\(\begin{array}{l} \beta=\frac{a}{1-\alpha} \\ \beta=\frac{0.96}{1-0.96}=24 \end{array}\)

Ground wave propagation: frequency less than 2 MHz
Sky wave propagation: 3 to 30 MHz
Space wave propagation: Above 30 MHz to 400 GHz
Satellite communication: uplink communication 6 GHz band downlink communication 4 GHz band

A = 1, B = 0, C = 1
y = A + B, y = (A + B).C
y = (1 + 0).1 ⇒ y = 1

\(Y_1=\overline{A+B}, y_2=\overline{A+B}=A+B, y=\overline{A+B}\)

N-type semiconductor: A n-type semiconductor is obtained by doping a pure Germanium (or Silicon) crystal with a dopant from group V pentavalent elements like Phosphorus, Arsenic, and Antimony. The dopant has five valence electrons while the Germanium atom has four valence electrons.
During the process of doping, a few of the Germanium atoms are replaced by the group V dopants. Four of the five valence electrons of the impurity atom are bound with the 4 valence electrons of the neighbouring replaced Germanium atom. The fifth valence electron of the impurity atom will be loosely attached to the nucleus as it has not formed the covalent bond.

Free electron which is loosely attached to the lattice:

Representation of donor energy level:
The energy level of the loosely attached fifth electron from the dopant is found just below the conduction band edge and is called the donor energy level. At room temperature, these electrons can easily move to the conduction band with the absorption of thermal energy. It is shown in the figure. Besides, an external electric field also can set free the loosely bound electrons and lead to conduction.

It is important to note that the energy required for an electron to jump from the valence band to the conduction hand in an intrinsic semiconductor is 0.7 eV for Ge and 1.1 eV tor Si, while the energy required to set free Ni a donor electron is only 0.01 eV for Ge and 0.05 eV for Si.
The V group pentavalent impurity atoms donate electrons to the conduction band and are called Onormpunties. Therefore, each impurity atom provides one extra electron to the conduction band in addition to the thermally generated electrons. These thermally generated electrons leave holes in valence band. Hence, the majority carriers of current in an n-type semiconductor are electrons and the minority carriers are holes. Such a semiconductor doped with a pentavalent impurity is called an n-type semiconductor

First Theorem :
The complement of the sum of two logical inputs is equal to the product of its complements.
\(\overline { A+B } \) = \(\bar { A } .\bar { B } \)
Proof:
(i) The Boolean equation for NOR gate is Y = \(\overline { A+B } \)
(ii) The Boolean equation for a bubbled AND gate is Y =\(\bar { A } .\bar { B } \)
(iii) Both cases generate same outputs for same inputs. It can be verified using the following truth

A	B	A+B	\(\overline { A+B } \)	Ā	\(\bar { B } \)	\(\bar { A } .\bar { B } \)
0	0	0	1	1	1	1
0	1	1	0	1	0	0
1	0	1	0	1	0	0
1	1	1	0	0	0	0

(i) From the above truth table, we can conclude \(\overline { A+B } \) = \(\bar { A } .\bar { B } \)
(ii) Thus De Morgan's first theorem is proved.
(iii) Hence, a NOR gate is equal to a bubbled AND gate

Second theorem :
The complement of the product of two is equal to the sum of its complements
\(\overline { A.B } \) = \(\bar { A } +\bar { B } \)
Proof:
(i) The Boolean equation for NAND gate is Y = \(\overline { A.B } \)
(ii) The Boolean equation for bubbled OR gate is Y = \(\bar { A } +\bar { B } \)
(iii) A and B are the inputs and Y is the output. The above two equations produces the same output for the same inputs. It can be verified by using the truth table.

A	B	A+B	\(\overline{\mathrm{A}. \mathrm{B}}\)	Ā	\(\bar { B } \)	\(\overline{\mathrm{A}}+\overline{\mathrm{B}}\)
0	0	0	1	1	1	1
0	1	0	1	1	0	1
1	0	0	1	0	1	1
1	1	1	0	0	0	0

(i) From the above truth table, we can conclude \(\overline { A.B } \) = \(\bar { A } +\bar { B } \)
(ii) Thus, De Morgan's second therom is proved.
(iii) Hence, a NAND gate is equal to a bubbled OR gate.

FuIl wave rectifier :
The positive and negative half cycles of the AC input signal pass through the full wave rectifier circuit and hence it is called the full wave rectifier
Construction:

(i) It consists of two p-n junction diodes, a center-tapped transformer, and a load resistor (R₁)
(ii) The centre is usually taken as the ground or zero voltage reference point.
(iii) Due to the centre tap transformer, the output voltage rectified by each diode is only one-half of the total secondary voltage.
Working:
During positive half cycle :
(i) When the positive half cycle of the ac input signal passes through the circuit, terminal M is positive, G is at zero potential and N is at negative potential.
(ii) This forward biases diode D₁ and reverse biases diode D₂.
(iii) Hence, being forward biased, diode D₁ conducts and current flows along the path MD₁AGC.
During negative half cycle:
(i) When the negative half cycle of the AC input signal passes through the circuit, terminal N becomes positive, C is at zero potential and M is at negative potential.
(ii) This forward biases diode D₂ and reverse biases diode D₁.
(iii) Hence, being forward biased, diode D₂ conducts and current flows along the path ND₂BGC.
(iii) During both positive and negative half cycles of the input signal, the current flows through the load in same direction.

(iv) The output signal corresponding to the input signal is shown in Figure. Though both half cycles of AC input are rectified, the output is still pulsating in nature.
(v) The efficiency (η) of full wave rectifier is twice that of a half wave rectifier and is found to be 81.2 %.

a) Information (Baseband or input signal):
i) Information can be in the form of a sound signal like speech, music, pictures, or computer data which is given as input to the input transducer.
b) Input transducer:
i) It converts the information which is in the form of sound, music, pictures or computer data into corresponding electrical signals.
ii) The electrical equivalent of the original information is called the baseband signal.
iii) The best example is the microphone that converts sound energy into electrical energy.
c) Transmitter
i) It feeds the electrical signal from the transducer to the communication channel
ii) It consists of circuits such as amplifier, oscillator, modulator, and power amplifier.
iii) Amplifier: The transducer output is very weak and is amplified by the amplifier.
iv) Oscillator: It generates high-frequency carrier wave (a sinusoidal wave) for long distance transmission into space. As the energy of a wave is proportional to its frequency, the carrier wave has very high energy.
v) Modulator: It superimposes the baseband signal onto the carrier signal and generates the modulated signal.
vi) Power amplifier: It increases the power level of the electrical signal in order to cover a large distance.
d) Transmitting antenna:
i) It radiates the radio signal into space in all directions.
ii) It travels in the form of electromagnetic waves with the speed of light.
e) Communication channel:
Communication channel is used to carry the electrical signal from transmitter to receiver with less noise or distortion.
Example: Wires, cables, optical fibres in wireline communication and free space in wireless communication.
f) Receiver:
i) The signals that are transmitted through the communication medium are received with the help of a receiving antenna and are fed into the receiver.
ii) The receiver consists of electronic circuits like demodulator, amplifier, detector etc. The demodulator extracts the baseband signal from the carrier signal.
iii) Then the baseband signal is detected and amplified using amplifiers.
iv) Finally, it is fed to the output transducer.
g) Repeaters:
i) Repeaters are used to increase the range or distance through which the signals are sent.
ii) It is a combination of transmitter and receiver.
iii) The signals are received, amplified, and retransmitted with a carrier signal of different frequency to the destination.
iv) The best example is the communication satellite in space
h) Output transducer:
i) It converts the electrical signal back to its original form such as sound, music, pictures or data.
ii) Examples of output transducers are loudspeakers, picture tubes, computer monitor, etc

(i) If the amplitude of the carrier signal is modified according to the instantaneous amplitude of the baseband signal, then it is called amplitude modulation Here the frequency and the phase, of the carrier signal remain constant. Amplitude modulation is used in radio and TV broadcasting.
(ii) The signal shown in Figure (a) is the baseband signal that carries information. Figure(b) show the high-frequency carrier signal and Figure (c) gives amplitude modulated signal. We can see that amplitude of the carrier wave is modified in proportion to the amplitude or the baseband signal.

(i) \((A+B)(A+B)=AA+AB+BA+BB\)     \((\because AA=A)\) 
\(=A+AB+B\) 
= A+ AB +B = A(1 + B) + B    \((\because AB+AB=AB)\)
= A + B   \((\because 1+B=1)\) 
(ii) \(A(\bar{A}+B)\)\(=A \bar{A}+A B=AB\)      \((\because A\bar{A}=0)\)
(iii) (A + B) (A + C) =  AA + AC + BA + BC
= A + AC + BA + BC
= A (1 + C) + BA + BC
= A + BA + BC
= A(1 + B) + BC   \((\because 1+C=1)\)
= A + BC  \((\because 1+B=1)\)

NAND and NOR gates are known as universal gates because any other logic gate can be made from NAND or NOR gates.

(i) The loop phase shift must be 0⁰ or integral multiples of \(2 \pi\)
(ii) The loop gain must be unity That is IA\(\beta\)I = 1
Here, A → voltage gain of the amplifier
\(\beta\) →  feed back ratio
(iii) There should be a positive feedback.

The process in which alternating voltage or alternating current is converted direct voltage or direct current is called rectification.

When a PN junction diode is forward biased, the depletion region decreases and the diode conduct once after the barrier potential is crossed, when it is reverse biased the depletion region increases and the diode does not conduct sci it is called as unidirectional device.

The process of adding impurities to the instrinsic semiconductor is called doping.

S.No Intrinsic semiconductor Extrinsic semiconductor (i)  A semiconductor in its pure form without impurity is called an intrinsic
semiconductor. When small amount of impurity is doped in pure semiconductors it becomes extrinsic semiconductors.
  (ii) The number of free electrons in conduction band is almost equal to number of holes in valence band. The number of free electrons and holes is never equal. (iii) Electrical conductivity is a function of temperature along. Its electrical conductivity depends upon the temperature as well as on the quantity of impurity atoms doped the structure.

ANS

\(\begin{array}{l} Y=(\bar{A}+\bar{B}) \cdot C \\ Y=(\bar{A} \cdot \bar{B}) \cdot C \end{array}\)         (Using De-Morgan's 1" theorem)      
Y =  ABC

LHS:
AC (1 + B) = AC .1 [1 + B = 1]
AC.1 = AC
Therefore AC + ABC = AC

Advantages of FM:
i) In FM, there is a large decrease in noise. This leads to an increase in signal-noise ratio.
ii) The operating range is quite large.
iii) The transmission efficiency is very high as all the transmitted power is useful
iv) FM Bandwidth covers the entire frequency range which humans can hear. Due to this, FM radio has better quality compared to AM radio.
Limitations of FM:
i) FM requires a much wider channel.
ii) FM transmitters and receivers are more complex and costly.
iii) In FM reception, less area is covered compared to AM.

i) Weather Satellites : They are used to monitor the weather and climate of Earth. By measuring cloud mass, these satellites enable us to predict rain and dangerous storms like hurricanes, cyclones, etc.
ii) Navigation satellites : These are employed to determine the geographic location of ships, aircraft or any other object.
iii) Communication satellites: Thev are used to transmit television, radio, internet signals etc. Multiple satellites are used for long distance communication.

Diode D₁ and D₄ is reverse biased [open]
Diode D₁ and D₃ are forward biased.
The resistances are in series
R = 1 + 10 + 1 - 12 Ω
Barier Potential, V = 0.7 + 0.7 = 1.4 V (Silicon diode)
Applying Kirchhoff's voltage Law,
0.7 + I(1) + I(10) + 0.7 + I(1) = 3V
12 I = 3 - 1.4
12 I = 1.6
\(I=\frac{1.6}{12}=\mathbf{0 . 1 3 3 A}\)