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Nonlinear Electronics 1: Nonlinear Dipoles, Harmonic Oscillators and Switching Circuits

Brahim Haraoubia

Chapter 4

Oscillator as a Nonlinear Device - all with Video Answers

Educators

Chapter Questions

Problem 1

The task is to realize an oscillator based on the feedback circuit schematically shown in Figure E1.1.
(FIGURE CAN'T COPY)
1) Find the expression of the transfer function $\left(\boldsymbol{\beta}=\mathbf{v}_2 / \mathbf{v}_1\right)$ of this circuit.
2) The active element used in the amplification has a real gain. It is built around an operational amplifier. Given these conditions, determine the overall electric diagram of the oscillator.
3) Find the oscillation frequency and the condition for sustained oscillation of this oscillator.

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Problem 2

Let us have an amplifier A of gain G and input resistance $R_c: R_c \longrightarrow \infty$. This amplifier is associated with a feedback loop formed of a passive quadripole $B$ that introduces attenuation $\beta$ (Figure E2.1).
(FIGURE CAN'T COPY)
1) Find the expression of ratio $\left(v_s / v_1\right)$.
2) What condition should be met for the voltage $v_s$ to maintain a finite value when $\mathrm{v}_1 \rightarrow 0$ ?
3) What is to be said about this system when the previous condition is verified?

Let us consider the passive quadripole in Figure E2.2.
(FIGURE CAN'T COPY)
4) Find the expression that defines the transfer function of this circuit. This quadripole occupies the position of the passive circuit designated by B in Figure E1.1 in order to realize an oscillator. Draw the diagram of this oscillator.
5) Find the oscillation frequency and the condition for sustained oscillation of this oscillator.

Numerical application: $\mathrm{R}_1=\mathrm{R}_2=2.2 \mathrm{k} \Omega ; \mathrm{C}_1=\mathrm{C}_2=0.22 \mu \mathrm{F}$.

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Problem 3

Given the circuit in Figure E3.1, the task is to study it and make it operate as an oscillator.
(FIGURE CAN'T COPY)
1) Separate the amplification circuit and the feedback circuit, which sets the oscillation frequency, and draw the diagram of each of these circuits.
2) Find the expression of the transfer function of each of these two circuits: feedback circuit, denoted by $\beta$, and amplification circuit, denoted by $A$.
3) What form does the expression of $\beta$ take if: $R_1=R_2=R$ and $C_1=C_2=C$ ?
4) Is it possible to apply the Barkhausen condition directly? Find the oscillation frequency of the output signal and the condition for sustaining these oscillations if $\mathrm{R}_1=\mathrm{R}_2=\mathrm{R}$ and $\mathrm{C}_1=\mathrm{C}_2=\mathrm{C}$.
5) Given these conditions, plot the voltages $v_{\mathrm{s}}$ and $v_{\mathrm{r}}$ on the same diagram. What conclusions can be drawn?

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Problem 4

An oscillator (Figure E4.1) is formed of an amplifier and a feedback circuit. The Bode plot of the feedback circuit is shown in Figure E4.2.
1) Assuming that the amplifier has a positive real gain, determine the oscillation frequency and the amplifier gain required to sustain this oscillation.
2) Considering now that the amplifier used has a negative real gain, answer the same questions formulated at point 1 .

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Problem 5

The operational amplifier circuit shown in Figure 5.1(a) is the object of our study.
1) Find the expressions of ratios $\left(v_1 / v_e\right)$ and $\left(v_2 / v_1\right)$. What is the overall voltage gain of the circuit shown in Figure E5.1(a)?
2) What is the phase shift introduced between $v_1$ and $v_e$ and then between $v_2$ and $v_e$ ?
3) Calculate the input resistance of the circuit shown in Figure E5.1(a).
4) The task is to realize an oscillator by connecting the circuits shown in Figures E5.1(a) and E5.1(b), as indicated. Find the oscillation frequency and the condition that $R_1$ and $R_2$ should satisfy for sustained oscillation.
(FIGURE CAN'T COPY)

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04:18

Problem 6

Let us consider the circuit shown in Figure E6.1. The operational amplifier is assumed ideal.
(FIGURE CAN'T COPY)
1) Prove that the impedance seen between the points $A$ and $M$ is negative.
2) The diagram (Figure E6.1) is transformed in order to realize the oscillator schematically presented in Figure E6.2. Find the oscillation frequency and the condition that R should satisfy in order to sustain this oscillation. Draw a simplified diagram of this oscillator.
3) Draw the curve of the output voltage when $R=\left(R_0 / 2\right)$.

Kajal Gautam
Kajal Gautam
Numerade Educator

Problem 7

Let us consider the passive circuit shown in Figure E7.1.
1) Calculate the ratio $\left(v_s / v_e\right)$
2) It is our intention to use this circuit to build an oscillator. Suggest an oscillator circuit whose feedback is the passive device shown in Figure E7.1.
3) Find the oscillation frequency and the condition for sustained oscillation of the suggested oscillator.
(FIGURE CAN'T COPY)
Figure E7.1.

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Problem 8

EXERCISE 8
The circuit to be studied is schematically shown in Figure E8.1.
1) Calculate the ratio $\left(v_s / v_e\right)$ putting it in the following form:

$$
\frac{v_{\mathrm{s}}}{v_{\mathrm{e}}}=\frac{k}{\left[1+\mathrm{j} \frac{\omega}{\omega_0}(x-k)-\left[\frac{\omega}{\omega_0}\right]^2\right]}
$$

2) Find the expressions of $x, k$ and $\omega_0$.
3) Draw the variations of the curve $A_{d B}=20 \cdot \log _{10}\left|v_s / v_e\right|$ and then deduce the condition that this circuit should meet in order to become an oscillator.
4) Let us now consider the passive circuit shown in Figure E8.2. Write its transfer function.
5) The passive circuit shown in Figure E8.2 is associated with an amplifier in order to obtain the circuit schematically shown in Figure E8.3. Find the relation between $R_1$ and $R_2$ that allows this device (Figure E8.3) to operate as an oscillator. Determine its oscillation frequency.

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04:25

Problem 9

Let us consider the two circuits schematically shown in Figures E9.1 and E9.2.
1) Express the ratios $\frac{v_{\mathrm{el}}}{v_{\mathrm{s} 1}}$ and $\frac{v_{\mathrm{e} 2}}{v_{\mathrm{s} 2}}$
2) Since these two circuits are oscillators, calculate the oscillation frequency and find the condition for sustained oscillation for each of these two circuits.
3) What conclusion can be drawn concerning this type of oscillators?
(FIGURE CAN'T COPY)
Figure E9.1.
(FIGURE CAN'T COPY)
Figure E9.2.

Kajal Gautam
Kajal Gautam
Numerade Educator

Problem 10

The task is to study the circuits represented in Figures E10.1, E10.2 and E10.3.
1) Calculate the ratio $\left(v_s / v_e\right)$ for the circuit shown in Figure E10.1.
2) Determine the gain for the circuits shown in Figures E10.2 and E10.3.
3) Determine a diagram of an oscillator that uses the device shown in Figure E10.1 as the feedback circuit and the appropriate circuit chosen among those represented in Figures E10.2 and E10.3 as the amplification element.
4) Calculate the oscillation frequency and find the condition for sustained oscillation of the chosen device.
(FIGURE CAN'T COPY)
Figure E10.1.
(FIGURE CAN'T COPY)
Figure E10.2.
(FIGURE CAN'T COPY)
Figure E10.3.

Numerical application:

$$
\mathrm{C}_1=2 \mathrm{nF} ; \mathrm{C}_2=100 \mathrm{nF} ; \mathrm{C}_3=200 \mathrm{pF} ; \mathrm{L}=100 \mu \mathrm{H} \text { and } \mathrm{R}_2=100 \mathrm{k} \Omega \text {. }
$$

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Problem 11

Let us consider the device shown in Figure E11.1. The blocks $B_1$ and $B_2$ are two passive quadripoles, the transfer functions of which are the quantities $\beta_1$ and $\beta_2$, respectively. $A_1$ is an ideal operational amplifier. Its open-loop gain is equal to $A_0$, the value of which is very high and therefore can be considered infinite.
(FIGURE CAN'T COPY)
Figure E11.1.
1) Establish the relationship among $A_0, \beta_1$ and $\beta_2$.

The quadripoles $B_1$ and $B_2$ are replaced by their equivalent circuit, as indicated in Figure E11.2.
(FIGURE CAN'T COPY)
Figure E11.2.
2) Find the expressions that define $\beta_1$ and $\beta_2$.
3) Considering that the circuit shown in Figure E11.3 replaces the two feedback networks $B_1$ and $B_2$, establish the diagram of the oscillator circuit obtained and determine the oscillation frequency and the condition for sustained oscillation.
4) Let us now consider that the bridge represented by the circuit schematically shown in Figure E11.4 takes the place of the bridge formed by $B_1$ and $B_2$. Determine the diagram of the obtained circuit as well as the oscillation frequency of this circuit and the condition for sustained oscillation.

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04:57

Problem 12

Let us consider the passive circuit schematically shown in Figure E12.1.
(FIGURE CAN'T COPY)
Figure E12.1.
1) Find the expression of the ratio $v_s / v_e$.
2) Represent the variations of the ratio $\left(\mathrm{v}_{\mathrm{s}} / \mathrm{v}_{\mathrm{e}}\right)$ as a function of module and phase. What conclusions can be drawn?
3) Let us now study the low-frequency oscillator circuit of the circuit schematically shown in Figure E12.2.

Determine the oscillation frequency and the condition for sustained oscillation of this device.
(FIGURE CAN'T COPY)

Narayan Hari
Narayan Hari
Numerade Educator

Problem 13

Let us consider the network shown in Figure E13.1.

Figure E13.1.
1) Using Kirchhoff's second law, find the parameters $Z_{i j}$ of the quadripole $\mathbf{Q}_p$ as a function of k and Z ( k is an imaginary number).

This device serves for building an oscillator as schematically shown in Figure E13.2. The active quadripole $\mathbf{Q}_{\mathrm{A}}$ is represented by its parameters $\mathrm{Y}_{\mathrm{ij}}$, which are real numbers (with $\mathrm{Y}_{11}=0$ and $\mathrm{Y}_{12}=0$ ).
(FIGURE CAN'T COPY)
Figure E13.2.

The operating equation of the oscillator is given by:

$$
\mathrm{Z}_{21}\left(\mathrm{Y}_{21}-\mathrm{Y}_{12}\right)+\mathrm{Y}_{11} \mathrm{Z}_{22}+\mathrm{Y}_{22} \mathrm{Z}_{11}+\Delta \mathrm{Y} \cdot \Delta \mathrm{Z}+1=0
$$

with $\Delta \mathrm{Z}=\mathrm{Z}_{11} \cdot \mathrm{Z}_{22}-\mathrm{Z}_{12} \cdot \mathrm{Z}_{21}$ and $\Delta \mathrm{Y}=\mathrm{Y}_{11} \cdot \mathrm{Y}_{22}-\mathrm{Y}_{12} \cdot \mathrm{Y}_{21}$
2) Let us consider $Z=R ; k Z=-j a Z(a \neq 0)$ is the impedance related to a capacitor C. Under these conditions, find the expression of k and a .
3) Write the impedances $Z_{i j}$ as a function of "a" and of R. Write the operating equation of the oscillator as a function of " a ", R and $\mathrm{Y}_{\mathrm{ij}}$.
4) Assuming that $Y_{i j}$ are real parameters, find the oscillation frequency of this circuit. Determine its expression when $\mathrm{Y}_{22} \cdot \mathrm{R} \gg 1$.

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Problem 14

Let us consider the oscillator shown in Figure E14.1. The capacitor $\mathrm{C}_0$ is considered a short circuit at working frequency. Resistance $R_E$ is very high compared to $\mathrm{h}_{11}$.
(FIGURE CAN'T COPY)
Figure E14.1.

The operating equation of the oscillator is:

$$
-Y_{21}\left(h_{21}-h_{12}\right)+Y_{11}+h_{22}+\Delta h Y_{22}+\Delta \mathrm{Yh}_{11}=0
$$

The active element is represented by parameters $\mathrm{h}_{\mathrm{ij}}$, and the passive element is represented by parameters $\mathrm{Y}_{\mathrm{ij}}$. Let us assume that $\mathrm{h}_{12} \cong 0$ and $\mathrm{h}_{22} \cong 0$.
1) What is the type of oscillator studied and what is the role of self-inductance $\mathrm{L}_1$ ? What is the role of capacitor $\mathrm{C}_0$ ?
2) What is the configuration in which the transistor is connected (common emitter, common base or common collector)? Justify your answer.
3) Find the dynamic equivalent diagram of this circuit evidencing the amplifier chain and the feedback loop. Let us assume that $\mathrm{h}_{12}=0$ and $\mathrm{h}_{22}=0$.
4) Calculate the parameters $Y_{i j}$ of the passive quadripole.
5) Write the operating equation of this oscillator and determine the oscillation frequency.

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Problem 15

Let us consider the diagram in Figure E15.1, which is the equivalent diagram of a quartz crystal.
(FIGURE CAN'T COPY)
Figure E15.1.
1) Find the admittance of the quartz crystal.
2) Find the condition that allows us to write this admittance in the following form:

$$
Y=j 2 \pi F\left(C_1+C_2\right) \frac{1-\left(F / f_p\right)^2}{1-\left(F / f_S\right)^2}
$$

3) Find the expression of frequency $f_p$ as a function of frequency $f_s$ as well as the values of $f_s$ and $f_p$. Given: $R=10 \Omega, C_1=1 \mathrm{fF}, C_2=5 \mathrm{pF}$ and $L=250 \mathrm{mH}$.
4) The quartz crystal operates at a frequency $F_0$, with $f_s<F_0<f_p$. Given these conditions, find the expression of the reactance presented by the quartz crystal, indicate its sign and deduce its nature.
5) The quartz crystal is used in oscillation; its quality factor (series resonance) is given by $\mathrm{Q}=\left(1 / 2 \pi \mathrm{RC}_1 \mathrm{~F}_0\right)$. What can be deduced from this, given that a good LC circuit has a quality factor $\mathrm{Q}_{\mathrm{LC}}=100$ ?

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Problem 16

Let us consider the diagram of the sinusoidal oscillator in Figure E16.1. The transistor quadripole is represented by its hybrid parameters, and the passive feedback quadripole is represented by its impedance parameters.
(FIGURE CAN'T COPY)
Without introducing the influence of $\mathrm{R}_{\mathrm{C}}$ and $\mathrm{R}_{\mathrm{B}}$, the operating equation of the oscillator is given by $Z_{21}\left(h_{21}-h_{12}\right)+Z_{22}+h_{11}+\Delta h Z_{11}+\Delta Z h_{22}=0$.
1) Find the equivalent diagram of this circuit. Explain the role of capacitor $\mathrm{C}_{\mathrm{c}}$. $\left(h_{12}=0\right.$ and $\mathrm{h}_{22}=0$ for the transistor).
2) An equivalent diagram of this oscillator is shown in Figure E16.2. Identify the elements of Figure E16.2 with the elements of the equivalent diagram you have proposed.
3) As indicated, the transistor is represented by its hybrid parameters and the passive quadripole by its Z parameters.
3.1) Find the $Z_{i j}$ parameters of the passive quadripole as a function of $Z_1, Z_2$ and Z , then of $\mathrm{C}_1, \mathrm{C}_2, \mathrm{~L}$ and $\omega$.
3.2) Write the operating equation in the presence of $R_B$ and $R_c$ -
3.3) If the influence of $R_B$ and of $R_C$ is now neglected (the values of resistances $\mathrm{R}_{\mathrm{C}}$ and $\mathrm{R}_{\mathrm{B}}$ are chosen sufficiently high), then find the oscillation frequency under these conditions.
(FIGURE CAN'T COPY)
Figure E16.2.

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Problem 17

Let us consider the circuit shown in Figure E17.1, related to the equivalent diagram of a quartz crystal:
(FIGURE CAN'T COPY)
Figure E17.1.
1) Calculate the impedance of this circuit and show that it can be written in the following form:

$$
Z=\frac{1+j C_{\mathrm{s}} \omega(\mathrm{r}+j L \omega)}{j\left(C_s+C_p\right) \omega-C_s C_p \omega^2(r+j L \omega)} \quad \text { with } \quad C_p \gg C_s
$$

2) If resistance $r$ is neglected $(r=0)$, then find the simplified expression of the impedance of quartz crystal as well as its value for a continuous signal.
3) Prove that the simplified expression can be written in the following form:

$$
\mathrm{Z}=\frac{1}{\mathrm{jC}_{\mathrm{eq}} \omega}\left[\frac{1-\left[\frac{\omega}{\omega_1}\right]^2}{1-\left[\frac{\omega}{\omega_2}\right]^2}\right]
$$

4) Find the expressions of $C_{e q}, \omega_1$ and $\omega_2$. Prove that $\omega_1<\omega_2$. What is the significance of $\omega_1$ and $\omega_2$ ?
5) Study the variations in terms of module and phase of the impedance Z as a function of $\omega$ and indicate the nature of $Z$ in the various ranges separated by $\omega_1$ and $\omega_2$.

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Problem 18

Let us consider the circuit shown in Figure E18.1, which is intended to be used in an oscillator circuit, where " $m$ " is a positive constant.
1) Calculate the ratio ( $\mathrm{v}_2 / \mathrm{v}_1$ ) using Kirchhoff's second law.
2) Plot the evolution $\left(\mathrm{v}_2 / \mathrm{v}_1\right)$ in terms of module and phase as a function of frequency. Indicate the sign of $\left(\mathrm{v}_2 / \mathrm{v}_1\right)$ when its imaginary part is zero.
3) This circuit is used in the realization of an oscillator that uses an operational amplifier or a bipolar transistor as amplifier element. Draw the diagram of this oscillator for each type of active component. Find the oscillation frequency and the gain of the amplifier that has been proposed for the two types of active components.
(FIGURE CAN'T COPY)
Figure E18.1.

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04:18

Problem 19

Let us consider the circuit shown in Figure E19.1 and the block diagram shown in Figure E19.2. The operational amplifier is ideal and operates under linear conditions.
1) Identify the amplifier and feedback blocks by drawing them separately, while indicating the position of voltages schematically shown in Figure E19.2 in the diagram shown in Figure E19.1.
2) Calculate the ratios $\left(v_2 / v_1\right)$ and $\left(v_3 / v_2\right)$ and find the relationship between $R_1$ and $R_2$ so that this circuit is an oscillator.
3) Under these conditions, find the oscillation frequency.
(FIGURE CAN'T COPY)
Figure E19.1.
(FIGURE CAN'T COPY)
Figure E19.2.

Kajal Gautam
Kajal Gautam
Numerade Educator