Klystron in Microwave Engineering

Microwave Communication in Microwave Engineering refers to the electromagnetic radiation in the frequency range of 1GHz to 300 GHz. The corresponding wavelengths are in the centimetric and millimetric ranges. Microwaves occupy a region in the electromagnetic spectrum that is founded by radio waves on the side of longer wavelengths and the infra-red (IR) waves on the side of shorter wavelengths. Some of advantages of microwaves in communication are :

• Increased bandwidth.
• Higher directivity for a given antenna size.
• Lower power requirements.
• Higher reliability.
• Better transmission (low attenuation).

Klystron in Microwave Engineering

A Klystron is a velocity modulated tube, in which the velocity modulation process produces a density modulated stream of electrons. Figure shows the '2-cavity klystron amplifier'. It is distinguished that a high velocity electron beam is created. It is a high positive potential with respect to cathode. Magnetic focusing is used here.
Klystron Amplifier
A 2-cavity klystron amplifier comprises of a cathode focusing, electrodes, two buncher grids divided by a extremely small space appearing a gap A, two catcher network with a small gap B trail by a collector. The input and output are taken from the tube via resonant cavities. The separation between buncher grids and catcher grids is called, 'drift space'. The electron beam passes gap A in the buncher cavity to which RF signal to be amplified is applied and is then allowed to drift freely without any interference from RF fields until it reaches the gap B in the output or catcher cavity.

The first grid controls the number of electrons in the electron beam and serves to focus the beam. The velocity of electron in the beam is determined by the beam accelerating potential. On leaving the region of first grid the electrons pass through the grids of buncher cavity. The grids of the cavity allow the electrons to pass through, but confine the magnetic fields within the cavity. The space between the grids is called 'inter-action space'. When electrons travel through this space, they are subjected to RF potentials at a frequency determined by the cavity reasonably frequency or the input frequency. The amplitude of this RF potential between the grids is determined by the amplitude of the incoming signal in case of the amplifier, or by the amplitude of the feedback signal from the second cavity if used as in oscillator. Then oscillations will be excited in the second cavity which is of a power much higher than in the buncher cavity, so that a large output can be taken.


A multicavity klystron can be used as a medium or high power amplifier in the UHF and microwave ranges, for both continuous and pulsed operations. The power ranges vary from 10kW to 25MW. The power gain ranges from 30dB to 60dB.

ISDN - Principles, Objectives, Services, Architecture, Channels

ISDN - Principles:

Integrated Services Digital Network (ISDN) is a high -speed, fully digital telephone service. It is an ITU-T (CCITT) standard for an end-to-end global digital communication system providing fully integrated digital services. The principles of ISDN from the point of ITU-T International Telecommunication Union - Telecommunications Standardisation Sector). CCITT (Consulative committee for International Telegraphy and Telephony are :

1. Support of voice and non-voice execution by means of a partial set of standardised services.
2. Support for switched and non-switched (committed lines) applications.
3. Reliance for 64 - kbps connections.
4. Aptitude in the network.
5. Layered protocol architecture.
6. Diversity of configurations are likely to execute ISDN.

ISDN - Objectives :

Activities currently under way are leading to the development of a worldwide ISDN. The key objectives of ISDN are :


2. Transparency : It should provide a transparent transmission service, thereby permitting users to develop applications and protocols with the confidence that they will not be affected by the underlying ISDN.

3. Separation of Competitive Functions

4. Leased and Switched Services : The ISDN should provide dedicated point-to-point services and switched services, thereby allowing the user to optimise implementation of switching and routing techniques.

5. Cost-related Tariffs

6. Smooth Migration: ISDN interfaces should evolve from current interfaces, and provide a migration path for users.

7. Multiplexed Support : ISDN should provide low-capacity support and multiplexed support.

ISDN - Services :

The purpose of the ISDN is to provide fully integrated services to users. These services are divided into three categories :

1. Bearer Services : They can be provided using circuit -switched, packet - switched, frame - relay or cell - relay networks.

2. Teleservices : In teleservicing, the network may change or process the contents of the data. These services be in contact to layers 4 to 7 of the OSI model. Teleservices rely on the facilities of the bearer services and are designed to accommodate complex user needs without the user having to be aware of the process details. Teleservices include telephony, teletex, telefax, videotex, telex and teleconferencing.

3. Supplementary Services : Supplementary services are those that provide additional functionality to the bearer services and teleservices. These services include reverse teleservices. These services include reverse charging, call waiting and message handling.

ISDN - Architecture :

Figure shows the block diagram of ISDN. ISDN supports a new physical connector for users, a digital subscriber loop (link from end user to central or end office), and modifications to all central office equipment. The area which most attention has been paid by standards organisations is that of user access. A common physical interface has been defined to provide, in essence, a DTE - DCE connection. The same interface should be usable for the telephone, computer terminal and videotex terminal.

Protocols are needed for the exchange of control information between user device and the network. Provisions must be made for high-speed interfaces to a digital PBX or a LAN.
The subscriber loop consists of twisted pair links between the subscriber and the central office carrying 4kHz analog signals.

The digital central office connects many ISDN subscriber loop signals to the IDN (Integrated Digital Network). It provides access to both the circuit - switched, packet switched networks, dedicated lines and time - shared, transaction oriented computer services.
Block Diagram of ISDN
ISDN — Channels

To allow flexibility, digital pipes between customers and the ISDN office (Subscriber loops) are organised into multiple channels of different sizes. The ISDN standard defines three channel types, each with different transmission rates. They are:

1. Bearer (B) channel - 64 kbps
2. Data (D) channel - 16 or 64 kbps
3. Hybrid (H) channel - 384 (H 0), 1536 (H 11) and 1920 (H 12) kbps.

B Channels : The B channels is the basic user channel. It can carry any type of digital data in full - duplex mode as long as the required data rate doesn't exceed 64 kbps. It carries transmissions end-to-end. It is not designed to demultiplex a stream midway in order to separate and divert transmissions to more than one recipient.

D Channels : The D channel serves two purposes - to carry signalling information to control circuit - switched calls on associated B channels at the user interface and for packet - switching or low - speed telemetry at times when no signalling information is waiting.

H Channels : H channels are provided for user information at higher bit rates. The user may employ such a channel as a high - speed trunk, or the channel may be subdivided according to the user's own TDM scheme. Some of the applications of H channels include fast facsimile, video, high - speed data, high - quality audio, teleconferencing and multiple - information streams at lower data rates.

BPSK System with Block Diagram

Binary Phase Shift Keying (BPSK) is a form of phase modulation using two different carrier phases to signal 1 and 0. BPSK is the simplest type of PSK. In binary phase shift keying (BPSK), the phase of a stable amplitude carrier signal is toggled amid two values with respect to the two likely signals m1 and m2 equivalent to binary 1 and 0, respectively. Normally, the phases are separated by 1800.

The two signals are therefore given by: 
S1 (t) = √( 2Eb/Tb) cos (2πfct)
S2 (t) = √( 2Eb/Tb) cos (2πfct +π ) = -√( 2Eb/Tb) cos (2πfct),

where fc is the carrier frequency. The relationship among bit information and waveform signal output and initial phase from modulator is illustrated as Figure.
This system may be represented using a single basis function
φ1 (t) = √( 2/Tb) cos (2πfct),    0 ≤ t ≤ Tb
with the two signal vectors amplitude given by

S1 = Eb
S2 = -Eb

Where Eb is the signal energy which is also in his case the energy per bit.

A coherent binary PSK system is therefore characterized by having a signal space that is one dimensional with a signal constellation consisting of two message points of equal and opposite amplitude. Such signals with equal energy and a cross correlation coefficient of -1 are called antipodal.

This signal space diagram is shown in Figure. If the symbols are equiprobable then the rule for deciding which symbol was transmitted is to choose the closest message point. The BPSK transmission system is illustrated as Figure.
BPSK System Block Diagram
From the Figure, we can describe the BPSK transmission system as 3 parts:
• Transmitter
• Receiver
• AWGN Channel.

Transmitter Part :

By using the block diagram in Figure above, if we generate, the binary number randomly we will get the information' sequences. Information generated: {b(i)} = 1, 0, 0,…..1.

This bit information sequences will modulate the carrier frequency, and the phase of the carrier frequency will shifted as function of binary information. 

The bit information "1" will not shift the carrier phase, and the bit information "0" will shift the carrier phase by 1800 or radians.

Receiver Part :

Basically the receiver part as similar with the transmitter part, but the function is contradictive. After demodulate process by using local signal oscillator and filtering by using LPF, the base band signal will be decided.

If the amplitude of the base band signal is < 0 it's decides that a binary "0" was send and if the amplitude of the baseband signal is 0, the receiver decides that binary "1" was send by the transmitter.

This description is made by assumption that all of the, process of modulator, Band Pass Filter, demodulator and filtering is work perfectly and inter symbol inter fervencies (ISI) is not happen.

The AWGN Channel :

The signal output from transmitter part is propagated through the channel, which is corrupted by the additive white Gaussian noise, and illustrated in the Figure.

Thus the received signal at the receiver part will have a form :

r(t) = s(t) + n(t); 0 t

where n(t) denotes a sample function of the additive white Gaussian noise (AWGN) process with variance value σ2 = No/2 Watt/Hertz. Here we made an assumption that the mean value of the AWGN is 0.


Recalling that our BPSK signal is s(t) = Ac m(t) cos(ωct) , we can use the convolution theorem to calculate its spectrum. 

The spectrum of cos(wct) is [δ(f - fc) + 6(f + fc)]/2, so
S(f)= (Ac/2)M(f - fc) + (Ac/2) M(f + fc)
while M ( f ) is called the fourier transform of m(t).

To calculate the specific sequence of logic states is to be known. If the bits are assumed random, then the power spectrum of m(t) is identical to the power spectrum of a single bit. Hence we have

|S(f)|2 = Tb sin c2 [Tb(f – fc)] 

This is illustrated in Figure and compared to the MSK spectrum.
BPSK Spectrum
The wider main lobe of BPSK as compared to MSK, and the much larger side lobes show that BPSK is not bandwidth efficient as MSK.

The bit rate of BPSK is Rb = Tb. The bandwidth is well approximated by B 1/ Ts = 1/ Tb

The spectral efficiency is  πBPSK = Rp/B = (1/Tb)/(1/Tb) = 1 bps/Hz

Digital Carrier Modulation Techniques

To transmit digital data over copper wires, they can be transmitted directly with two voltage levels +A and -A because digital data is a string 1's and 0's. But if they are to be transmitted through space using antenna, it is necessary to modulate the incoming data on to a sinusoidal carrier wave. The data may be the output of PCM system or digital computer output. Amplitude modulation, frequency modulation and phase modulation are commonly used Digital Carrier Modulation Techniques for an analog signal. But, in digital communication, the modulation process involves switching or keying the amplitude, phase or frequency of the carrier according to the incoming digital data. As the modulating signal consists of only two levels, the digital carrier modulation techniques are known as Amplitude Shift Keying (ASK), Phase Shift Keying (PSK) and Frequency Shift Keying (FSK).
Various Shift Keying Systems
Modulation Techniques:

In amplitude shift keying, the amplitude of the carrier is switched between two levels. ON and OFF. The resultant waveform consists of ON pulse representing binary '1' and OFF pulse representing binary '0'. The ASK waveform is shown in Figure.

In frequency shift keying, the frequency of the carrier is switched between to frequencies. The resultant wave form consists of high frequency carrier representing binary '1' and low frequency carrier representing binary '0'. The FSK waveform is shown in Figure (a).

If the carrier phase is switched between two levels, we get phase shift keying. The PSK waveform is shown in Figure (c). In PSK and FSK methods and amplitude of the carrier remains constant.

All the above systems use a pair of signals to represent binary symbols 1 and 0. So, these systems are known as Binary Shift Keying systems. (BPSK, BFSK, BASK). Generally, the modulator produces one signal of an available set of M = 2N district signals in response to N bits of source data at a time. This type of digital modulation is known as M - ray shift keying system. (M - ray PSK, M - ray FSK, M - ray ASK). Binary phase shift keying is a special case of M - ray shift keying with M = 2 and N = 1. In ASK, PSK and FSK, a single parameter of the carrier i.e., amplitude, frequency and phase of the carrier undergoes modulation. Sometimes, the amplitude and phase of the carrier are varied to produce single modulation techniques, amplitude phase shift keying (APSK). This type of techniques is known as hybrid modulation technique.

Types of Pulse Modulation

Pulse Modulation is divided into two as Analog and Digital.

Analog Modulation is divided into Pulse Amplitude Modulation, Pulse Time Modulation, Pulse Width Modulation or Pulse Duration Modulation and Pulse Position Modulation

Digital Modulation is divided into Pulse Code Modulation and Delta Modulation. All the Pulse Modulations are explained below with the help of diagram.

The types of Pulse Modulation are shown in the Figure below


In pulse amplitude modulation (PAM), the information, signal is sampled at the sampling rate and the carrier pulse In pulse is discretely amplitude - modulated at the sampling frequency. For faithful transmission, the signal should It-119 The information is transferred to e sampled sufficiently. the pulse at the sampling frequency.

PAM is not often used because the amplitude of pulses not constant and goes against the basic advantage of NI is system. Therefore, in a practical PAM system, the pulses are made to perform frequency modulation of the carrier rather than amplitude modulation.

PAM Generation :

A PAM modulator circuit is shown in Figure. This circuit is a simple emitter follower. In the absence of clock signal, the output follows the input. The modulating signal is applied as the input signal. Another input to the base of the transistor is the clock signal. The frequency of the clock signal is made equal to the desired carrier pulse train frequency. The amplitude of the clock signal is so chosen that the high level is at ground (OV), and the low level is at some negative voltage which is sufficient to bring the transistor in the cut-off region. Thus when the clock signal is high, the circuit behaves as an emitter follower, and the output follows the input modulating signal. When the clock signal is low, the transistor is cut-off and the output is zero.

Thus the output waveform, shown in Figure is the desired PAM signal.

PAM Detection:

A PAM demodulator circuit is just an envelope detector followed by a low pass filter. The diode and R - C combination network act as the envelope detector. This is followed by a second order OP-AMP low pass filter (LPF) to have a good filtering characteristic. Thus, for the received PAM Signal as the input signal, the desired demodulated signal is show in figure is the output.
PAM Modulator

In this system, we have fixed amplitude and starting time of each pulse, but the width of each pulse is made proportional to the amplitude of the signal at that instant.
PWM Generation:

The clock signal of the desired frequency is applied flow which negative edge triggering pulses are derived with the help of a diode and a R1 – C1 combination which works as differentiator.

These negative trigger pulses applied to the pin 2 of IC 555, which is working as the monostable multivibrator makes an excellent voltage-to-time converter. These pulses decide the starting time of the PWM pulses. The end of pulses depends upon R2 - C2 combination on the modulating signal applied at pin 5.

The width of the pulses depends upon the value of the modulating signal and thus the pin 3 output is the desired PWM.

PDM uses in communication:

1. It is often used in the high powered audio amplifiers used to modulate AM transmitters.
2. It has also been used for telemetry systems.
3. Though still an analog mode, it is more robust than PAM because it is insensitive to amplitude changes due to noise and distortion.
4. PWM still works if synchronization between transmitter and receiver fails.

Disadvantages of PDM :

1. PDM pulses are of varying width and therefore of varying power content.

PWM Modulator

Generation and demodulation of PPM

The Pulse position modulation obtained from PWM as shown in Figure in PWM, it is seen that each pulse has a leading edge and a trailing edge. Their position depends on pulse width, which is determined by the signal amplitude at the instant. In order to it obtain the PPM signal what is required is first a PWM signal and removal of the leading edges of PWM pulses.
In demodulation of PPM, it is first converted into PWM, this is done with a flip-flop or bistable multivibrator. One input of the multivibrator get trigger pulses from a local generator which is synchronized by trigger pulses gets from the transmitter, in addition to these triggers are utilized to switch off one of the stages of the flip-flop. The PPM pulse are fed to the other base of the flip-flop and switch that stage ON (actually by switching the other one OFF). The resulting PWM pulse train is then demodulated.

The PWM signal is applied to pin no.2 through the diode R1 - C1 combination. Thus the input to pin no.2 is the negative trigger pulse which corresponds o the trailing edges of PWM wave form. The IC 555 timer is working in a monostable mode and the width of the pulse is constant (governed by R2 - C2 combination).

The negative trigger pulses decide the starting time of the output pulses and thus the output at Pin no.3 is the desired pulse position modulated (PPM) signal.


PCM is the mainly used digital modulation system. In PCM the accessible range of signal voltages, separated into levels, and each is given a binary number. Every sample is then symbolized by the binary number representing the level adjoining to its amplitude, and this number is conveyed in serial form. In linear PCM, levels are separated by equal voltage gradations.
The number of levels available depends on the number bits used to express the sample value. The number of levels is given by

N = 2m

Here N  is the number of levels, and
m is the number of bits per sample

Problem:  Estimate the number of levels if the number of bits per sample is as follows.

(a) 8 (as in telephony)
(b) 16 (as in compact disc audio systems)

Solution :

a) The number of levels with 8 bits per sample
N = 2m = 28 = 256

b) The number of levels with 16 bits per sample N = 2m = 216 = 65,536.

This process is called 'quantizing'.

For example, if the signal amplitude is 3.2 volts at one instant it will be sent as a digit 3 only but as 3.2 volts pulse. As there are 8 levels (23) as taken in this example the bits required to transmit the levels are 3. The digit 3 will be sent as 011. That is, OPP. Where O represents no pulse and represent pulse. However, if we take yet another level 5 volts the binary digit is 101, POP this will be sent as 101 or POP. This makes the demodulation of the signal easier.

Generation of PCM Signals :

In short in the pulse code modulation process: the signal is continuously sampled. It is quantized, coded and then transmitted. Quantizing is done to the nearest standard amplitude. The code is binary number when sufficient quantizing levels are used the signal will resemble the same as is from an analog system.

A group of pulses representing a sample is termed a word. The word consists of a supervisory or signalling bit and the binary bits. So a sample is expressed by n + 1 bits where 2n is the chosen number of standard levels.

Demodulation of PCM:

The PCM signal is given to a quantizer. This stage eliminates the noise and requantizes the signal. The output of quantizer is given to a decoder. The decoder is a Digital to Analog converter. It performs the reverse operation of the encoder. The PAM signal is reconstituted. A filter is used to reject the unwanted frequency components. The output of the filter will be the original signal.

Advantages of PCM:

1. PCM is immune to noise and interference.
2. It is easy to generate PCM signals.

Disadvantages of PCM :

1. PCM requires very complex circuitry.
2. PCM requires large band width compared to analog signal.

Applications of PCM:

1. It is used for telephony.
2. It is used in data transmission.
3. It is used in space communications.

Digital Transmission System Block Diagram

Many of the signals used in modern communication are digital (for example, the codes for alphanumeric characters and the binary data used in computer programs). In addition, digital techniques are often used in the transmission of analog signals. Digitizing a signal often results in improved transmission quality with a reduction in distortion and an improvement in signal-to-noise ratio. This topic looks mainly at the digital transmission of analog signals such as voice and video signals.
In Figure, the following Digital Transmission System Block Diagrams are shown.

(a) Analog Signal and Base Band Transmission
(b) Analog Modulation using Modulation and Demodulation
(c) Digital Signal Transmitted on Digital Channel
(d) Digital Signal Transmitted by Modem
(e) Analog Signal Transmitted Digitally
(f) Analog Signal Digitized and Transmitted by Modem

The functions of Digital Transmission System with examples are defined below.

In Fig (a) an analog signal is sent over a channel with no modulation. This is an example in an ordinary public-address system consisting of a microphone (converts sound energy into electrical signal), an amplifier (amplifies electrical signal) and a loud speaker (which converts electrical signal into sound energy) using twisted pair wire as a channel.

Fig (b) shows analog transmission using modulation and demodulation. Broadcast radio and television are good examples.

Fig (c) and (d) start with a digital signal (for example, a data file from a computer)

In Fig (c) the link can handle some kind of digital pulse signal directly.

In Fig (d) the channel cannot transmit pulses directly. In these cases the digital signal has to be modulated into a carried at One end and demodulated at the other. The modem is a combination, of modulator-demodulator.

Fig (e) and Fig (f) show an analog signal that is digitized at the transmitter and converted back to analog form at the receiver. The difference between these systems is that in Fig (e) the transmission is digital, while in the transmission channel cannot carry pulses, so modulation and demodulation are required.

Measurement of Inductance

Before examining the bridge methods for measuring inductance it is important to know the general form of the AC bridge and know the general balance equation for the bridge.

(a) General form and General Equation for Balance of AC Bridge:

The general form of an AC bridge is shown in Figure. It consists of four bridge arms consisting of impedances Z1, Z2, Z3, and Z4 that are not specified. An alternating voltage is applied to the bridge at points B and D. The detector shown is a head phone set, connected between points A and C.
When the potential difference from A to C is zero the bridge will be balanced. For this condition the voltage drop across BA must be equal to the voltage drop across BC in both magnitude and phase.

Using complex notation we can write :

EAB = EBC or I1Z1 = I2Z2   -----------------1

To satisfy balance condition the currents are:

I1 = E/(Z1 + Z3)   ------------- 2
I2 = E/(Z2 + Z4)   --------------3

Substituting equations (2) and (3), in equation (1), gives
Z1 Z4 = Z2Z3  ---------------- 4

Using admittances in place of impedances :
Y1 Y4= Y2Y3 ----------------- 5

Equation (4) is the general equation for balance of the AC bridge. It can be stated from this equation that the product of impedances of one pair of opposite arms must be equal to the product of the impedances of the other pair of opposite arms, the impedances expressed in the complex notation. Representing in this manner means that both the magnitude and the phase angles of the impedances and phase angle must be taken into consideration. Equation (4) is convenient to be used when dealing with series elements of a bridge circuit. Equation (5) is used in dealing with parallel elements in the bridge.

Representing in polar form the impedances can be written as Z = Z θ; where Z represents the magnitude and θ represents the phase angle of the complex impedance. Equation (4) can be represented in the form:

(Z1 θ1)(Z4 θ4) = (Z2 θ2)(Z3 θ3)      ----------------------------6

Hence the balance condition can be represented as:

Z1Z4 θ1 + θ4 = Z2Z3 θ2 + θ3  --------------------  7

From equation (7), it can be observed that two conditions are to be satisfied simultaneously to balance an AC bridge.

The first condition is :

Z1 Z4 = Z2Z3  ------------------ 8

That is the magnitude of the impedances must be equal as shown in equation (8). The second condition is that the phase angles of the impedances should satisfy the relation:

θ1 + θ4 = θ2 + θ3 ------------------ 9

The phase angles are positive for inductive impedance and negative for capacitive impedance. Working in terms of rectangular coordinates we can express

Z1 = R1+ jX1
Z2 = R2 + jX2
Z3 = R3 + jX3

And Z4 = R4 + jX4

From equation (8), for balance

Z1 Z4 = Z2 Z3

or (R1 + jX1) (R4 + jX4) = (R2 + jX2) (R3 + jX3)
or R1 R4 – X1 X4 + j (X1 R4 + X4 R1) = R2 R3 - X2 X3 + j (X2 R3 + X3 R2) ------------------ 10

Equation (10) is a complex equation. A complex equation will be satisfied only if real and imaginary parts of each side are separately equal in the equation. Therefore for balance:

R1 R4 – X1 X4 = R2 R3 - X2 X3 --------------------- 11
also X1R4 + X4 R1 = X2 R3 + X3 R2 ----------------------12

The above two independent conditions for balance must be satisfied. To satisfy both conditions for balance and for convenience of adjustment the bridge must have two variable elements in its configuration. For better results each of the balance equations must contain one variable element alone. Then the equations are called independent equations. If the variables chosen to balance the bridge do not yield independent equations the bridge has poor convergence of balance and gives an effect called 'sliding balance '. Sliding balance is a condition of interaction between the two controls of a bridge.

In an AC bridge circuit the balance equations are independent of frequency. This is an advantage, as the frequency of the source need not be known. One point is to be noted here pertaining to the balance condition of an AC bridge. If a bridge is balanced for a fundamental frequency it should also be balanced for the harmonic frequencies also. The reason is that inductance or capacitance is frequency conscious and hence the balance made at fundamental frequency is not valid at harmonic frequencies. In order to avoid such difficulties the wave form of the source should be pure and free from harmonics. The detector can be tuned to the fundamental frequency in all such cases. Some bridges use the frequency of the source to an advantage and in such cases the purity of waveform is of prime importance.

Maxwell Bridge, Advantages and Disadvantages

The Maxwell Bridge is shown in the figure below. It measures an unknown inductance in terms of a known capacitance. The unknown inductance with its resistance is represented as LX and RX respectively. The parallel combination of the variable resistance R1 and the capacitor C1 forms a ratio arm. A variable resistance R3 and resistance R2 take the positions of the two other arms of the bridges.
Maxwell Bridge Circuit Diagram
The balance condition is obtained as shown below:

Taking  Zx = Z2Z3Y1
Z2 = R2
Z3 = R3
Y1 = 1/R1 + jωC1

Therefore Zx = Rx + jωLx = R2R3 (Y1)

= Rx + jωLx = R2R3 (1/R1jωC1)
= Rx + jωLx = ((R2R3)/R1) + R2R3 jωC1

Equating real terms:
Rx = R2R3/R1

Equating imaginary terms:
Lx = R2R3C1

Hence we have two variables R1 and C1 that appear in one of the two balance equations and hence the two equations are independent. The expression for Q factor:

Q = ωLx/R1 = ωC1R1

The Maxwell Bridge Advantages and Disadvantages are as follows.

(a) Advantages of Maxwell Bridge:

1. The two balance equations are independent if we chose R1 and C1 as variable elements.

2. The frequency does not appear in any of the two equations.

3. This bridge gives simple expression for unknown values of Lx and Rx in terms of known bridge elements. Physically R2 and R3 can be each say 10, 100, 1000 or 10,000 Ω. and their value is selected to give suitable value of product R2 R3 that appears in both the balance equations. C1 can be a decade capacitor and R1 can be a decade resistor. If the product R2 R3 = 106, then the inductance is given by
Lx = C1 x 106.
Therefore when the balance is achieved the value of C1 directly in micro-farads gives the value of inductance in henry.

4. This bridge is useful for measurement of wide range of inductance at power and audio frequencies.

(b) Disadvantages of Maxwell Bridge :

1. The bridge requires a variable standard capacitor which can be very costly if the degree of accuracy is high. In some cases fixed value of capacitance are used. The balance adjustment may be done by either varying R2 or R1. As R2 appears in both the balance equations, the balance adjustment will be difficult. Other method is to place extra resistance in series with the inductance under measurement and then varying the resistance R1.

2. This bridge can measure values of inductance of medium Q coils. The reason is that the phase angles of arms 2 and 3 put together must be equal to the phase angles of arms 1 and 4 added up and that must be equal to 0°.

To accommodate the measurement of high Q coil the phase angle of the capacitive arm must be nearly 90° (negative), as the phase angle of a high Q coil will be nearly 90°. This needs a very large value of resistance R1 which is practically not possible.

Low Q coils also cannot be measured with this bridge due to the interaction of R1 and R3. When R3 is adjusted for inductive balance, the resistive balance is upset. This is termed sliding balance. That is when once balanced with R1 and then with R3 if we try to balance again with R1 we get new balance point. The balance appears to be moving or sliding towards the final point after several adjustments. Therefore this bridge is to be balanced first with R3 for inductance balance and then with R1 for resistive balance. The process is to be repeated to get the final balance.

Hence we note that this Maxwell bridge is suitable for measuring medium Q coils alone.

Wheatstone Bridge Design and Working

Wheatstone Bridge Design and Working:

The schematic diagram of the Wheatstone bridge is shown in Figure. It consists of four resistance arms. A battery is used to supply the bridge. A sensitive ammeter or a galvanometer is used as a null detector. The magnitude of the current flowing through the null detector is dependent on the potential difference between points 'c' and 'd'. The bridge will be balanced if there is no potential difference between the points 'c' and 'd'.\
Wheatstone Bridge

The balance condition occurs when the voltage across the resistance R1 equals the voltage across the resistance R2 with reference to the positive terminal of the battery. Similarly the bridge balance will occur when the voltage across the resistance arms R1 and R4, are equal. Thus when

I1R1 = I2R2 -------------- (1)
When the null indicator indicates ‘0’

I1 = I3 = E/(R1 + R3)  ---------------- (2)

I2 = I4 = E/ (R2 + R4)  ---------------- (3)

Substituting the values of I1 and I2 in equation (1), and simplifying yields

R1 /(R1 + R3) = R2/(R2 + R4)  --------------- (4)

From equation (4), we get

R1R4 = R2R3  --------------- (5)

The above equation is the balance equation of the Wheatstone bridge.

(a) Components of the Wheatstone Bridge:

As has been mentioned above, the bridge consists of four resistance arms as shown in the Figure. Resistors R1 and R2 are named as the ratio arms. The resistance R3 is called the standard arm. There will be no resistance present in the fourth resistance arm. The unknown resistance R4 takes the place of the fourth arm.

Ratio control switch will be provided in the practical Wheatstone bridge. The switch permits switching of the resistance of the ratio arms in decade steps. The resistance of the standard arm will also be provided with a switching arrangement usually in four steps for balance adjustment. Wheatstone bridge can be used to measure resistance from 1 ohm to few mega ohm.

(b) Measurement of Unknown Value of Resistance:

The unknown resistance will be connected to the terminals that form the fourth arm of the bridge. The ratio of the ratio arms may be adjusted to suit easy balance. Using the standard arm switches the bridge will be brought to balance condition where the null indicator indicates '0'. The value of unknown resistance can be estimated from the settings of the standard arm switches. This is because we know from equation 5, that

R1R4 = R2R3

As the unknown resistance takes the place of R4 , the value of the unknown resistance Rx, can be expressed as, Rx = R3(R2/R1)

The value of resistance indicated by this bridge, is dependent on the proper calibration of the null detector, either galvanometer or sensitive current meter.

LVDT Working Principle

LVDT (Linear Variable Differential Transformer) Working Principle:

The differential transformer is an electromechanical transducer which produces an electrical output proportional to the displacement of a movable core. These are essentially variable reluctance transducers. They have an advantage over the synchro, in that there is no need for brushes and slip rings. So the friction is reduced to a minimum. The disadvantage is that it can be used over only a limited range of input position. The basic principle involved is shown in the schematic diagram below. These are called Linear Variable Differential Transformers (LVDT). The LVDT is shown in Figure.
The primary is supplied form an AC source. The two identical secondary coils are connected in phase opposition. The voltage induced in the two secondary coils is dependent upon their respective induction with the primary winding.

 This is governed by the position of the core. There exists a null position of the movable core when both secondary coils induce the same voltage i.e., e1 = e2. Then e0 the output voltage is equal to zero. A slight movement of the core upwards from this position will produce variation in the coupling and e1 will be greater than e2. Consequently e0 being equal to el - e2 will be non zero.

It is in fact proportional to the displacement x1. If the displacement takes place in the downward direction from the null position, e2 will be greater than e1. Thus there will be an output with 180°, out of phase with the voltage produced by upward motion. The differential transformers are also known as pick off. The excitation of such devices normally ranges from 0 V R.M.S. to 15 V R.M.S. The frequency range is from 60 Hz to 20 kHz,

(a) Advantages of LVDT:

1. Good linearity [linear up to 5 mm, practically 0.005% linear]
2. The resolution is infinite
3. Presents high output
4. Good sensitivity [of the order of 40 V/mm]
5. Power consumption is low
6. Rugged in construction
7. Low hysteresis         
8. Offers no friction

(b) Disadvantages:

1. To obtain fair differential output displacement required is large
2. As they are sensitive to stray magnetic fields shielding required
3. The receiving instrument must work on AC.
4. Demodulator is necessary if the receiver has to work on DC.
5. It is affected by temperature
6. It has limited dynamic response by the mass of the core and applied voltage.

Semiconductor Strain Gauge, Capacitance Microphone & Inductance Transducer

(a) Semiconductor Strain Gauge:

The special characteristic of the semiconductor strain gauge is the high sensitivity, offering gauge factors of 5 to 200. Their response is nonlinear. They are sensitive to temperature variation, The gauge is made up of semiconducting material. The constructional details of uniaxial and rosette strain gauges are shown in Figure.

The constructional details of the gauge elements depend on the strain to be measured. That is whether it is uniaxial, biaxial or multidirectional.
Constructional Details of Strain Gauge
For uniaxial working narrow and long elements are used. For multidirectional working rosette gauge elements are available.

Advantages of Semiconductor Strain Gauge :

1. High gage factor of the order of around +130
2. Excellent hysteresis characteristics less than 0.5%
3. Good frequency response up to 1012 Hz
4. Long life around 10 x 10' operations
5. Small size


1. Highly sensitive to temperature
2. Poor linearity
3. Costly

(b) Capacitance Microphone/Transducer:

The capacitance transducer which is nothing but a capacitance microphone is used for measuring the pressure in vacuum. The constructional details of capacitance transducer are shown in Figure.

Capacitance Transducer
The constructional details of the condenser microphone are shown in Figure. A stretched metal diaphragm is mounted near a metal electrode. This is the fixed electrode or static plate. There is a physical separation between the diaphragm and the fixed electrode. That is both are insulated from one another. The unit is usually housed in an air tight aluminium casing. An air cavity is provided on the rear side as shown in figure. The contacts from the diaphragm and the fixed electrodes are terminated over an isolated tag on the body of the microphone. A protective cover is provided over the diaphragm side. The capacitance offered by this type of microphone is of the order of 0.0003 µF.

Changes in pressure vibrate the diaphragm. That is they displace the diaphragm from the static position. As the diaphragm and the static plate are separated by the dielectric [Air], when the diaphragm is displaced from its static position in the thickness of the dielectric changes. As the capacitance is inversely proportional to the thickness of the dielectric, there will be change in the capacitance. This change in capacitance is proportional to the pressure variations. The change in the capacitance can be measured. So the pressure variation can be know in terms of the capacitance.

Advantages of Capacitance Transducer

1. Simple to construct
2. Compact in size
3. Not expensive


1. Its response is not linear
2. It requires advanced and complex design for practical use.

(c) Inductance Transducer:

The inductive transducer is used in the measurement of force. The change in the inductance ratio of a pair of coils or by the change of inductance in a single coil indicates the magnitude of force.

A ferromagnetic armature will be displaced by the force under the measurement. The force under consideration varies the reluctance of the magnetic circuit. The Figure shows an inductance transducer. The double coil arrangement and single coil arrangement are shown in figure.

Inductance Transducer
The diaphragm, or armature will be displaced by the applied force. This causes variation in the reluctances of the magnetic circuit. The change in value of inductance with and without displacement of the diaphragm or armature is a measure of the applied force.

Potentiometric and Resistance Transducer


A potentiometric transducer is an electromechanical device. It is a passive transducer. It requires external power source for its functioning. It contains a resistance element that is provided with a slider or wiper. The motion of the wiper may be translatory or rotational. Combination of translator and rotational motion leads to a helical movement; such potentiometers are called heliports. Figure shows the three types of potentiometric transducers.

The translatory resistive elements are linear devices [shown in Figure]. Rotational resistive devices are circular [shown in Figure].They are used for the measurement of angular displacement.

Helipots are multi turn rotational devices. They can be used for either translatory or rotational motion. Helipot is shown in Figure.

Advantages of Potentiometric Transducer:

They are economical
1. Easy to operate
2. Ability to large amplitudes of displacement
3. High efficiency
4. Large Output


I. Sufficient force is required to slide the wiper
2. Noise will be produced due to wear out


This is a type of passive transducer. The mechanical displacement is converted into a change in resistance. Such a transducer is called a strain gauge.

A strain gauge is a thin, slice like device. This slice or wafer looking strain gage, will be attached to the job to measure the strain applied to the job. Resistance wire of small diameter is used in the strain gauge. Copper Nickel alloy called Constant and is used to make the resistance wire. Etched foils are also used to form the gauge.

The gauge resistance changes with length, as the job undergoes tension or compression. The change in resistance will be proportional to the applied strain. Special wheatstones bridge is used to measure this change in resistance. Gauge factor, indicates the sensitivity of a strain gauge. It is defined as the unit change in resistance per unit change in length. It is denoted by the letter 'K'.

Gage factor K = (ΔR/R) / (Δl/l)
Where K = Gauge factor
R = Nominal gauge resistance
ΔR = Change in gage resistance. (without application of stress)
l =  Normal specimen length
Δl = Change in specimen length

It is to be noted that large stress results in a relatively very small change in resistance. Hence the bridge used for measurement of resistance must be highly sensitive.

The metallic strain gauge are formed from either resistance wire or etched foil of metal sheets. They are small in size. They can be used in high temperature applications. Their leakage is low. Foil type gauges are superior to wire types. Foil types can be used under extreme temperature conditions. They can be used with prolonged loading conditions. Self induced heat is easily dissipated in foil types. The following materials are used in the manufacture of resistance transducers (strain gauges) :

Constantan *** Nickel Alloy-Low temp Coefficient
Nichrome V *** Nickel Chrome Alloy-Static strain measurements up to 375°C
Dynaloy *** Nickel Iron Alloy-Dynamic strain applications
Stabile *** Modified Nickel Iron Alloy-Good stability
Platinum-Tungsten Alloy *** Offers high stability and large resistance to fatigue at higher temperatures. Temperature range up to 800°C.