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Scattering Matrix in Microwave Engineering

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6. In a two port network, if the reference plane are shifted from one and two to 1’ and 2’ the new S matrix is given by

## Scattering Matrix in Microwave Engineering

###
**Microwave Hybrid Circuits:**

A microwave network or microwave hybrid circuit consist of several microwave devices such as sources, attenuators, filters, amplifiers etc coupled together by transmission lines for the transmission of microwave signal. The point of interconnection of two or more devices is known as a junction. The measurement of z,y,h and ABCD parameter is difficult at microwave frequencies due to following reasons.

1. Non availability of voltage and current measuring equipments.

2. Short circuit not easily achieved for wide range of frequencies.

3. Presence of active devices make the circuit unstable so microwave circuits are analyzed using scattering parameters or ‘S’ matrix. S matrix relates the amplitude of reflected waves with incident waves.

###
Scattering Matrix in Microwave Engineering** :**

It is a square matrix which gives all the combinations of power relationship between input and output ports of a microwave junction. The elements of ‘S’ matrix are known as scattering parameters or scattering coefficients.

Microwave 2 port network |

Consider the microwave 2 port network.

a

_{1}– amplitude of incident wave at port 1
a

_{2}– amplitude of incident wave at port 2
b

_{1}– amplitude of reflected wave at port 1
b

_{2}– amplitude of reflected wave at port 2
The incident and reflected waves can be related using ‘S‘ matrix as

[b] = [s] [a]

b

_{1}= s_{11}a_{1}+ s_{12}a_{2}
b

_{2}= s_{21}a_{1}+ s_{22}a_{2}
s

_{11}is the reflection coefficient at port 1 when, a_{2}= 0
s

_{11}= b_{1}/a_{1}where a_{2}= 0
s

_{22}is the reflection coefficient at port 2 when, a_{1}= 0
s

_{22}= b_{2}/a_{2}where a_{1}= 0
s

_{12}is the attenuation of wave travelling from port 2 to port 1 when, a_{1}= 0
s

_{12}= b_{1}/a_{2}where a_{1}= 0
s

_{21}is the attenuation of wave travelling from port 1 to port 2 when, a2 = 0
s

_{21}= b_{2}/a_{1}where a_{2}= 0
In a microwave network if the incident power is P

_{i}, reflected power is P_{r}, output power is P_{a}then, the losses defined are
1. Insertion loss = 10 log (P

_{i}/P_{o})
2. Transmission loss or attenuation = 10 log((P

_{i}- P_{r})/P_{o})
3. Reflection loss = 10 log (P

_{i}/(P_{i}- P_{r}))
4. Return loss = 10 log (P

_{i}/P_{r})###
**Properties of S matrix:**

1. S matrix is always a square matrix of order n x n.

2. Under perfect match condition the diagonal elements of S matrix are zero.

3. S matrix is always symmetric. ie, S

_{ij}= S_{ji}
4. S matrix is an unitary matrix. Ie, [S][S]

^{*}= I. where, I is an identity matrix.
5. The sum of product of each term of any row or column multiplied by complex conjugate of corresponding term of another row or column is zero.

6. In a two port network, if the reference plane are shifted from one and two to 1’ and 2’ the new S matrix is given by

This property is known as phase shift property.

7. Since S matrix is symmetric [S]

^{T}= [S]
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###

## Z Transform of U(n)

**Find the Z Transform and ROC of U(n) ?**

The Z transform of a discrete time signal x(n) is given by,

Here given x(n) = u(n)

Therefore,

We know that U(n) = 1; n ≥ 0

= 0; n < 0

Therefore,

X(z) = Z

^{0}+ Z^{-1}+Z^{-2}+ Z^{-3}+ Z^{-4}+………….
= 1 + Z

^{-1}+Z^{-2}+ Z^{-3}+ Z^{-4}+………….
It is clear that the infinite series is a Geometric Progression (GP)

The sum of the GP is given by

Sum = First Term / (1 – Common Ratio)

The common ratio (r) is given by

r = second term/first term

= third term/second term

So, r = Z

^{-1}/1 = Z^{-2}/ Z^{-1 }= Z^{-1}
Hence, the sum of the series is given by

###
**ROC of U(n)**

The ROC of U(n) is given by

|r| < 1

|Z

^{-1}| < 1
|1/Z| < 1

|Z| > |1|

ROC of U(n) |

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## Velocity Modulation in Klystron Amplifier

**Klystron Amplifiers:**

The 2 cavity klystron amplifier is widely used microwave amplifier, operated by the principles of voltage and current modulators.

**Basic Operations:**

A high velocity electron beam produced by the accelerating anode is passed through a buncher cavity, drift space, catcher cavity and finally collected by the collector terminals. The electrons injected from the cathode is accelerated by applying a DC voltage ‘Vg’. They arrive at the first cavity that is the buncher cavity or input cavity with uniform velocity. At the buncher cavity these electrons encounter signal voltage or gap voltage. The electrons that pass through the zeros of the gap voltage pass with unchanged velocity. The electrons that pass through positive half cycles of the gap voltage undergo acceleration in velocity. The electrons that pass through negative half cycles of the gap voltage undergo retardation in velocity. (As a result of these the electrons get bunched together as they travelled through the drift space). The variation in electron velocity in drift space is called velocity modulation. (The buncher cavity velocity modulates the electron beam). This electron beam induces a RF current in this field is opposite to the input cavity. Thus the kinetic energy is transferred from the electrons to the field capture cavity. The second cavity is called capture cavity since it captures energy from the bunch electron beam. The electrons emerging from the capture cavity are collected by the collector terminal.

**Velocity Modulation in Klystron Amplifier:**

The velocity of electrons before entering the buncher cavity is given by,

V

_{o}= √2ev_{0}/m
Where m is the mass of the electron

e is the charge of electron

v

_{0}is the cathode potential
On substituting the values of e and m, the equation reduces to

V

_{o}= 0.596 x 10^{6}√ (v_{0}) m/sec ------------- 1
When the microwave signal is applied to the input terminal, the gap voltage is given by

V

_{s}= V_{1}sinωt ----------------- 2
V

_{1}is the amplitude of the signal.
The average transit time through the gap at a distance ‘d’.

Τ = d/V

_{o}= t_{1}– t_{0}-------------- 3
Where t

_{0}is the line at which beam reaches the buncher cavity. t_{1}is the time at which the beam leaves the buncher cavity.
The average transit angle, θ

_{g}= ω(d/V_{o}) = ω(t_{1 }– t_{0}) ------------ 4
The average microwave voltage in the buncher cavity is

= V1/T ω (cos ωt

_{0}– cos ωt_{1}) ----------------- 5
From eq (4)

ωd/V

_{0}= ω(t_{1}– t_{0})
ωt

_{1}= ω(d/V_{o}+ t_{0}) ------------- 6
Subsituting eq (6) in eq (5)

<V

_{s}> = V_{1}/Tω [cosωt_{0}– cos(ωd/V_{o}+ ωt_{0})] ------------- 7
Let ωt

_{0 }+ ωd/2V_{o}= ωt_{0}+ θ_{g}/2 = A
ωd/2V

_{o }= θ_{g}/2 = B
A + B = ωt

_{0}+ θ_{g}/2 + θ_{g}/2 = ωt_{0}+ θ_{g }
A – B = ωt

_{0}+ θ_{g}/2 - θ_{g}/2 = ωt_{0}_{ }

A + B = ωt

_{0}+ θ_{g }, A – B = ωt_{0 }----------------- 8
Substitute eq (8) in eq (7)

<V

_{s}> = V_{1}/Tω [cos(A - B) – cos(A + B)]
= 2V

_{1}/Tω. sin A sin B
= 2V

_{1}/Tω. sin (ωt_{0}+ θ_{g }) sin (θ_{g}/2)
Let ωT = θ

_{g}_{ }

Therefore,

_{ }<V_{s}> = 2V_{1}/ θ_{g}. sin (ωt_{0}+ θ_{g }) sin (θ_{g}/2)
<V

_{s}> = V_{1 }βi sin (ωt_{0}+ θ_{g}/2)**βi = sin (θ**

_{g}/2)/ (θ_{g}/2)
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## Bunching Process in Microwave

**Bunching Process in Microwave:**

Once the electrons leave the buncher cavity, they drift with a velocity along the space between two cavities. The effect of velocity modulation produces, bunching of electron beam or current modulation. The electrons that pass the buncher cavity with zero voltage travel with unchanged velocity and become the bunching centre. Electrons that pass the bunching cavity during positive half cycles of microwave input become faster and electrons that pass during the negative half cycle become slower.

t

_{a}= time at which maximum retardation occur
t

_{b}= time at which electrons have uniform velocity
t

_{c}= time at which maximum acceleration occur
Bunching centre is the point at which electron density is maximum. The distance to the bunching centre,

ΔL = V

_{o}(t_{d}- t_{b}) ---------------------- (1)
t

_{c}= t_{b}+ (π/2ω),
t

_{b}= t_{a}+ (π/2ω),
t

_{a}= t_{b}- (π/2ω) ------- (2)
The distance of electron at ta,

Δt = V

_{min}(t_{d}- t_{a}) = V_{min}(t_{d}– t_{b}+ (π/2ω)) ----------- (3)
The distance of electron at ta,

Δt = V

_{max}(t_{d}- t_{c}) = V_{max}(t_{d}– t_{b}– (π/2ω)) ----------- (4)
Let V

_{min}= V_{o}[1 – β_{i}V_{i}/2V_{o}] ------------ (5)
V

_{max}= V_{o}[1 + β_{i}V_{i}/2V_{o}] ------------ (6)
Substitute eqn 5 in eqn 3

Δt = V

_{o}[1 + β_{i}V_{i}/2V_{o}] (t_{d}– t_{b}+ (π/2ω))
Δt = [V

_{o}+ β_{i}V_{i}V_{o}/2V_{o}] (t_{d}– t_{b}+ (π/2ω))
Δt = V

_{o}t_{d }- V_{o}t_{b}+ V_{o}(π/2ω) - t_{d}β_{i}V_{i}/2 + t_{b}β_{i}V_{i}/2 - β_{i}V_{i}π/4ω ------ (7)
Substitute eqn 6 in eqn 4

Δt = V

_{o}[1 + β_{i}V_{i}/2V_{o}] (t_{d}– t_{b}– (π/2ω))
Δt = V

_{o}t_{d}– V_{o}t_{b}– V_{o}(π/2ω)) + t_{d}β_{i}V_{i}/2 – t_{b }β_{i}V_{i}/2 - β_{i}V_{i}π/4ω ------ (8)
The necessary condition at which electrons meet at a distance ΔL is,

Equating eqn 7 and eqn 8

V

_{o}t_{d }- V_{o}t_{b}+ V_{o}(π/2ω) - t_{d}β_{i}V_{i}/2 + t_{b}β_{i}V_{i}/2 - β_{i}V_{i}π/4ω =
V

_{o}t_{d}– V_{o}t_{b}– V_{o}(π/2ω)) + t_{d}β_{i}V_{i}/2 – t_{b }β_{i}V_{i}/2 - β_{i}V_{i}π/4ω
We get,

V

_{o}π/ω = 2β_{i}V_{i}/2 (t_{d}– t_{b})
We have,

(β

_{i}V_{i})(t_{d}– t_{b}) = V_{o}π/ω
t

_{d}– t_{b = }V_{o}π/ω β_{i}V_{i }------------------ (9)
Subsitute, eqn (9) in eqn (1)

ΔL = V

_{o}(V_{o}π/ω β_{i}V_{i })
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## Measurement of Distortion

There are several methods of measurement of distortion. The important methods are

1. The use of tuned circuit to tune to the frequency component. This is called tunable selective type of measurement.

2. Heterodyne distortion meter or Heterodyne wave analyser.

3. Fundamental suppression method of distortion measurement.

**(a) Distortion Measurement Using Tunable Selective Amplifier :**

The arrangement for measurement of distortion using tunable selective amplifier is shown in Figure. Input signal reaches the attenuator. The first stage is an emitter follower stage that couples the attenuator to the input of the tunable amplifier. The tunable amplifier is of the R.C. timed type. The output of the amplifier is connected to the electronic voltmeter for indication.

Tunable Selective Type of Distortion Factor Meter |

The measurement procedure involves in applying the signal to the input terminals, tuning the R.C. tuned amplifier to the harmonic frequencies and noting the reading in the electronic voltmeter. The meter can be calibrated in terms of R.M.S. distortion.

The attenuator works as a range multiplier and allows application of large signal amplitudes to be analyzed without overloading the amplifier. The calibration can be done by an auxiliary signal generator which applies a known voltage to the input. If the amplifier is designed with constant gain for all the frequencies of the tuning, the attenuator can be set to give the harmonic components directly as a fraction of the fundamental amplitude.

The choice of the R.C. tuned amplifier is because of its wide tuning range and as it does not use coils that complicate the measurement. Also the bandwidth of the response characteristic of the R.C. tuned amplifier is substantially constant percentage over the entire tuning range.

**(b) Distortion Measurement Using Heterodyne Wave Analyser :**

The block diagram of the heterodyne wave analyser is shown in Figure. A balanced mixer is used which receives its input from the tunable oscillator. The other input to this balance mixer is from the attenuator. The wave to be analysed is passed through the attenuator to the balanced mixer. As the balanced mixer has two inputs, and as it is a nonlinear device, it produces a heterodyne signal. The output of the balanced mixer is applied to a highly selective multistage amplifier. This amplifier has a predetermined fixed response frequency that is higher than any of the frequencies contained in the unknown wave. The output of the selective amplifier is indicated by an electronic voltmeter or equivalent.

Block Diagram of Hetrodyne Wave Analyser |

To determine the amplitude of the harmonic component, the local oscillator is adjusted to get the heterodyne frequency produced by the mixer to be equal to the resonant frequency of the selective amplifier. As the input is available at the input of the balanced mixer, the component to be determined has its frequency transformed to the predetermined value of the selective amplifier. Though the other frequency components also produce the corresponding difference signal (beat signal) in the output of the balance mixer, they will be rejected by the highly selective amplifier.

The amplitude of the unknown component of the input signal can be measured by the electronic multimeter. It can as well be calibrated in terms of amplitude. The frequency of the unknown component can readily be known from the frequency to which the local oscillator is tuned to get the required beat signal. Heterodyne wave analysers may use crystal filters or ordinary resonant circuit with high "Q".

**(c) Fundamental Supression Method of Distortion Measurement :**

R.M.S. distortion can be measured by suppressing the fundamental frequency component of a signal and measuring the remaining part of the signal. When the output is measured by a thermocouple, or square wave electronic voltmeter, the R.M.S. value is realised correctly. However rectifier type of instruments works well with a small error.

The fundamental of a signal can be suppressed using a high pass filter, designed to pass the harmonic only. The filter design must attenuate the fundamental. Alternately bridged T networks balanced to the fundamental frequency and unbalanced for the harmonics can be used. Resonance bridge, Bridge T, Wien Bridge can be employed for this purpose.

The block diagram of fundamental suppression type of distortion factor meter is shown in Figure. The arrangement consists of an attenuator bridge T network and a switch. The output of the bridge T network is given to the output indicator.

Block Diagram for Fundamental Supression Method |

The switch S is set to the first position "A" and the input signal is applied. Now the bridge T network is adjusted for the fundamental frequency. Therefore the indication will be low. The switch is now set to position "B" and the attenuator is adjusted to give the same indication as before. It is to be noted that in this position the bridge T network is shorted. The attenuator reading gives the R.M.S. distortion defined in decibles.

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## Amplitude, Frequency and Phase Distortion

Signal distortion can be of three types. They are Amplitude, Frequency and Phase Distortion

Amplitude distortion is due to the nonlinearity of the devices used in the system. The wave shape changes resulting in the generation of harmonics. Hence amplitude distortion is also sometimes referred as the harmonic distortion.

Frequency distortion is one in which the gain of the system vanes with frequency of the applied input voltage. This is primarily caused by the presence of reactive elements m the system.

**1. Amplitude distortion.**

Amplitude distortion is due to the nonlinearity of the devices used in the system. The wave shape changes resulting in the generation of harmonics. Hence amplitude distortion is also sometimes referred as the harmonic distortion.

**2. Frequency distortion.**

Frequency distortion is one in which the gain of the system vanes with frequency of the applied input voltage. This is primarily caused by the presence of reactive elements m the system.

**3. Phase distortion.**

Phase distortion is the phase shift between the output and input of the system. It is related to the time of transmission through the amplifier in accordance with the relation:

Phase shift (lag) in radians = ωγ + nπ

where γ is called the time delay in seconds and ω is 2π times the frequency, and n is an integer which will be odd if the system has an inherent tendency to reverse the phase of all components of the applied voltage. It will be even if there is no tendency for phase reversal. It is to be noted here that the phase shift of 180° given by an amplifier is not termed as distortion as any signal undergoes this inversion.

where γ is called the time delay in seconds and ω is 2π times the frequency, and n is an integer which will be odd if the system has an inherent tendency to reverse the phase of all components of the applied voltage. It will be even if there is no tendency for phase reversal. It is to be noted here that the phase shift of 180° given by an amplifier is not termed as distortion as any signal undergoes this inversion.

Harmonic distortion present in the output of an audio generator can be measured, using distortion factor meter. It measures the R.M.S. harmonic distortion, represented by a ratio of the amplitude of the harmonic to that of the fundamental frequency, expressed as a percentage. Then harmonic distortion is represented as :

D

_{2}= B_{2}/B_{1}, D_{3}= B_{3}/B_{1}, D_{4}= B_{4}/B_{1}_{ }

Where D

_{n}(n = 2, 3, 4, ...) represents the distortion of the nth harmonic, B_{n}is the amplitude of the nth harmonic and B1 is the amplitude of the fundamental.
The total harmonic distortion or distortion factor is defined as :

D = √ (D

_{2}^{2}+ D_{3}^{2}+ D_{4}^{2}+………..)
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## Pulse Generator Block Diagram and Explanation

Before we speak on a pulse generator we have to differentiate between a square wave and pulse. The pulse and the square wave differ primarily in their duty cycle. Duty cycle is defined as the ratio of the average value of the pulse over one cycle in the peak value of the pulse.

As the average value and peak value are inversely related to their time duration the duty cycle can be defined in terms of the pulse width and the period or pulse repetition time.

Therefore duty cycle = Pulse width/Period

In a square wave the output voltage will have equal on and off times, such that the duty cycle is 0.5 or 50 %. The duty cycle remains unchanged even if the frequency is changed.

In case of a pulse the duty cycle is not constant, it may vary. Short durations of pulses give a low duty cycle. Short duration of pulse has the advantage that the dissipation of power in the component under test is low.

**(a) Pulse Characteristics and Terminology:**

The characteristics are explained hereunder:

(i) Raise Time : It is defined as the time required for the pulse to increase from 10% to 90% of its normal amplitude.

(ii) Fall Time : It is defined as the time required for the pulse to decrease from 90% to 10 % of its maximum amplitude

(iii) Over Shoot : An over shoot is defined as excess initial raise of amplitude beyond the correct value. It may be visible as a pip or ringing.

(iv) Droop or Sag : Sag is said to occur when the maximum amplitude of the pulse is not constant but decreases slowly,

**(b) Types of Pulse Generators :**

There are two types of pulse generators. They are :

Active Pulse Generators

Passive Pulse Generators

The active pulse generators are relaxation oscillators. Multivibrators and blocking oscillators are the relaxation type pulse generators.

The passive pulse generators generate a sine wave in original and suitable wave shaping will be done to get the required wave shape.

**(c) Pulse Generator :**

Pulse generators usually have their range from I Hz to 10 MHz. A linearly calibrated dial will be provided. There will be provision for variation in the duty cycle. There will be two independent output terminals. The pulse generator can be free running or can also be synchronised with external signals.

**(d) Explanation of the Block Diagram of a Pulse Generator :**

The block diagram of a pulse generator is shown in figure. The frequency control circuit controls the sum of the two currents from the current sources. It applies control voltages to the base of the current control transistors in the two current generators. There are two current sources, ramp capacitor, schmitt trigger and the current switching circuit in the generating loop.

The current source gives a constant current for changing the capacitor (ramp capacitor). The ratio of these two currents is determined by the setting of the symmetry control. This control latter determines the duty cycle of the output waveform. The capacity of the ramp capacitor is selected by the multiplier switch. The last two controls provide decade switching and vernier control of the frequency of the output.

The upper current source supplying a constant current to the ramp capacitor, charges this capacitor at a constant rate and the ramp voltage increases linearly. When the positive slope of the ramp voltage reaches the upper limit set by the internal circuit components, the schmitt trigger changes states.

Block Diagram of a Pulse Generator |

The trigger circuit's output goes negative changing i.e. reversing the conditions of the current control switch and the capacitor starts discharging. The discharge rate is linear, controlled by the lower current source. When the negative ramp reaches a predetermined lower level, the schmitt trigger switches back to its original state. This now provides the positive trigger output that reverses the condition of the current switch again cutting off the lower current source and switching on the upper current source. One cycle of operation is complete now. The entire process is repeated. The schmitt trigger circuit provides a negative pulse at a continuous rate.

The output of the schmitt trigger circuit is passed to the trigger output circuit and to the 50Ω and 600Ω amplifiers. The trigger output circuit differentiates the square wave output from the Schmitt trigger, inverts the resulting pulse and provides a positive triggering pulse. The 60Ω amplifier is provided with an output attenuator to allow a vernier control of the signal output voltage. In addition to its free running mode of operation the generator can be synchronised or locked in to an external signal. This is accomplished by triggering the schmitt trigger circuit by an external synchronization pulse. The power supply is a regulated power supply and supplies all the subsystems of the pulse generator.

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The function generator is a generating instrument. It is also called the sweep function generators. They produce non sinusoidal signals and asymmetrical signals. These signals can be mathematically described as a function of time. Therefore they are called function generators.

## Function Generator Block Diagram Explanation

Triangular, ramp, square rectangular, and also sine wave voltages can be obtained from these function generators. They work in the frequency range from 0.001 Hz to around 20 MHz

(a) The Block Diagram:

The block diagram is given in Figure.

Block Diagram of a Function Generator |

**(b)**

**Explanation**

**of the Block Diagram and Working :**

Usually function generators derive their frequency from a non-sinusoidal oscillator. The block diagram shows a triangular wave generator as the signal source. The triangular wave is generated by charging a capacitor using two constant current sources. Electronic switching will be iced to connect the positive current source and negative current source to the capacitor for required eriods to generate a linear ramp signal. This triangular signal is amplified.

There are two shaper circuits that produce either the sine wave or square wave. A switch is used to select the required wave shape. The output from the shapers or direct from the source generator is given to the amplifier. If required, an off set voltage can be added to the wave from the off set generator. The attenuated output is available from the output socket. This function generator can give triangular, square and sine wave signals.

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