The conductivity of an intrinsic semiconductor depends on temperatures as $\mathrm{\sigma }={\mathrm{\sigma }}_0{e}^{-∆E/2kT}$, where ${\mathrm{\sigma }}_0$ is a constant. Find the temperature at which the conductivity of an intrinsic germanium semiconductor will be double of its value at $T=300 \mathrm{K}$. Assume that the gap for germanium is $0.650 \mathrm{eV}$ and remains constant as the temperature is increased.

1. Energy Bands in Solids:

(i) Valence band: The energy band formed by a series of energy levels containing valence electrons is known as valence band.

(ii) Conduction band: The higher energy level band is called the conduction band.

(iii) Conduction band: The higher energy level band is called the conduction band.

(iv) Forbidden energy gap $\left(\mathrm{\Delta }{E}_{\mathrm{g}}\right)$: Energy gap between the conduction band and valence band $\mathrm{\Delta }{E}_{\mathrm{g}}={\left(CB\right)}_{\min }-{\left(VB\right)}_{\max }$

Note: No free electron is present in the forbidden energy gap.

(v) Fermi energy level: The highest energy level which can be occupied by an electron in valence band at $0 \mathrm{K}$ is called fermi level.

2. Semiconductors:

(i) Holes in Semiconductors: When an electron is removed from a covalent bond, it leaves a vacancy behind. An electron from a neighbouring atom can move into this vacancy, leaving the neighbour with a vacancy. In this way, the vacancy formed is called hole and can travel through the material and serves as an additional current carrier. Hole acts as a virtual charge, although there is no physical charge on it. The mobility of hole is less than the electron.

(ii) Conductivity of semiconductor: $\sigma =e\left[n_{\mathrm{e}}{\mu }_{\mathrm{e}}+n_{\mathrm{h}}{\mu }_{\mathrm{h}}\right]$ where, ${\mu }_{\mathrm{e}}=\frac{v_{\mathrm{e}}}{E}=$ mobility of electron and ${\mu }_{\mathrm{h}}=\frac{v_{\mathrm{h}}}{E}=$ mobility of holes

3. Types of Semiconductor:

4. P-N Junction Diode:
When a $P$-type semiconductor is suitably joined to an N-type semiconductor, the resulting arrangement is called P-N junction or P-N junction diode.

(i) Depletion region: On account of the difference in concentration of charge carriers in the two sections of P-N junction, the electrons from N-region diffuse through the junction into P-region and the hole from P region diffuse into N region.

→ Width of the depletion layer $\propto \frac{1}{\mathrm{Doping}}$

(ii) Potential barrier: The potential difference created across the $P-N$ junction due to the diffusion of electrons and holes is called as potential barrier.

→ For $\mathrm{Ge}$, $V_B=0.3 \mathrm{V}$ and for silicon, $V_B=0.7 \mathrm{V}$

5. Biasing of a P-N junction diode:

6. Reverse Breakdown:
If the reverse-biased is too high, then a breakdown of P-N junction diode occurs. It occurs in two ways,

(i) Zener breakdown: Breaking of covalent bonds due to a strong electric field in the reverse direction.

(ii) Avalanche breakdown: Breaking of covalent bonds due to collision between high-speed electron and valence electron.

Both Zener and avalanche breakdown occur simultaneously.

7. Zener Diode:
It is a heavily doped diode that works as a voltage regulator in reverse bias. It allows different values of current at reverse breakdown voltage. Zener voltage depends on the material of the semiconductor and doping concentration.

8. Zener Diode as Voltage Regulator:

$I=\frac{V_{\mathrm{in}}-V_{\mathrm{z}}}{R_{\mathrm{s}}}; I_{\mathrm{L}}=\frac{V_{\mathrm{out}}}{R_{\mathrm{L}}}; I_{\mathrm{L}}=I-I_2$

→ If $V_{\mathrm{in}}\uparrow$ by keeping $R_{\mathrm{L}}=$ constant, then I↑ and I2↑ but $V_{\mathrm{out}}=$ constant

→ If $R_{\mathrm{L}}\uparrow$ by keeping $V_{\mathrm{in}}=$ constant, then I= constant, $I_{\mathrm{L}}\downarrow$ and $I_{\mathrm{z}}\uparrow$ but $V_{\mathrm{out}}=$ constant

9. Half wave rectifier:

When the P-N junction diode rectifies half of the $AC$ wave, it is called a half-wave rectifier.

(i) Average output in one cycle: $I_{\mathrm{avg}}=\frac{I_0}{\pi }$ and $V_{\mathrm{avg}}=\frac{V_0}{\pi }; I_0=\frac{V_0}{r_{\mathrm{f}}+R_{\mathrm{L}}}$ ($r_{\mathrm{f}}=$ forward-biased resistance)

(ii) RMS output: $I_{\mathrm{rms}}=\frac{I_0}{2}, V=\frac{V_0}{2}$

(iii) The ratio of the effective alternating component of the output voltage or current to the $DC$ component is known as the ripple factor.

$r=\frac{I_{\mathrm{AC}}}{I_{\mathrm{DC}}}={\left[\left(\frac{I_{\mathrm{rms}}}{I_{\mathrm{DC}}}\right)2-1\right]}^{\frac{1}{2}}=1.21$

(iv) Peak inverse voltage $\left(PIV\right)$: The maximum reverse-biased voltage that can be applied before the commencement of Zener region is called the $PIV.$ When a diode is not conducting, $PIV$ across it is V0.

(v) Efficiency: It is given by $\% \eta =\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times 100=\frac{40.6}{1+\frac{ r_{\mathrm{f}}}{R_{\mathrm{L}}}}$

(vi) Form factor $=\frac{ I_{\max }}{I_{\mathrm{DC}}}=\frac{\pi }{2}=1.57$

(vii) The ripple frequency $\left(\omega \right)$ for half-wave rectifier is same as that of $AC$.

10. Full Wave Rectifier:

It rectifies both halves of $AC$ input signal.

(i) Average output: $V_{\mathrm{avg}}=\frac{2V_0}{\pi }, I_{\mathrm{avg}}=\frac{2I_0}{\pi }$

(ii) RMS output: $V_{\mathrm{rms}}=\frac{V_0}{\sqrt{2}}, I_{\mathrm{rms}}=\frac{I_0}{\sqrt{2}}$

(iii) Ripple factor: $r=0.48=48\%$

(iv) Ripple frequency: This ripple frequency of full-wave rectifier =2×(frequency of input $AC$)

(v) Peak inverse voltage $\left(PIV\right)$: It’s value is 2V0.

(vi) Efficiency: $\% \eta =\frac{81.2}{1+\frac{r_{\mathrm{f}}}{R_{\mathrm{L}}}}$ for $r_{\mathrm{f}}\ll R_{\mathrm{L}}, \eta =81.2\%$

11. Transistor:

Junction transistors are of two types:

(i) NPN transistor: It is formed by sandwiching a thin layer of P-type semiconductor between two N- type semiconductors.

(ii) PNP transistor: It is formed by sandwiching a thin layer of N-type semiconductor between two P- type semiconductors.

Different modes of operation of a transistor:

→ Relation between currents in a transistor is $i_{\mathrm{e}}=i_{\mathrm{b}}+i_{\mathrm{c}}; i_{\mathrm{b}}=5\%i_{\mathrm{e}}$ and $i_{\mathrm{c}}=95\%i_{\mathrm{e}}$.

12. Transistor as CB amplifier:

(i) $AC$ current gain: ${\alpha }_{\mathrm{AC}}=\frac{\mathrm{Small} \mathrm{change} \mathrm{on} \mathrm{collector} \mathrm{current} \left(∆i_{\mathrm{c}}\right)}{\mathrm{Small} \mathrm{change} \mathrm{in} \mathrm{collecor} \mathrm{current} \left(∆i_{\mathrm{e}}\right)}, V_{\mathrm{B}}=$ constant

(ii) $DC$ current gain: ${\alpha }_{\mathrm{DC}}=\frac{\mathrm{Collector} \mathrm{current} \left(i_{\mathrm{c}}\right)}{\mathrm{Emitter} \mathrm{current} \left(i_{\mathrm{e}}\right)}$. Value of ${\alpha }_{\mathrm{DC}}$ lies between $0.95$ to $0.99$

(iii) Voltage gain: $A_{\mathrm{v}}=\frac{\mathrm{Change} \mathrm{in} \mathrm{output} \mathrm{voltage} \left(∆P_{\mathrm{o}}\right)}{\mathrm{Change} \mathrm{in} \mathrm{input} \mathrm{power} \left(∆P_{\mathrm{i}}\right)}$ $\Rightarrow A_{\mathrm{v}}=a_{\mathrm{AC}}\times$Resistance gain

(iv) Power gain $=\frac{\mathrm{Change} \mathrm{in} \mathrm{output} \mathrm{power}\mathrm{ }\left(∆P_{\mathrm{o}}\right)}{\mathrm{Change} \mathrm{in} \mathrm{input} \mathrm{power} \left(∆P_{\mathrm{c}}\right)}$ $\Rightarrow \mathrm{Power} \mathrm{gain}={\alpha }_{\mathrm{AC}}^2\times \mathrm{Resistance} \mathrm{gain}$

13. Transistor as CE Amplifier:

(i) AC current gain: ${\beta }_{\mathrm{AC}}=\left(\frac{\Delta i_{\mathrm{c}}}{\Delta i_{\mathrm{b}}}\right), V_{\mathrm{CE}}=$ constant

(ii) DC current gain: ${\alpha }_{\mathrm{DC}}=\frac{i_{\mathrm{c}}}{i_{\mathrm{b}}}$

(iii) Voltage gain: $A_{\mathrm{v}}=\frac{\mathrm{\Delta }{V}_0}{\mathrm{\Delta }{V}_{\mathrm{i}}}={\beta }_{\mathrm{AC}}\times$ Resistance gain

(iv) Power gain $=\frac{\mathrm{\Delta }{P}_0}{\mathrm{\Delta }{P}_{\mathrm{i}}}={\beta }_{\mathrm{AC}}^2\times$ Resistance gain

→Relation between α and β: α=α1-α or α=β1+β

14. Logic Gate and Truth Table:
(i) NOT gate:

(ii) Truth tables for OR, AND, NOR and NAND gates:

15. Basic Boolean postulates and laws:

(i) Boolean Postulates: $0+A=A, 1·A=A, 1+A=1 0·A=0$ and $A+\overline{A}=1$

(ii) Identity law: A+A=A, and $A·A=A$

(iii) Negation law: $\overline{\left(\overline{A}\right)}=A$

(iv) Commutative law: A+B=B+A, and $A·B=B·A$

(v) Associative law: $\left(A+B\right)+C=A+\left(B+C\right)$, and $\left(A·B\right)·C=A·\left(B·C\right)$

(vi) Distributive law: $A·\left(B+C\right)=A·B+A·C$, and $\left(A+B\right)·\left(A+C\right)=A+BC$

(vii) Absorption laws: $A+A·B=A, A·\left(A+B\right)=A·\overline{A}·\left(A+B\right)=\overline{A}·B$

(viii) Boolean identities: $A+\overline{A}B=A+B, A\left(\overline{A}+B\right)=AB$,

$A+BC=\left(A+B\right)\left(A+C\right), \left(\overline{A}+B\right)·\left(A+C\right)=\overline{A}C+AB$

(ix) De Morgan’s theorem: It states that the complement of the whole sum is equal to the product of individual complements and vice versa i.e. $\overline{A+B}=\overline{A}·\overline{B}$ and $\overline{A·B}=\overline{A}+\overline{B}$.

16. Basic components of a communication system:

Faithful transmission of information from one place to another place is called communication.

17. Transmitter:

A transmitter converts the message signal produced by the information source into a form (e.g. electrical signal) that is suitable for transmission through the channel to the receiver.

18. Communication channel:

A communication channel is a medium (transmission line, an optical fibre or free space etc) that connects a receiver and a transmitter. It carries the modulated wave from the transmitter to the receiver.

19. Receiver:

It receives and decodes the signal into its original form.

20. Important terms used in communication:

Transducer: A transducer is a device that converts one form of energy into another. For example microphone, photodetectors and piezoelectric sensors are types of the transducer.

Signal: Signal is the information converted in electrical form. Signals can be analog or digital. Sound and picture signals in TV are analog.

It is defined as a single-valued function of time that has a unique value at every instant of time.

(i) Analog Signal: A continuously varying signal (Voltage or Current) is called an analog signal. A decimal number with a system base 10 is used to deal with analog signals.

(ii) Digital Signal: A signal that can have only discrete stepwise values is called a digital signal. A binary number system with base 2 is used to deal with digital signals.

(iii) Noise: There are unwanted signals that tend to disturb the transmission and processing of message signals. The source of noise can be inside or outside the system.

(iv) Attenuation: It is the loss of strength of a signal while propagating through a medium. It is like the damping of oscillations.

(v) Amplification: It is the process of increasing the amplitude (and therefore the strength) of a signal using an electronic circuit called the amplifier. Amplification is absolutely necessary to compensate for the attenuation of the signal in communication systems.

(vi) Range: It is the largest distance between the source and the destination upto which the signal is received with sufficient strength.

(vii) Repeater: A repeater acts as a receiver and a transmitter. A repeater picks up the signal which is coming from the transmitter, amplifies and retransmits it with a change in carrier frequency. Repeaters are necessary to extend the range of a communication system as shown in figure A communication satellite is basically a repeater station in space.

21. Bandwidth:

Bandwidth of signals: Different signals used in a communication system such as voice, music, picture, computer data etc. all have different ranges of frequency. The difference of maximum and minimum frequency in the range of each signal is called the bandwidth of that signal.

Bandwidth can be of message signal as well as of transmission medium.

(i) Bandwidth for analog signals: Bandwidth for some analog signals are listed below:

(ii) Bandwidth for digital signal: Basically digital signals are rectangular waves and these can be splitted into a superposition of sinusoidal waves of frequencies $v_0, 2v_0, 3v_0, 4v_0, n{v}_0,$ where n is an integer extending to infinity. This implies that the infinite bandwidth is required to reproduce the rectangular waves. However, for practical purposes, higher harmonics are neglected for limiting the bandwidth.

22. Bandwidth of Transmission Medium:

Different types of transmission media offer different band width of which some are listed below,

23. Ground Wave Propagation:

(i) The radio waves which travel through the atmosphere following the surface of the earth are known as ground waves or surface waves and their propagation is called ground wave propagation or surface wave propagation. These waves are vertically polarized in order to prevent short-circuiting of the electric component. The electric field due to the wave-induced charges on the earth's surface. As the wave travels, the induced charges in the earth also travel along with it. This constitutes a current on the earth's surface. As the ground wave passes over the surface of the earth, it is weakened as a result of energy absorbed by the earth. Due to these losses, the ground waves are not suited for very long-range communication. Further, these losses are higher for high frequency. Hence, ground wave propagation can be sustained only at low frequencies $\left(500 \mathrm{kHz} \mathrm{to}1500 \mathrm{kHz}\right)$.

(ii) The ground wave transmission becomes weaker with the increase in frequency because more absorption of ground waves takes place at a higher frequency during propagation through the atmosphere.

(iii) The ground wave propagation is suitable for low and medium frequency i.e. upto $2 \mathrm{MHz}$ only.

(iv) The ground wave propagation is generally used for local band broadcasting and is commonly called a medium wave.

(v) The maximum range of ground or surface wave propagation depends on two factors:

(vi) The frequency of the radio waves and (ii) Power of the transmitter.

24. Sky Wave Propagation:

(i) The sky waves are the radio waves of frequency between $2 \mathrm{MHz}$ to $30 \mathrm{MHz}$.

(ii) The ionospheric layer acts as a reflector for a certain range of frequencies ($3 \mathrm{MHz}$ to $30 \mathrm{MHz}$). Therefore it is also called ionospheric propagation or short wave propagation. Electromagnetic waves of frequencies higher than $30\mathrm{ }\mathrm{MHz}$ penetrate the ionosphere and escape.

(iii) The highest frequency of radio waves which when sent straight (i.e. normally) towards the layer of ionosphere gets reflected from the ionosphere and returns to the earth is called critical frequency. It is given by N is the number density of electron per ${\mathrm{m}}^3$.

25. Space wave propagation:

(i) The space waves are the radio waves of very high frequency (i.e. between $30 \mathrm{MHz}$ to $300 \mathrm{MHz}$ or more.

(ii) The space waves can travel through the atmosphere from transmitter antenna to receiver antenna either directly or after reflection from the ground in the earth's troposphere region. That is why space wave propagation is also called tropospheric propagation or line of sight propagation.

(iii) The range of communication of space wave propagation can be increased by increasing the heights of transmitting and receiving antennas.
(iv) Height of transmitting Antenna:

The transmitted waves, travelling in a straight line, directly reach the received end and are then picked up by the receiving antenna as shown in the figure. Due to the finite curvature of the earth, such waves cannot be seen beyond the tangent points S and T.

${\left(R+h\right)}^2=R^2+d^2$

As $R\gg h,$So $h^2+2Rh=d^2$

⇒d=2Rh

Area covered for TV transmission: A=πd2=2πRh

Population covered = population density×area covered.

If the height of the receiving antenna is also given in the question, then the maximum line of sight is,

$d_{\mathrm{M}}=\sqrt{2R{h}_{\mathrm{T}}}+\sqrt{2R{h}_{\mathrm{R}}}$

Where,

R= the radius of the earth (approximately $6400 \mathrm{km}$)

$h_{\mathrm{T}}=$ height of transmitting antenna

$h_{\mathrm{R}}=$ height of receiving antenna

26. Modulation:

The phenomenon of superposition of information signal over a high-frequency carrier wave is called modulation. In this process, the amplitude, frequency or phase angle of a high-frequency carrier wave is modified in accordance with the instantaneous value of the low-frequency information.

27. Need for Modulation:

(i) To avoid interference: If many modulating signals travel directly through the same transmission channel, they will interfere with each other and result in distortion.

(ii) To design antennas of practical size: The minimum height of the antenna (not of the antenna tower) should be $\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, where λ is the wavelength of modulating signals. This minimum size becomes impractical because the frequency of the modulating signal can be upto $5 \mathrm{kHz}$ which corresponds to a wavelength of $\raisebox{1ex}{$\left(3\times {10}^8\right)$}\!\left/ \!\raisebox{-1ex}{$\left(5\times {10}^3\right)$}\right.=60 \mathrm{km}$. This will require an antenna of the minimum height of $\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$4$}\right.=15 \mathrm{km}$. This size of an antenna is not practical.

(iii) Effective Power Radiated by an Antenna: A theoretical study of radiation from a linear antenna (length l shows that the power radiated is proportional to (frequency)² i.e. ${\left(\raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\right)}^2$. For a good transmission, we need high powers and hence this also points out the need for using the high-frequency transmission.

Amplitude Modulation:

Modulation factor, $\mathrm{m}=\frac{\mathrm{amplitude} \mathrm{of} \mathrm{modulating} \mathrm{wave}}{\mathrm{amplitude} \mathrm{of} \mathrm{normal} \mathrm{carrier} \mathrm{wave}}$

If $V_{\mathrm{m}}=V_{\mathrm{m}}\cos \left({\omega }_{\mathrm{m}}t\right)$ and $V_{\mathrm{c}}=V_{\mathrm{c}}\cos \left({\omega }_{\mathrm{c}}t\right)$ then $m=\frac{V_{\mathrm{m}}}{V_{\mathrm{c}}}$

As the amplitude of the carrier wave varies at a signal frequency $f_{\mathrm{m}}$ so the amplitude of $\mathrm{AM}$ wave $=V_{\mathrm{c}}+m{V}_{\mathrm{c}}cos\left({\omega }_{\mathrm{m}}t\right)$ & frequency of $AM$ wave $=\frac{{\omega }_{\mathrm{c}}}{2\pi }$

Therefore $V=\left[V_{\mathrm{c}}\left(1+m\right)\cos \left({\omega }_{\mathrm{m}}t\right)\right]\cos \left({\omega }_{\mathrm{c}}t\right)$

$\Rightarrow V=V_{\mathrm{c}}\cos \left({\omega }_{\mathrm{c}}t\right)+\frac{m{V}_{\mathrm{c}}}{2}\cos \left[\left({\omega }_{\mathrm{c}}+{\omega }_{\mathrm{m}}\right)t\right]+\frac{m{V}_{\mathrm{c}}}{2}\cos \left[\left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)t\right]$

28. Frequency spectrum of AM wave:

29. Power in AM wave:

Power of carrier wave $:P_{\mathrm{c}}=\frac{V_{\mathrm{c}}^2}{2R}$ where R= resistance of antenna in which power is dissipated.

Total power of sidebands: $P_{\mathrm{sideband} }=2\times \frac{1}{2R}{\left(\frac{m{V}_{\mathrm{c}}}{2}\right)}^2=\frac{m^2}{2}{P}_{\mathrm{c}}$

Total power of AM wave $=P_{\mathrm{c}}\left(1+\frac{m^2}{2}\right)$

Fraction of total power carried by sidebands =m22+m2

30. Frequency Modulation (FM):

When the frequency of the carrying wave is changed in accordance with the instantaneous value of the modulating signal, it is called frequency modulation.

31. Modulation factor or index and carrier swing (CS)

(i) Modulation factor : $m=\frac{ \mathrm{maximum} \mathrm{frequency} \mathrm{deviation} }{\mathrm{Modulating} \mathrm{frequency}}=\frac{\mathrm{\Delta }\mathrm{f}}{{\mathrm{f}}_{\mathrm{m}}}$

$\mathrm{\Delta }f=f_{\max }-f_{\mathrm{c}}-f_{\mathrm{c}}-f_{\min }; V_{\mathrm{FM}}=V_{\mathrm{c}}\cos \left[{\omega }_{\mathrm{c}}t+m_{\mathrm{i}}\cos \left({\theta }_{\mathrm{m}}t\right)\right]$

(ii) Carrier Suing (CS):

The total variation in frequency from the lowest to the highest is called the carrier swing $\Rightarrow CS=2\times \mathrm{\Delta }f$

(iii) Side Bands:

FM wave consists of an infinite number of side frequency components on each side of the carrier frequency $f_c, f_{\mathrm{c}}\pm f_{\mathrm{m}}, f_{\mathrm{c}}\pm 2f_{\mathrm{a}}, f_{\mathrm{c}}\pm 3f_{\mathrm{m}},$& soon.

32. Modem:

The name modem is a contraction of the terms Modulator and Demodulator. Modem is a device that can modulate as well as demodulate the signal.

33. FAX (Facsimile Telegraphy):

FAX is an abbreviation for facsimile which means exact reproduction. The electronic reproduction of a document at a distance place is called Fax.