Gauss Law detailed explanation

Maxwell Equations and Light

 And Maxwell said

and there was Light.

                                                                

How to draw Root Locus diagram?

1) There are 4 poles and no zeros. P=4 and Z=0. N=4. There will be 4 branches in root locus

2) All 4 branches will start from open loop poles and terminate at infinity.

3) The branches will terminate at infinity along straight line asymptotes whose angles are determined by

where q=0..N-1

Angle of asymptotes =45, 135,225,315

4) The asymptotes meet the real axis at Centroid

Centroid= Sum of real parts of poles-sum of real parts of Zeros/(P-Z)=0-1-2-3/4=-1.5

5) Breakaway point and Break in  point  is calculated by solving dk/ds=0

Solving this gives s=-1.5, -0.381, -2.619. (Note : -1.5 is not valid breakaway point point because it doesnt lies on root locus)

6) The value of k and the point at which the root locus branch crosses the imaginary axis is determined by applying Routh Criterion to the characteristic equation. The roots at the intersection point are imaginary.

 s^4 | 1     11          k

 s^3 | 6       6            0

 s^2 | 10         k          0   

 s^1 | (60-6k)/10   0       

 s^0 |   k                        

60-6k=0 --> kmax=10

Auxiliary  equation : 10s^2+k=0

At k=10, s=+j and s=-j                  

7) Root Locus 

  

This is the root locus diagram for the given transfer function

Summary

The system is absolutely stable for 0<k<10 and at k=10, the system is marginally stable and for k>10, system is unstable.

Different types of Controllers in Control System

1) Proportional Controller : Transfer Function of proportional controller is G(s)=Kp. Mainly used to vary transient response of a system

2) Integral Controller : Transfer function of Integral controller is Gc(s)=KI/S. It is used to decrease steady state error of a system. But in doing so, the stability of a system decreases

3) Derivative Controller : G(s)=KD*S. It is opposite of Integral controller. It is used to increase the stability of a system. The stability of a control system can be enhanced by adding ZEROS. And since the type of a system decreases, the steady state error increases.

4) PI Controller : G(s)=Kp+ KI/S= (SKp+KI)/S. This controller will reduce the steady state error without effecting stability.

5) PD Controller : G(S)= KD.S+Kp. This controller is used to increase the stability without effecting steady state error. Since type is not changed and a zero is added.

6) PID controller = Kp+ KD.S+ KI/S

 It is used to increase the stability and decrease the steady state error of the control system.

How Edwin Armstrong became the most influential person in radio history

                                                                                   

                     

The Germans were using certain high frequency signals to communicate with each other during world war 1. 

During that time, if the other countries wanted to know, well in advance as ' what the Germans are planning', they had to receive the signal and amplify it accordingly. The plan would be to design a tuned filter. Back in the days, it was easy to build a receiver circuit to track the signal, but there wasn't a proper Amplifer. 

Why Amplifier?

  Amplifier is a device which increases the voltage, current or power of the signal. In common terms, to increase the volume of the signal(Voice supposedly).

   When United States entered world war 1, Armstrong joined Army signal corps and was sent to Paris, and was placed in charge of Research section. While travelling to France, Armstrong met captain H.J Round, an engineer with British Marconi Company. Armstrong learned that the British were ahead of Americans in development of Vaccum tables capable of handling frequency signals ranging from 500KHz to 3.5MHz, a frequency range that it was suspected, the Germans were using for communication. In that way, the British had kept track of many Germany ships and also had broken their codes and nearly read all the messages.

 Germans continued to communicate with each other from different ships planning the destruction of allied countries.  Americans tried to read those messages, but ultimately fell short to build a proper receiving circuit/Amplifier stage and were like

Can't hear anything. Increase the volume

They appointed Armstrong and assigned him to detect short wave enemy communication.

                                       

Armstrong realized that if radio direction-finding (RDF) receivers could be operated at a higher frequency, this would allow better detection of enemy shipping. However, at that time, no practical "short wave" (defined then as any frequency above 500 kHz) amplifier existed, due to the limitations of existing triodes.

It had been noticed some time before that if a regenerative receiver was allowed to go into oscillation, other receivers nearby would suddenly start picking up stations on frequencies different from those that the stations were actually transmitted on. Armstrong (and others) eventually deduced that this was caused by a "supersonic heterodyne" between the station's carrier frequency and the oscillator frequency. Thus if a station was transmitting on 300 kHz and the oscillating receiver was set to 400 kHz, the station would be heard not only at the original 300 kHz, but also at 100 kHz and 700 kHz.

Armstrong realized that this was a potential solution to the "short wave" amplification problem, since the beat frequency still retained its original modulation, but on a lower carrier frequency. To monitor a frequency of 1500 kHz for example, he could set up an oscillator at, for example, 1560 kHz, which would produce a heterodyne difference frequency of 60 kHz, a frequency that could then be more conveniently amplified by the triodes of the day. He termed this the "Intermediate Frequency" often abbreviated to "IF".

The signal with intermediate frequency had retained its original modulation(It means, "The secret message" was still encrypted in the carrier signal, albeit on a lower frequency). 

But there was an advantage this time, the united states Army had proper equipment(Amplifier) to amplify the signals in the IF frequency range which was not the case in the short wave range.

The Military of US adopted the methodology and were able to increase the volume of signal(Amplify) and were able to hear 'what the Germans were planning'?.

If you are wondering how superheterodyne receiver looks like. Here you go.

One of the important advantages of Superheterodyne receiver is  : Irrespective of the frequency transmitted by any station, the IF(Intermediate frequency) remains same, so the amplifier design remains robust and doesn't depend on the frequency of the station.

Almost all modern radio receivers employ this method and even today, Edwin Armstrong is regarded as one of the greats in Radio history.

Frequency Modulation examples/problems

1) A music signal with frequency components from 50 Hz to 21000 Hz is Frequency modulated.  If the maximum allowed frequency deviation is 50 kHz. (i)       What is the modulation index?
ii)What is the signal bandwidth using Carson’s rule?

Solution: i) Modulation index = frequency deviation/modulating frequency = 50 kHz/21 kHz = 2.38

               

(ii) Bandwidth is 2 (modulating frequency + frequency deviation) = 2 (21 kHz + 50 kHz) = 142 kHz

2)A 200 MHz carrier is frequency modulated by a 10 V peak-to-peak signal of 10 kHz. The instantaneous carrier frequency varies between 199.90 and 200.10 MHz.

 Calculate (i) The modulator sensitivity.

                        (ii) The modulation index.

                        (iii) The signal bandwidth using Carson’s rule.

(i)

2 * Frequency deviation = 200.10 – 199.9 MHz = 200 kHz.

Frequency deviation = 100kHz

This is caused by a 10 V signal.

Therefore sensitivity is 100/10 kHz/V = 10 kHz/V

(ii)

The modulation index is 200 kHz / 10 kHz = 20.

(iii)

BW      =          2 (frequency of modulating signal + frequency deviation)                

            =          2 ( f_m + k * f_d)                                                                                     

            =          2 ( 1 + b ) * f_m                                                                                     

Using any of the above

            BW = 420kHz

3) A speech signal with frequency components from 300 Hz to 4 kHz is Frequency modulated.  If the maximum allowed frequency deviation is 30 kHz (i)                 What is the modulation index? (ii)               What is the signal bandwidth using Carson’s rule?

(i)

Modulation index = frequency deviation/modulating frequency = 30 kHz/4 kHz = 7.5

(ii)

Bandwidth is 2 (modulating frequency + frequency deviation) = 2 (30 kHz + 4 kHz) = 68 kHz

Need for Modulation

1) To reduce Antenna height

The antenna height required to transmit a signal is proportional to operating wavelength. For efficient transmission, the minimum antenna height required should be lambda/4. So if we want to transmit a low frequency signal(f), we have v=f*lambda. lambda is inversely proportional to frequency, so low frequency means large wavelength(lambda)--> Antenna required will be very high. Modulation converts the given signal to high frequency signal.

2) To reduce noise

The effect of noise will be more at some frequencies and the effect will be less at some other frequencies. Modulation shifts the spectrum to other band of frequencies where noise effect will be less.

3) Conversion of bandlimited signal to narrowband signal

 Modulation converts a bandlimited signal to a narrowband signal that has ratio between highest frequency to lowest frequency less, so that we can transmit a signal with single antenna which wouldn.'t be possible if the ratio was high.(Which is the case if modulation is not performed on the respective signal)

4) Multiplexing

More than one signal can be transmitted through the same communication channel. By modulation it is possible. We will see about it in detail in the coming posts.
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