕\mathit{𝑅𝐶}& \\ & & A=-{\left(\sqrt{1+\tan <!--\; FUNCTION\; APPLICATION\; -->{(30^{\circ })}^2}\right)}^6\approx -2.37& \end{array}$$

• for $N=7$, the loop phase can equal $N\cdot 360^{\circ }$ for 2 finite and non-zero (radian) frequencies. For the two solutions it can easily be derived (using symmetry) that both the resulting (radian) frequency and the required loop gain for one solution is (much) higher than that for the other solution:

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=-{(180∕7)}^{\circ }& \\ & & {\omega }_{\mathit{𝑜𝑠𝑐}}\approx \mathit{𝑡𝑎𝑛}(25.7^{\circ })∕\mathit{𝑅𝐶}& \\ & & A\approx -{\left(\sqrt{1+\tan <!--\; FUNCTION\; APPLICATION\; -->{(25.7^{\circ })}^2}\right)}^7\approx -2.07& \end{array}$$

  and

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=-{(540∕7)}^{\circ }& \\ & & {\omega }_{\mathit{𝑜𝑠𝑐}}\approx \mathit{𝑡𝑎𝑛}(77.1^{\circ })∕\mathit{𝑅𝐶}& \\ & & A\approx -{\left(\sqrt{1+\tan <!--\; FUNCTION\; APPLICATION\; -->{(77.1^{\circ })}^2}\right)}^7\approx -36000& \end{array}$$

  Note that this means that if the circuit is designed to oscillate at ${\omega }_{\mathit{𝑜𝑠𝑐}}\approx \mathit{𝑡𝑎𝑛}(25.7^{\circ })∕\mathit{𝑅𝐶}$, the amplifier gain is set to sustain this oscillation. At the same time, this amplifier gain is (much) to low to sustain the second solution: this second solution will then not yield oscillations.

  The other way round, if the circuit is designed to sustain the second solution, for which ${\omega }_{\mathit{𝑜𝑠𝑐}}\approx \mathit{𝑡𝑎𝑛}(77.1^{\circ })∕\mathit{𝑅𝐶}$, the amplifier gain is such that the loop gain at the first solution is much larger than one. The magnitude of that oscillation mode will hence rapidly grow to (theoretically) infinity. This implies that there is no way to get stable harmonic oscillation for the second solution: the first solution will always be stronger.

• ...

Figure 8.15 shows the polar plots for phase shift oscillators with 3, 4, 5 and 6 low-pass sections. On the left the full polar plot is shown while a zoomed in version is shown on the right hand side. As all circuits are designed to be harmonic oscillators, all curves cross the $(1,0)$ point. Note that each $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ curve rotates $N\cdot -90^{\circ }$ for $0\le \omega \le \infty$.

8.3.7 More-than-three stage low pass phase shift oscillator (positive gain)

In the previous section, some RC phase shift oscillators with more than 3 stages having amplifiers with negative gain were treated briefly. This specific section shows some ins and outs for low-pass RC phase shift oscillators with positive gain amplifiers.

Let’s first assume that $N$ low pass sections are used that do not load each other significantly. This can be accomplished by e.g. including voltage buffers between RC-sections as shown in e.g. Figure 8.10 or by significantly scaling up the impedance of consecutive RC-sections as assumed in section 8.3.3. Assuming an amplifier with a real positive gain and leveraging the symmetry in the circuit it can readily be derived that:

• for $N=4$ the phase shift of the loop gain $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ lies between $0^{\circ }$ and $4\cdot 90^{\circ }=-360^{\circ }$. For this situation, oscillation may occur for the following two conditions. Note that both conditions do not implement a sensible harmonic oscillator as the oscillation (radian) frequency would be $\omega =0$ respectively $\omega \to \infty$.

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=0^{\circ }\mathrm{\text{ with }}A=1& \\ & & \angle \mathrm{\text{RC-section}}=-90^{\circ }\mathrm{\text{ with }}A\to \infty & \end{array}$$

• for $N=5$ the phase shift of the loop gain $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$ lies between $0^{\circ }$ and $5\cdot 90^{\circ }=-450^{\circ }$. Then harmonic oscillation may occur for the next two conditions.

  $$\begin{array}{rcll}& & \angle \mathrm{\text{RC-section}}=0^{\circ }\mathrm{\text{ with }}A=1& \\ & & \angle \mathrm{\text{RC-section}}=-72^{\circ }\mathrm{\text{ with }}A={\left(\sqrt{1+\tan <!--\; FUNCTION\; APPLICATION\; -->{(72^{\circ })}^2}\right)}^5\approx 355& \end{array}$$

  The first solution would correspond to oscillating at DC, which usually is not regarded as oscillation. The second solution yields a possible solution for harmonic oscillation at a finite, non-zero frequency.

  Trying to get the 5-stage solution to oscillate at ${\omega }_{\mathit{𝑜𝑠𝑐}}=\tan <!--\; FUNCTION\; APPLICATION\; -->(72^{\circ })∕\mathit{𝑅𝐶}$ will fail. This is because the DC “oscillation” requires a lower gain $A=1$ to achieve “oscillation”. Setting the $A\approx 355$ will result in a very rapidly diverging (to $\pm \infty$) output signal for the DC “oscillation” mode.

  Figure 8.16 shows polar plots of $A_{\mathit{𝑙𝑜𝑜𝑝}}(\mathit{𝑗𝜔})$, with $A=355$ and hence dimensioned to oscillate .at ${\omega }_{\mathit{𝑜𝑠𝑐}}=\tan <!--\; FUNCTION\; APPLICATION\; -->(72^{\circ })∕\mathit{𝑅𝐶}$ The right hand side figure is a zoomed version of the full polar plot on the left hand side. These plots illustrate that although the curve crosses $(1,0)$, a real positive $A_{\mathit{𝑙𝑜𝑜𝑝}}(0)=355$ exists that causes the previously mentioned divergent behavior.

• for $N>5$, essentially the same behavior as described at $N=5$ occurs: there are multiple frequencies where the oscillation condition can be satisfied, but always the $\omega =0$ solution will win. Consequently no harmonic oscillation can be achieved for low-pass RC phase shift circuits with positive voltage gain.

8.4 Where and how to derive the loop gain - part 1

To analyze a harmonic oscillator (and possibly to dimension that circuit to operate as you want it to operate), an equation for the loop gain of the circuit must be derived. For this, you have to open the loop somewhere, apply an input signal and get the “output signal” of that opened loop.

For circuits having an ideal voltage amplifier it may be natural to cut the loop directly in front or directly after the amplifier. From a fundamental point of view, opening the loop and injecting a sine and deriving a proper loop gain requires that:

• the load impedance “seen” by the node just before the cut is not changed. This can easily be done cutting just in front of an ideal voltage amplifier that has $r_{\mathit{𝑖𝑛}}\to \infty$.
• the source impedance “seen” by the node just after the cut is not changed. This can be accomplished cutting just after an ideal voltage amplifier that has $r_{\mathit{𝑜𝑢𝑡}}=0\mathrm{\Omega }$.

Cutting the loop somewhere in the frequency dependent part of the oscillator circuit schematic (in Figure 8.17 the part in the greyed region) would change impedance levels and therefore would not yield a valid loop gain derivation. For example cutting the loop just after $R_1$ would omit frequency dependency of the combination of $R_1$ and $C_1$ and would change impedance levels “seen” by the rest of the $\mathit{𝑅𝐶}$-network as compared to that in the closed loop configuration.

Noting that an ideal voltage amplifier is an ideal two port block consisting of an ideal voltage input port and a voltage controlled voltage source, a generalized guideline is that a loop can safely be opened (for loop gain derivation purposes) inside such an ideal controlled two port component. Then the following configuration is obtained, where the voltage controlled voltage source is driven by $v_{\mathit{𝑖𝑛}}$ and the input signal to this amplifier is $v_{\mathit{𝑖𝑛}}^{\prime }$, that are equal in the original closed loop configuration

8.5 Startup issues

The oscillation condition $A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})=1$ only gives information on the frequency of the oscillator and conditions for the required gain in the circuit. At $A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})=1$ a sine at $\omega ={\omega }_{\mathit{𝑜𝑠𝑐}}$ is sustained indefinitely: the amplitude does not grow, nor does the sine die out. Example signals somewhere in an arbitrary harmonic oscillator that oscillates at 10 kHz are shown in Figure 8.19. Both signals are perfectly fine signals at the same node. The two signals differ in signal amplitude by a factor 10.

The oscillation condition does not state anything on how the system will start to oscillate nor on the oscillation amplitude. It only states that any signal at the oscillation frequency remains unchanged forever at its initial amplitude. From here, a interesting question arises:

• How does the amplitude of the generated sine arrive to a target (desired) steady state?

Forcing or designing for a specific oscillation amplitude can be done leveraging the properties of stability (rephrases for harmonic oscillators:

• For $|A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}}|=1$ the oscillation amplitudes in a harmonic oscillator do not change
• For $|A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}}|>1$ the oscillation amplitudes in a harmonic oscillator increases (exponentially)
• For $|A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}}|<1$ the oscillation amplitudes in a harmonic oscillator decreases (exponentially)

Whereas the constant amplitudes are illustrated in Figure 8.19, the changing amplitudes are illustrated in Figure 8.20.

These three types of behavior are used to obtain oscillation at a specific (target) amplitude by ensuring $|A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}}|=1$ at the target amplitude and having a (slightly) higher or lower loop gain magnitude at respectively a lower and higher amplitude. Three examples of suitable (magnitude of) loopgain at the oscillation frequency are given in Figure 8.21. Note that this does assume a non-zero initial signal in the oscillator; this is addressed in section 8.8.

Gain-amplitude curves (1) and (2) in Figure 8.21 can be used both on the signal amplitude $V_{\mathit{𝑜𝑠𝑐}}$ and (a bit ugly) on the momentary signal value $v_{\mathit{𝑜𝑠𝑐}}(t)$. If a signal’s amplitude is used to control the loop gain, then a proper estimator for that amplitude is required. This can be e.g. using a peak detector (assuming a sinusoidal signal), using signal power, using a Fourier transform, ... This is worked out a little in §8.6. Working on a momentary signal $v_{\mathit{𝑜𝑠𝑐}}(t)$, all curves implement to some degree (soft or hard) clipping which necessarily distorts the signal(s). Some examples of clipping types of amplitude control are shown in §8.7

8.6 Amplitude control using the actual amplitude

An amplitude-dependent gain can be created in a number of ways. Obviously, for every implementation, some kind of amplitude estimator is required. That amplitude information must then be used to control the (amplifier’s) gain in the oscillator circuit. An example is given below, assuming a Wien bridge oscillator. Implementing the amplifier as a non-inverting opamp configuration as discussed in section 7.2.1, adjusting gain $A$ can be done by adapting resistor $R_3$ or $R_4$ in its feedback path. If enough power is dissipated in $R_3$ or $R_4$ to cause amplitude-dependent heating, these resistors may be implemented as Positive Temperature Coefficient (PTC) resistor or as Negative Temperature Coefficient (NTC) resistor respectively. The resistance of a PTC resistor increases with temperature. Implementing $R_3$ with a PTC resistor and designing such that a signal amplitude translates in heating the PTC effectively implements an amplitude stabilizing feedback. Implementing $R_4$ as an NTC resistor does the same.

There are many other ways to implement a tunable voltage gain, including making the bias current of the amplifier (or an amplifier stage) dependent on $V_{\mathit{𝑜𝑠𝑐}}$ and thereby implementing gains that result in loop gains similar to those in Figure 8.21, curves (1) or (2).

8.7 Amplitude control using clipping

Whereas amplitude control that actually uses amplitude information is the proper way, and yields the most clean signal, it does require a proper estimation of the amplitude and requires a proper way to tune a (linear) gain. A more crude way is to soft or hard limit (clip) a signal. Neglecting some other effects, clipping lowers and limits the harmonic content of the signal at ${\omega }_{\mathit{𝑜𝑠𝑐}}$ which effectively lowers the loop gain.

This is illustrated for the loopgain-amplitude curves in Figure 8.21.

• For very low signal amplitudes, $A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})>1$ for the curve marked with (3) in Figure 8.21, (replicated below) the oscillation signal amplitudes grows exponentially until the signal amplitude reaches $V_{\mathit{𝑡𝑎𝑟𝑔𝑒𝑡}}$ as defined in the same figure.

  With increasing amplitude, an increasing part of the signal now gets clipped. This results in a signals as shown in the time-domain figure below. Assuming no clipping, for a signal with amplitude larger than $V_{\mathit{𝑡𝑎𝑟𝑔𝑒𝑡}}$ the gray sinusoidal signal would result (indicated by “unclipped”). The actual, clipped, signal is shown in blue and indicated by “clipped”. The frequency component at ${\omega }_{\mathit{𝑜𝑠𝑐}}$ is shown in black and indicated by “$1^{\mathit{𝑠𝑡}}$ harmonic”.

  
  This latter signal has a smaller amplitude than the unclipped signal, corresponding to a lower (average across the whole signal) voltage gain, at the cost of distortion. In this book no in-depth treaty is given on this type of distortion, nor on its (small) impact on the actual oscillation frequency. You could do the math yourself using Fourier.

• For very low signal amplitudes, also $A_{\mathit{𝑙𝑜𝑜𝑝}}(j{\omega }_{\mathit{𝑜𝑠𝑐}})>1$ for Figure 8.21, curve (1), and the oscillation signal amplitudes grows (not exponentially in time) until the signal amplitude gets close to $V_{\mathit{𝑡𝑎𝑟𝑔𝑒𝑡}}$. Due to the quite smooth “clipping” the resulting signal gets compressed but does not show the nasty distortion as described above:

  

8.8 Initial signal at about the oscillation frequency

In a harmonic oscillator, a larger-than-unity loop gain takes care of increasing signal amplitudes. To actually get oscillation, a non-zero initial amplitude at (or close to) the oscillation frequency. This initial amplitude can be ensured in a number of ways:

• Every actual circuit, in operation or not, has an almost infinitely wide spectrum of frequencies, due to the thermal agitation of the matter from which the circuit is built. These vibrations cause fluctuations in electric behavior of the components, resulting in an infinitely wide spectrum of electric vibrations, generally called “thermal noise” or “white noise”. The spectral density of this white noise can be in the order of a few $\mathit{𝑛𝑉}∕\sqrt{\mathit{𝐻𝑧}}$ at room temperature. As this thermal noise spectrum ranges from very low frequencies to very high frequencies, including the oscillation frequency this thermal noise about the oscillation frequency may serve as starting amplitude from which an oscillator can start up.
• Besides the thermal noise level, also switching-on an oscillator may induce a voltage step somewhere in the oscillator. A steep voltage step or any unbalance in an oscillator may serve as initial value from which an oscillator can start. Note that with a step (or similar signal) somewhere in or around the oscillator induces also a signal at the oscillation frequency; which can be seen from e.g. a Fourier transform of a step.

8.9 Trivial extension to low-Q oscillators using transconductors

All harmonic oscillators discussed so far in this chapter were built around an ideal voltage amplifier: a block with a frequency-independent voltage gain $A$, an infinite input impedance and a zero output impedance. At higher frequencies, and almost always in integrated circuits, the natural active building block is however not a voltage amplifier but a transconductor, drawn as the $g_m$-block in the circuits below. A transconductor implements a voltage-controlled current source (VCCS) with as element equation

with — ideally — an infinite input and output impedance. Switching to this building block seems to require a complete redo of all the loop-gain derivations. It does not: the step from a voltage amplifier to a transconductor is nothing more than a Thévenin–Norton transformation, and all results of the previous sections carry over unchanged.

Thévenin–Norton equivalence. With the Thévenin–Norton equivalence, a voltage source that delivers a voltage $V$ with series impedance $Z$ is equivalent to a current source delivering a current $I=V∕Z$ with a parallel impedance $Z$. The easiest way is however to assume resistive-only impedances. This latter equivalence is depicted in Figure 8.23: a voltage source $A{v}_{\mathit{𝑖𝑛}}$ in series with an impedance $R$ is (seen from its output terminals) equivalent to a current source $A{v}_{\mathit{𝑖𝑛}}∕R$ in parallel with that same impedance $R$. This equivalence also holds with voltage-controlled sources, e.g. with a voltage amplifier and a transconductor.

Consequently, in oscillators that include a voltage amplifier with a series $R$, these two can conveniently be replaced by a transconductance with a parallel $R$.

A transconductor can implement a voltage amplifier with $R_{\mathit{𝑜𝑢𝑡}}$. As derived for common drain circuits and common collector configurations, a transconductor in (negative) feedback configuration behaves as a voltage source with an output resistance $r_{\mathit{𝑜𝑢𝑡}}=1∕g_m$. In schematic this yields the following equivalence, where (left) the transconductance is represented by its (trapezoidal) circuit symbol. To avoid confusion about the current direction, $i_{\mathit{𝑔𝑚}}$ is explicitly shown.

Consequences for the oscillators. With the above explained equivalency, we can simply replace an ideal voltage amplifier with series resistor by a transconductor that is either in a feedback configuration or that has a shunt resistor. The loop gain then keeps exactly the same form as before under the substitution $A_v+R\to g_m+\mathit{𝑓𝑒𝑒𝑑𝑏𝑎𝑐𝑘}$ respectively $A_v+R_{\mathit{𝑠𝑒𝑟𝑖𝑒𝑠}}\to g_m+R_{\mathit{𝑠h𝑢𝑛𝑡}}$. As a result the oscillation frequency — fixed by the frequency-dependent network and not by the amplifier — is essentially unchanged, and only the amplitude and start-up condition is reinterpreted. Note that with transconductors in feedback configuration, the equivalent voltage gain is positive.

As a first example, Figure 8.25 redraws the Wien bridge oscillator of Figure 8.3 with a transconductor. Because the transconductor delivers a current, the resistor $R_1$ is now placed as a shunt at the transconductor output node. Via the total feedback networks ($R_1$, $C_1$, $R_2$ and $C_2$) the frequency-selective feedback to the non-inverting input is implemented. The oscillation condition is obtained exactly as before — equate the loop gain to unity for one finite, non-zero frequency — now with $g_m{R}_1$ in place of $A$.

As next example, the distributed three-stage phase-shift oscillator in Figure 8.10 can be morphed into the transconductor version of Figure 8.26. Each transconductor now drives only a shunt capacitor. The transconductor with negative feedback implements the (positive) voltage gain with output resistance $R=1∕g_m$. The three identical $g_m$–$C$ stages behave exactly as the original three buffered RC-stages do. Note that an extra inverting amplifier is required for oscillation at a non-zero frequency⁵¹ .