9 Low-Q harmonic oscillators

Q 9.1

The op-amps in the phase-shift oscillator below are all ideal: symmetrical, $A_0\to \infty$, $R_{\mathit{in}}\to \infty$ and $R_{\mathit{out}}\to 0\mathrm{\Omega }$. The circuit is designed to create a harmonic signal.

(a)

Derive an expression for the signal transfer of each of the 3 (identical) stages at the right hand side of the oscillator.

(b)

Derive an expression for the total (combined) signal transfer of the 3 (identical) stages.

(c)

Draw a Bode plot (magnitude and phase) of the transfer function of one of the 3 stages AND of the combined 3 stages. For these Bode plots you may use e.g. $R=1\mathrm{\Omega }$ and $C=1F$.

(d)

Node $z$ will be connected to either $x$ or $y$. The node that is NOT connected to $z$ will be connected to ground. Determine which of the nodes $x$ and $y$ of OA1 has to be the inverting one to obtain oscillation.

(e)

Derive an equation for the (radian) frequencies/frequency for which this transfer is purely real

(f)

Determine the value of $\beta$ such that the oscillation condition is satisfied.

Q 9.2

Consider the harmonic oscillator circuit shown below. The amplifier in this circuit schematic has a finite gain A.

(a)

Derive an expression for the transfer function of the feedback network, i.e. from the output $v_{\mathit{out}}$ to the differential input voltage $v_{+}-v_{-}$ of the amplifier.

(b)

For which $\omega$ is the derived signal transfer real?

(c)

Derive an expression for the amplifier gain A that is needed for harmonic oscillation.

Q 9.3

Consider the RC phase shift oscillator shown below, which consists of 5 identical sections with resistance $R$, capacitance $C$ and a negative voltage gain A.

(a)

The circuit schematic of this oscillator is symmetric; this property can be leveraged to simplify calculations. Determine the phase shift per stage, required for harmonic oscillations.

(b)

Derive an expression for the (radian) oscillation frequency.

(c)

Derive an expression for the amplifier gain A, needed for harmonic oscillation

Q 9.4

The circuit schematic below shows the circuit of an RC phase shift oscillator. In this circuit you can assume that the RC sections hardly influence each other because $C_1=10\cdot C_2=100\cdot C_3$ and $R_1=\frac{R_2}{10}=\frac{R_3}{100}$. You may assume that there is no influence whatsoever as simplification.

(a)

Derive an expression for the voltage transfer of the feedback network $\beta$ consisting of the three RC stages (that is from the output of the amplifier to the input of the amplifier).

(b)

Draw a Bode plot (magnitude and phase) of the feedback network.

(c)

Derive an expression for the oscillation frequency $f_{\mathit{osc}}$ or the radian oscillation frequency ${\omega }_{\mathit{osc}}$.

(d)

Derive an expression for the amplifier gain A needed for harmonic oscillation.

(e)

Draw a polar plot of A$\beta$ and indicate relevant angular frequencies $\omega =0$, $\omega \to \infty$, and $\omega ={\omega }_{\mathit{osc}}$.

Q 9.5

The circuit below is an harmonic oscillator, consisting of three identical stages. The transistors can be assumed to have infinite current gain $\mathit{\alpha fe}$ and infinite output resistance. Furthermore, you may use the numerical values $\frac{\mathit{kT}}{q}=25\mathit{mV}$ and $\frac{q}{\mathit{kT}}=40V^{-1}$ in your expressions.

(a)

Derive an equation for the small signal voltage gain of a single stage.

(b)

Show that the small-signal open-loop gain can be written as $A_{\mathrm{OL}}\approx -{\left(\frac{40V_{\mathit{CC}}-24}{1+\mathit{j\omega RC}}\right)}^3$.

(c)

Derive an expression for the oscillation frequency $f_{\mathit{osc}}$.

(d)

Derive an expression for the supply voltage $V_{\mathit{CC}}$ for stable oscillation.

(e)

Explain what happens:

i.

if $V_{\mathit{CC}}$ is lower than what is derived in c),

ii.

and if $V_{\mathit{CC}}$ is higher than what is derived in c).

(f)

If we build the same circuit with 4, 5 or 6 identical stages (BJT+R+C), which of these can also be harmonic oscillators (with $V_{\mathit{CC}}$ adjusted accordingly to enable potential harmonic oscillation) and which cannot? Explain/prove/illustrate your answer for each number of stages.

Q 9.6

See the harmonic oscillator circuit below. The circuit contains two amplifiers, one with voltage gain A and the other with voltage gain equal to 1.

(a)

Derive an expression for the oscillation (angular) frequency of this oscillator, expressed in e.g. A, $C_1$, $C_2$, $R_1$, and $R_2$.

(b)

Derive the value for the voltage gain A required for harmonic oscillation, expressed in e.g. $C_1$, $C_2$, $R_1$, and $R_2$.

Q 9.7

Given is the harmonic oscillator circuit below. The circuit contains two amplifiers, one with voltage gain A and the other with voltage gain equal to 1.

(a)

Derive an expression for the oscillation (angular) frequency of this oscillator, expressed in e.g. A, $C$, $L$, $R_1$, and $R_2$.

(b)

Derive the value for the voltage gain A required for harmonic oscillation, expressed in e.g. $C$, $L$, $R_1$, and $R_2$.

Q 9.8

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The opamps are ideal, with a real valued voltage gain $A\to \infty$ and with no frequency dependency.
• The voltage gain of the voltage buffer at the top of the schematic is a real valued factor $A_{\mathit{buf}}$; this value can be positive or negative.
• The value of all resistors is the same (R), the capacitors have identical values (C) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 3 stages at the bottom of the figure. Each stage starts at the output node of the previous opamp/amplifier and ends at the output node of the opamp in the stage.
Combined with the $A_{\mathit{buf}}$ this forms the loop gain.

(c)

Determine whether the given circuit can function as harmonic oscillator for a specific (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, go to question (d). Otherwise go to question (e).

(d)

IF harmonic oscillation possible at question (c): derive an equation for the oscillation frequency AND (after that) derive an equation for the required value of $A_{\mathit{buf}}$. Both these equations should be functions of the various component values in the schematic — they may be ugly equations. After finishing (d) skip (e).

(e)

IF no harmonic oscillation possible at question (c): swap a resistor and the reactive component in one of the stages in such a way that harmonic oscillation can be achieved, and then redo questions (c) and (d). IF there is no way to get the circuit to oscillate harmonically: prove that!

Q 9.9

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The opamps are ideal, with a real valued voltage gain $A\to \infty$ and with no frequency dependency.
• The voltage gain of the voltage buffer at the top of the schematic is a real valued factor $A_{\mathit{buf}}$; this value can be positive or negative.
• The value of all resistors is the same (R), the capacitors have identical values (C), and all inductors have identical values (L) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 3 stages at the bottom of the figure. Each stage starts at the output node of the previous opamp/amplifier and ends at the output node of the opamp in the stage.
Combined with the $A_{\mathit{buf}}$ this forms the loop gain.

(c)

Determine whether the given circuit can function as harmonic oscillator for a specific (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, go to question (d). Otherwise go to question (e).

(d)

IF harmonic oscillation possible at question (c): derive an equation for the oscillation frequency AND (after that) derive an equation for the required value of $A_{\mathit{buf}}$. Both these equations should be functions of the various component values in the schematic — they may be ugly equations. After finishing (d) skip (e).

(e)

IF no harmonic oscillation possible at question (c): swap a resistor and the reactive component in one of the stages in such a way that harmonic oscillation can be achieved, and then redo questions (c) and (d). IF there is no way to get the circuit to oscillate harmonically: prove that!

Q 9.10

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The voltage gain of the voltage buffer at the left hand side of the schematic is a real valued factor $A_v$; this value can be positive or negative. The voltage gain of the other voltage buffers is 1; for all voltage buffers $r_{\mathit{in}}\to \infty \mathrm{\Omega }$ and $r_{\mathit{out}}=0\mathrm{\Omega }$.
• The value of all resistors is the same (R), the capacitors have identical values (C), and all inductors have identical values (L) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 4 stages in the figure. Combined these form the loop gain.

(c)

Assume — only for this specific question — that $R=1\mathrm{\Omega }$, $L=1H$, $C=1F$ and $A_v=1$. Draw a Bode plot (magnitude and phase) of the loopgain.

(d)

Estimate from the Bode plot, and for the special case that $R=1\mathrm{\Omega }$, $L=1H$ and $C=1F$ whether the circuit can or cannot be used to implement an harmonic oscillator when $A_v$ can be chosen freely.

(e)

Determine whether the given circuit can function as harmonic oscillator for a specific (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, go to question (d). Otherwise go to question (e).

(f)

IF harmonic oscillation possible at question (c): derive an equation for the oscillation frequency AND (after that) derive an equation for the required value of $A_{\mathit{buf}}$. Both these equations should be functions of the various component values in the schematic — they may be ugly equations. After finishing (d) skip (e).

(g)

IF no harmonic oscillation possible at question (c): swap a resistor and the reactive component in one of the stages in such a way that harmonic oscillation can be achieved, and then redo questions (c) and (d). IF there is no way to get the circuit to oscillate harmonically: prove that!

Q 9.11

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The voltage gain of the voltage buffer at the left hand side of the schematic is a real valued factor $A_v$. The voltage gain of the other voltage buffers is 1; for all voltage buffers $r_{\mathit{in}}\to \infty \mathrm{\Omega }$ and $r_{\mathit{out}}=0\mathrm{\Omega }$.
• The value of all resistors is the same (R), the capacitors have identical values (C), and all inductors have identical values (L) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 4 stages in the figure. Combined these form the loop gain.

(c)

Determine whether the given circuit can function as harmonic oscillator for a specific (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, go to question (d). Otherwise go to question (e).

(d)

IF harmonic oscillation possible at question (c): derive an equation for the oscillation frequency AND (after that) derive an equation for the required value of $A_{\mathit{buf}}$. Both these equations should be functions of the various component values in the schematic — they may be ugly equations. After finishing (d) skip (e).

(e)

IF no harmonic oscillation possible at question (c): swap a resistor and the reactive component in one of the stages in such a way that harmonic oscillation can be achieved, and then redo questions (c) and (d). IF there is no way to get the circuit to oscillate harmonically: prove that!

Q 9.12

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The voltage gain of the voltage buffer at the left hand side of the schematic is a real valued factor $A_v$. The voltage gain of the other voltage buffers is 1; for all voltage buffers $r_{\mathit{in}}\to \infty \mathrm{\Omega }$ and $r_{\mathit{out}}=0\mathrm{\Omega }$.
• The value of all resistors is the same (R) and the capacitors have identical values (C) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 4 stages in the figure. Combined these form the loop gain $A_{\mathit{loop}}$.

(c)

Draw a Bode plot (magnitude and phase) of the loop gain; for this question you may assume $R=1\mathrm{\Omega }$, $C=1F$ and $A_v=1$ when making this plot.

(d)

Determine whether the given circuit can function as harmonic oscillator for a specific positive (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$.

(e)

Determine whether the given circuit can function as harmonic oscillator for a specific negative (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$.

Q 9.13

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The voltage gain of the voltage buffer at the left hand side of the schematic is a real valued factor $A_v$ in questions (a)-(d); this value can be positive or negative. The voltage gain of the other voltage buffers is 1; for all voltage buffers $r_{\mathit{in}}\to \infty \mathrm{\Omega }$ and $r_{\mathit{out}}=0\mathrm{\Omega }$.
• The value of all resistors is the same (R), the capacitors have identical values (C), and all inductors have identical values (L) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 4 stages in the figure. Combined these form the loop gain.

(c)

Draw a Bode plot (magnitude and phase) of the loop gain; for this question you may assume $R=1\mathrm{\Omega }$, $C=1F$ and $A_v=1$ when making this plot.

(d)

Determine whether the given circuit can function as harmonic oscillator for a specific positive (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$.

(e)

Determine whether the given circuit can function as harmonic oscillator for a specific negative (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$.

Q 9.14

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

The opamps are ideal, with a real valued voltage gain $A\to \infty$ and with no frequency dependency.

The value of all resistors is the same ($R$), except for the rightmost resistor (that one has value $R_1$). Whenever applicable, all capacitors have identical values ($C$), and all inductors have identical values ($L$).

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

The circuit can be decomposed into 3 stages. Each stage starts at the output node of the previous opamp and ends at the output node of the opamp in the stage.
Derive the voltage transfer for each of the 3 stages.

(c)

What range of phase shift can be achieved by the three stages, for $0\le \omega \le \infty$? Note that your answer consists of three phase ranges, one per stage.

(d)

Determine whether the given circuit can function as harmonic oscillator. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $R_1$. Otherwise demonstrate clearly (mathematically) that this circuit cannot oscillate harmonically.

A similar but different circuit schematic is shown below. Also this circuit is intended to be an harmonic oscillator.

(e)

Also this circuit can be decomposed into 3 stages (and an extra voltage gain stage having a real valued voltage gain $A_{\mathit{buf}}$). Each stage starts at the output node of the previous opamp and ends at the output node of the opamp in the stage.
Derive the voltage transfer for each of the 3 stages.

(f)

What range of phase shift can be achieved by the three stages, for $0\le \omega \le \infty$? Note that your answer consists of three phase ranges, one per stage.

(g)

Determine whether the given circuit can function as harmonic oscillator, assuming a real valued $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$. Otherwise demonstrate clearly (mathematically) that this circuit cannot oscillate harmonically.

Q 9.15

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The opamps are ideal, with a real valued voltage gain $A\to \infty$ and with no frequency dependency.
• The value of all resistors is the same ($R$). All capacitors have identical values ($C$).

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

The circuit can be decomposed into 3 stages that have frequency dependent transfer functions (identified by their gray background in the figure), and one ideal voltage buffer stage with a real valued voltage gain $A_{\mathit{buf}}$.

(b)

Show (with a derivation) that the voltage transfer for both the first stage and for the second stage is $H(\mathit{j\omega })=\frac{1-\mathit{j\omega RC}}{1+\mathit{j\omega RC}}$.

(c)

Derive the phase shift of each of the three stages for $\omega =0$ and for $\omega \to \infty$(rad/s). Note that your answer consists of six phases, two per stage.

(d)

Draw in one Bode plot (magnitude and phase) the voltage transfer of each of the three stages. You may assume (for this plot) $R=1\mathrm{\Omega }$ and $C=1F$.

(e)

Determine whether the given circuit can function as harmonic oscillator using a positive $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$. Otherwise demonstrate clearly (mathematically) that this circuit cannot oscillate harmonically.

Explicit hint: using the findings in the previous 2 questions or using standard forms for transfer functions simplifies significantly.

(f)

Determine whether the given circuit can function as harmonic oscillator using a negative $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$. Otherwise demonstrate clearly (mathematically) that this circuit cannot oscillate harmonically.

Explicit hint, the same one: using the findings in the previous 2 questions or using standard forms for transfer functions simplifies significantly.

Q 9.16

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The opamp is ideal, with a real valued voltage gain $A\to \infty$ and with no frequency dependency.
• The voltage buffer has a finite voltage gain $A_v$.
• The value of all resistors is the same ($R$); all capacitors have identical values ($C$).

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Determine whether the given circuit can function as harmonic oscillator using a positive, real valued $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$. Otherwise demonstrate clearly (mathematically) that this circuit cannot oscillate harmonically.

(c)

Determine whether the given circuit can function as harmonic oscillator using a negative $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$. Otherwise demonstrate clearly (mathematically) that this circuit cannot oscillate harmonically.

(d)

In the previous question, a real valued $A_{\mathit{buf}}$ was assumed. For (one of) the case(s) that the circuit oscillates harmonically, show whether the oscillation frequency increases or decreases is the buffer has a (small) negative phase shift (due to finite bandwidth). Showing this can be done by a mathematical treaty, graphics constructions, ... but needs to exceed guessing.

Q 9.17

(a)

Explain why $A\cdot \beta (j{\omega }_{\mathit{osc}})=1$ needs to be satisfied in order for a circuit using positive feedback to function as a harmonic oscillator. You may derive and make use of the closed-loop transfer of the following closed-loop system in your explanation:

(b)

Given is the following second-order Sallen-Key low-pass filter:

Show that the transfer function of this filter is $H(\mathit{j\omega })=\frac{1}{1+2\mathit{j\omega RC}+j^2{\omega }^2{R}^2{C}^2}$ for $A\to \infty$.

(c)

Given is the following second-order filter:

Show that the transfer function of this filter is $H(\mathit{j\omega })=\mathit{j\omega }\frac{L}{R}\cdot \frac{(1-\mathit{j\omega RC})}{1+\mathit{j\omega }\frac{L}{R}}$.

(d)

Now we want to see if it is possible to make a harmonic oscillator if we connect the low-pass filter from above in series with a voltage buffer and a couple of other opamp circuits in a closed-loop, see the circuit diagram below. The opamps are ideal, with a real valued $A\to \infty$. The voltage gain of the voltage buffer at the top of the schematic is a real valued negative factor $A_{\mathit{buf}}$.

Note that the value of all resistors is the same (R), that the capacitors have identical values (C), and that all inductors have identical values (L).

Derive the transfer function for every stage.

(e)

Draw the corresponding Bode plots (both magnitude and phase plots). You may assume that the corner frequencies ${\omega }_c$ are the same for all stages.

(f)

Calculate the total loop gain and draw the Bode plot (magnitude and phase) of this total loop gain (assume $A_{\mathit{buf}}=1$ and assume the same corner frequencies ${\omega }_c$).

(g)

Can the circuit operate as a harmonic oscillator for $A_{\mathit{buf}}>0$? If so, at what frequency and for what $A_{\mathit{buf}}$?

(h)

Can the circuit operate as a harmonic oscillator for $A_{\mathit{buf}}<0$? If so, at what frequency and for what $A_{\mathit{buf}}$?

Q 9.18

Given is the circuit below. This circuit is intended to be a harmonic oscillator. The voltage gain of the voltage buffer at the top of the schematic is a positive real valued factor $A_{\mathit{buf}}$. The value of all resistors is the same (R), and that the capacitors have identical values (C).

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Which one of the following Nyquist plots corresponds to the open loop transfer function $A_{\mathit{loop}}(\mathit{j\omega })$ of the circuit above? You may assume $A_{\mathit{buf}}>0$ here.

(c)

Can this circuit oscillate harmonically for a $A_{\mathit{buf}}>0$? If so, at what frequency?

(d)

Can this circuit oscillate harmonically for a $A_{\mathit{buf}}<0$? If so, at what frequency?