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

Brahim Haraoubia

Chapter 1

Nonlinear Two-terminal Devices - all with Video Answers

Educators

Chapter Questions

01:53

Problem 1

The diode circuit shown in Figure E1.1 is the object of our study. As a first step, the diode is represented by its real model. The following are given: $I_S=10^{-12} \mathrm{~A}$ (reverse saturation current); (KT/e $=26 \mathrm{mV})$; "K", the Boltzmann constant, "T", the temperature and " e ", the elementary charge. The expectation is to have a current $\mathrm{I}_1=1 \mathrm{~mA}$ across the circuit from a voltage $\mathrm{v}_{\mathrm{e}}=2 \mathrm{~V}$.
(FIGURE CAN'T COPY)
1) Calculate the voltage across the diode when it is represented by its exponential model.
2) Determine the expression and value of $R$ that would impose the current $I_1$.
3) Calculate the dynamic resistance $R_d$ of the diode.
4) The diode is represented by a simplified model $\left(V_D=V_0=0.6 \mathrm{~V}, R_d=0\right.$ and $R_i$ is infinite), where $V_0$ is the threshold voltage, $R_d$ is the forward dynamic resistance and $R_i$ is the reverse resistance. Calculate the current $I_2$ that flows through the mesh (the value of resistance R calculated at point 2 will be used). Compare the currents $\mathrm{I}_2$ and $\mathrm{I}_1$. Draw the conclusions.
5) Using the diode model described at point $4, v_e=4 \sin (2 \pi f t+\varphi)$, represent $v_c, v_R$ and $\mathrm{v}_{\mathrm{S}}$.

Dominador Tan
Dominador Tan
Numerade Educator

Problem 2

The circuit is given in Figure E2.1.
1) Write the equation of the load line.
2) Plot the direct characteristic of the diode together with the load line and evidence the operating point in the following cases:
2.1) Real diode; 2.2) diode in second approximation $\left(V_0=0.6 \mathrm{~V} ; R_d=0\right.$ and $R_i$ is infinite).
3) The diode is considered ideal $\left(V_0=0 ; R_d=0\right.$ and $R_i$ is infinite). The studied circuit is shown in Figure E2.1. The voltage $\mathrm{v}_{\mathrm{e}}$ is sinusoidal $\left(\mathrm{v}_{\mathrm{e}}=\mathrm{V}_{\mathrm{m}} \cdot \sin (2 \pi \mathrm{ft})\right.$ ).
3.1) Find the expression of $v_s(t)$ and represent $v_s(t)$.
3.2) Represent $v_s=f\left(v_c\right)$.
4) Let us consider that the diode is represented by its model in Figure E2.2. Answer the same questions as those in question 3.
(FIGURE CAN'T COPY)
5) Let us now consider the circuit shown in Figure E2.3. The diode is assumed ideal.
5.1) Fill in Table E2.1.
(FIGURE CAN'T COPY)
5.2) Plot $\mathrm{v}_{\mathrm{s}}=\mathrm{f}\left(\mathrm{v}_{\mathrm{c}}\right)$.

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Problem 3

Let us consider the circuit shown in Figure E3.1. Diodes $\mathrm{D}_1$ and $\mathrm{D}_2$ are identical.
(FIGURE CAN'T COPY)
Voltages $\mathrm{v}_1$ and $\mathrm{v}_2$ are continuous. The following are given: $\mathrm{v}_1=12 \mathrm{~V}, \mathrm{v}_2=8 \mathrm{~V}$ and $\mathrm{V}_3=5 \mathrm{~V}$. The threshold voltage of the diodes is $\mathrm{V}_0=0.6 \mathrm{~V}$ and $\mathrm{R}_1=\mathrm{R}_2=\mathrm{R}_{\mathrm{L}}=10 \mathrm{k} \Omega$.
1) Determine the operating state of the two diodes and find the expression of the current through load resistance $\mathrm{R}_{\mathrm{L}}$ and its numerical value. Determine the expression of $\mathrm{v}_{\mathrm{s}}$ and its value.
2) Now, voltages $v_1$ and $v_2$ are sinusoidal. They have the same frequency and their initial phase is zero. Their amplitudes are different. $\mathrm{v}_1=12 \sin (2 \pi \mathrm{ft})$ and $\mathrm{v}_2=8 \sin (2 \pi \mathrm{ft})$ and $\mathrm{V}_3=5 \mathrm{~V}$. Find the expression of the output voltage and its representation as a function of time.

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03:17

Problem 4

Let us consider the circuit shown in Figure E4.1. The diode has the following characteristics: threshold voltage $\mathrm{V}_0=0.6 \mathrm{~V}$, forward dynamic resistance $\mathrm{R}_{\mathrm{d}}=0$ and infinite reverse resistance.
(FIGURE CAN'T COPY)
1) As a first step, let us have $v_1=1 \mathrm{~V}, v_2=4 \mathrm{~V}$ and $R_2=R_3=1 \mathrm{k} \Omega$. Find the expression of voltage $v_3$ as a function of the circuit elements and the value of the current across resistance $\mathrm{R}_3$.
2) The following are given: $v_1=3 \mathrm{~V}, v_2=3 \mathrm{~V}$ and $R_1=R_2=R_3=1 \mathrm{k} \Omega$. The same question formulated at point 1 should be answered.
3) Let us now set $R_2=5 \mathrm{k} \Omega$ and $R_1=R_3=1 \mathrm{k} \Omega$. Points $A$ and $B$ are joined and the periodic signal $s(t)$ is applied (Figure E4.2). Draw the graphic representation of signal $\mathrm{v}_3(\mathrm{t})$.
(FIGURE CAN'T COPY)

Hubert Agamasu
Hubert Agamasu
Numerade Educator

Problem 5

An ideal diode $\mathrm{D}\left(\mathrm{V}_0=0, \mathrm{R}_{\mathrm{d}}=0\right.$ and $\mathrm{R}_{\mathrm{i}}$ is infinite $)$ is inserted in the circuit shown in Figure E5.1.
(FIGURE CAN'T COPY)
1) Draw the representation of its characteristic $I_D=f\left(V_D\right)$.
2) Assuming that voltage $v=12 \mathrm{~V}$ and $R=1 \mathrm{k} \Omega$, find the expression and the value of the maximum current flowing through diode D .
3) Write the equation of the load line and represent it on the same diagram as the characteristic $I_D=f\left(V_D\right)$.
4) Let us now consider the diagram in Figure E5.2. The task is to fill in Table E5.1, knowing that diodes $\mathrm{D}_1$ and $\mathrm{D}_2$ are considered ideal (for the diodes: evidence
5) Signal $v_1$ is sinusoidal with peak amplitude $3 \mathrm{~V} ; v_2=2 \mathrm{~V}$. evolutions of $\mathrm{v}_{\mathrm{S} 1}$ and $\mathrm{v}_{\mathrm{S} 2}$ as functions of time.

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04:30

Problem 6

In the circuit shown in Figure E6.1, $D_1$ and $D_2$ are assumed ideal $\left(V_0=0, R_d=0\right.$ and $R_i$ is infinite) and $v_1=5 \sin (2 \pi f t)$.
5) Signal $v_1$ is sinusoidal with peak amplitude $3 \mathrm{~V} ; v_2=2 \mathrm{~V}$. Represent the evolutions of $v_{\mathrm{S} 1}$ and $v_{\mathrm{S} 2}$ as functions of time.
1) Find the expression of voltage across $R_1$.
2) Represent the evolution in time of the voltage across resistance $R_1$.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
02:29

Problem 7

Let us consider a forward-polarized PN silicon junction. The direct characteristic of this junction follows the equation: $I_D=I_{S 0}\left[e^{\frac{V_D}{V_{\mathrm{To}}}}-1\right] ; V_{T 0}=26 \mathrm{mV}$ at $T=300 \mathrm{~K}$. At $\mathrm{T}_1=350 \mathrm{~K}$, the following values are noted: $\mathrm{V}_{\mathrm{D}}=0.3 \mathrm{~V}, \mathrm{I}_{\mathrm{D}}=20 \mathrm{~mA}$.
1) Calculate $V_{T 1}$.
2) Calculate the reverse saturation current $I_{S 1}$. Compare $I_{S 1}$ and $I_D$ and draw the conclusions.
3) Knowing that $I_S=I_{S 0}\left[e^{a\left(T-T_0\right)}\right]$, where $a=0.06$ for the semiconductor employed, calculate the reverse saturation current $\mathrm{I}_{\mathrm{S} 0}$ for a temperature $\mathrm{T}_0=300 \mathrm{~K}$. Compare $\mathrm{I}_{\mathrm{S} 1}$ and $\mathrm{I}_{\mathrm{S} 0}$ -
4) Calculate the temperature variation $\Delta \mathrm{T}$ in ${ }^{\circ} \mathrm{C}$ so that the reverse saturation current $I_S$ doubles its value.

Chai Santi
Chai Santi
Numerade Educator

Problem 8

The circuit schematically represented in Figure E8.1 is given in which $R_1=R_2=$$100 \Omega$.
(FIGURE CAN'T COPY)
1) Represent on the diagram voltages $V_{R 1}, V_{R 2}$ and $V_D$ (across $R_1, R_2$ and $D$, respectively), as well as currents $I_1, I_2, I_D$ (across $R_1, R_2$ and $D$, respectively).
2) Find the value of $v_c$ starting from which $D$ is conducting.
3) Resistance $R_2$ is removed (Figure E8.2). Under these conditions, determine the equation of the load line, its graphic representation and the coordinates of the operating point for $\mathrm{v}_{\mathrm{e} 1}=1.2 \mathrm{~V}$.
(FIGURE CAN'T COPY)
4) $\mathrm{V}_{\mathrm{cl}}$ is a triangular voltage (Figure E8.3). D is considered ideal $\left(V_0=0, R_d=0\right.$ and $R_i$ is infinite) in this case. Represent one below the other the evolutions of $v_{\mathrm{el}}$, $\mathrm{v}_{\mathrm{R} 1}$ and $\mathrm{v}_{\mathrm{D}}$.
(FIGURE CAN'T COPY)

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03:17

Problem 9

Let us consider the circuit shown in Figure E9.1.
(FIGURE CAN'T COPY)
1) Initially, $v_e$ is a continuous voltage, the diode is considered ideal (threshold voltage $V_0$ is zero, forward dynamic resistance $R_d$ is zero and reverse resistance $R_i$ is infinite). Calculate the current I across the diode for the following cases: $\mathrm{V}_{\mathrm{e}}=2 \mathrm{~V}$, $v_e=4 \mathrm{~V}, v_e=-0.5 \mathrm{~V}, v_{\mathrm{e}}=-2 \mathrm{~V}$ and $v_e=-4 \mathrm{~V}$. Then, deduce the curve $\mathrm{I}=\mathrm{f}\left(\mathrm{v}_{\mathrm{e}}\right)$.
2) Now, the diode features a conduction threshold $V_0=0.6 \mathrm{~V}$, resistance $R_d=0$ and $R_i$ is infinite. Calculate the current $I$ across the diode for the cases: $v_e=4 \mathrm{~V}$, $\mathrm{v}_{\mathrm{e}}=2 \mathrm{~V}, \mathrm{v}_{\mathrm{e}}=1 \mathrm{~V}, \mathrm{v}_{\mathrm{e}}=-0.5 \mathrm{~V}, \mathrm{v}_{\mathrm{e}}=-2 \mathrm{~V}$ and $\mathrm{v}_{\mathrm{e}}=-4 \mathrm{~V}$. Then, deduce the curve $I=f\left(v_{\mathrm{e}}\right)$.
3) The input voltage considered in this case study is sinusoidal $\left(\mathrm{v}_{\mathrm{e}}=4 \sin 2 \pi \mathrm{Ft}\right)$. Draw the representations of the variations of voltage across $R_2$ in correspondence with the input voltage.

Hubert Agamasu
Hubert Agamasu
Numerade Educator

Problem 10

Given is the circuit shown in Figure E10.1.

$$
\begin{aligned}
& \mathrm{R}_1=100 \Omega \\
& \mathrm{R}_2=50 \Omega \\
& \mathrm{R}_{\mathrm{L}}=1000 \Omega \\
& \mathrm{V}=2 \mathrm{v}
\end{aligned}
$$

The diode is ideal.
(FIGURE CAN'T COPY)
1) Input voltage $\mathbf{v}_e$ has two values: $\mathbf{v}_{\mathbf{e}}=1$ and 4 V . Determine for each value of $\mathbf{v}_{\mathbf{e}}$ the equivalent diagram of the circuit shown in Figure E10.1, as well as the expression of the output voltage $\mathrm{v}_{\mathrm{s}}$.
2) Output voltage ranges between -10 and 10 V and varies in 1 V steps. Represent $v_s=f\left(v_c\right)$.

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Problem 11

Let us consider the circuit shown in Figure E11.1. Diode D is assumed ideal.
(FIGURE CAN'T COPY)
1) Find the equivalent Thévenin generator ( $R_{T H}$ and $V_{T H}$ ) between points $A$ and M .
2) Determine the expression of current $I$ as a function of $V, R_1, R_2$ and $R_3$. Numerical application: $\mathrm{V}=10 \mathrm{~V} ; \mathrm{R}_1=\mathrm{R}_2=20 \mathrm{k} \Omega ; \mathrm{R}_3=10 \mathrm{k} \Omega$.
3) Let us now consider the circuit shown in Figure E11.2. Diode D is assumed ideal. Find the Thévenin equivalent generators between $B$ and $M$ and then between N and M . Draw the equivalent diagram under these conditions.
(FIGURE CAN'T COPY)
4) Then, deduce the value of current $I_1$ and the state of diode $D$.

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02:51

Problem 12

The circuit to be studied is schematically shown in Figure E12.1. The signal in Figure E12.2 is applied at circuit input.
(FIGURE CAN'T COPY)
(FIGURE CAN'T COPY)
1) On the same diagram, draw the evolution of the output signal $v_s$ in correspondence with signal $\mathrm{v}_{\mathrm{e}}$ applied at the input assuming that the switch K is open.
2) The same question as 1 ), but now the switch $K$ is closed between $t_1$ and $t_2$.

Rajesh Singh
Rajesh Singh
Numerade Educator

Problem 13

The circuit to be studied is presented in Figure E13.1. The voltage applied at input is sinusoidal.
1) Find the expression of the output signal.
2) Represent the evolution in time of voltages $v_e$ and $v_s$.
3) Find the circuit function when $R_2=0$.
4) Let us now consider that $R_2=R_1=R$; represent in this case the evolution in time of voltages $\mathrm{v}_{\mathrm{e}}, \mathrm{v}_{\mathrm{A}}$ and $\mathrm{v}_{\mathrm{s}}$.
5) Represent $v_{\mathrm{s}}=f\left(v_{\mathrm{e}}\right)$ and deduce from it the function performed by this circuit.

$$
\mathrm{v}_{\mathrm{e}}=\mathrm{V}_{\mathrm{M}} \cdot \sin (2 \pi \mathrm{Ft})
$$

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01:43

Problem 14

Let us consider the circuit shown in Figure E14.1. The input voltage is sinusoidal: $\mathrm{v}_{\mathrm{e}}=\mathrm{V}_{\mathrm{M}} \sin (\omega \mathrm{t})$.
(FIGURE CAN'T COPY)

1) Find the definition expressions of voltages $v_A$ and $v_s$.
2) Represent one below the other the evolutions in time of signals $v_e, v_A$ and $v_s$.

Jilin Wang
Jilin Wang
Boston University