Essentials of indicating instruments

In this section we will discuss certain features which are common to all electrical measuring instruments.
We will first consider various torques acting on its moving system. In an indicating instrument, it is
essential that the moving system is acted upon by three distinct torque (or forces) for satisfactory
working. There torques are:

1. A deflecting or operating torque, Td
2. A controlling torque, Tc
3. A dampling torque, Tv.

Deflecting (Or the Operating) Torque

The deflecting torque, causes the moving system of the instrument to move from its zero position. It
may be produced by utilizing any one of the effects of current or voltage in the instrument such as
magnetic effect, electromagnetic induction effect, heating effect, electrostatic effect etc. The actual
method of producing a deflecting torque depends upon the type of the instruments.
The deflecting torque has to supply the following torque-components presents in an instrument.
(a) The torque required to overcome the torque due to the inertia of the moving system,
J (d2θ/dt2), where J is the moment of inertia and θ is the movement (rotation in radians).
(b) The torque required to overcome the controlling torque, Tc (≡ kcθ).
(c) The torque required to overcome the damping torque, Tv v
k d
dt
= θ   
 
, where kv is damping
torque constant.
(d) The torque requirred to overcome the frictional (coulomb) torque. This component is minimized
by appropriate design considerations.

Controlling Torque

The controlling torque developed in an instrument has two functions:
(a) It limits the movement of the moving system and ensures that the magnitude of the deflections
always remains the same for a given value of the quantity to be measured.
(b) It brings back the moving system to its zero position where the quantity being measured is
removed or made zero.
The controlling torque is dependent on the magnitude of deflection produced. The moving system
is deflected from zero to such a position that the controlling torque at that deflected position is equal to
the deflecting torque. The controlling torque increases in magnitude with the deflection till it balances
the deflecting torque. That is, for a steady deflection,
Controlling torque, Tc = Deflection or operating torque, Td ...(12.10)
The controlling torque is entered in all commercial instruments by any one of the following
three ways.
• By means of one or two coiled springs. The corresponding insturment is termed spring
controlled instruments (mostly used system).
• By the action of gravity due to suitably placed weights on the moving system. Such instruments
are known as gravity controlled instruments.
• By means of a permanent magnet (magnetic control system).
Spring control is now almost universal in indicating instruments. Gravity control is employed in a
few cases, notably in special laboratory types, and magnetic control is applied to some galvanometers
and certain moving iron instruments (the polarized form). We will discuss the first two methods of
obtaining the controlling torque in a measuring instrument as given below.
Spring Control
Figure 12.1(a) shows a spindle free to turn between two pivots. The moving system is attached to the
spindle. Two phosphor-bronze hair springs A and B wound in opposite directions are also shown whose
inner ends are attached to the spindle. The outer end of spring A is connected to a leaver which is
pivoted the adjustment of which gives zero setting. However, the outer end of B is fixed.
When the pointer is deflected one spring unwinds itself while the other is twisted. This twist in the
spring produces restoring (controlling) torque, which is proportional to the angle of deflection of the
moving systems.
Let E be the young-modulus for the material of the spring and θ (radians) be the deflection of the
moving system to which one end of the spring is attached. Then, the controlling torque developed in the
spiral spring is given by
Tc =
2
12
Ebt
l
θ ...(12.11)
or TC = ks θ ...(12.12)
where l = Total length of spring strip (m)
b = depth of the strip (m)
t = thickness of the strip (m)
ks = spring constant
The controlling spring must meet the following requirements:
• The stress developed in the spring must be well below the elastic limit of the spring material at
the maximum deflection of the moving system. This is essential to avoid fatigue and to preserve
stability over a long period. For this, we must have
l
r
=
2 max
E
S
θ
...(12.13)
where Smax = maximum stress which must not exceed. For a full scale deflection θ = 90°, the
ratio l/t is about 3000 in a good instruments.
• It springs are used as leads of current to the instrument, their cross-sectional area must be
sufficient to carry the current without overheading them failing which the consistency will be
impaired. The spring material should also have the following properties:
* It should have low resistance
* The temperature coefficient should also be low.
• The springs must be of non-magnetic material.
In a permanent magnet moving coil type instrument the deflecting torque is proportional to the
current passing through them. Thus the operating torque, Td , is directly proportional to the
current,
Td = KI ...(12.14)
Then for spring control instrument, the controlling torque, TC, is
Tc = Ksθ ...(12.15)
The pointer comes to rest when the deflecting torque (Td ) and the controlling or restoring
torque (Tc ) are equal, i.e., Td is equal and opposite to Tc.
At equilibrium, Td = Tc
Therefore, KI = Ksθ
∴ I = KS
K
θ ...(12.16)
This equation shows that the current is directly proportional to the deflection and since
Eqn. (12.16) is a linear relation, the scale with spring controlled instrument for deflecting
torque given by Eqn. (12.14) will be uniform throughout the scale.

Gravity Control

In gravity controlled instruments a small adjustable weight is attached to the
spindle of the moving system such that the deflecting torque produced by the instrument has to act
against the action of gravity. Thus a controlling torque is obtained. This weight is called the control
weight. Another adjustable weight is also attached is the moving system for zero adjustment and balancing
purpose. This weight is called Balance weight.
When the control weight is in vertical position as shown in Fig. 12.2 (a), the controlling torque is
zero and hence the pointer must read zero. However, if the deflecting torque lifts the controlling weight
from position A to B such that the spindle rotates by an angle θ, then due to
gravity a restoring (or controlling) torque is exterted on the moving system.

The controlling (or restoring) torque, Tc, is given by
Tc = Wl sin θ = kg sin θ ...(12.17)
where W is the control weight; l is the distance of the control weight from the axis of rotation of the
moving system; and kg is the gravity constant.
Equation (12.18) shows the controlling torque can be varied quite simply by adjustment of the
position of the control weight upon the arm which carries it.
Again, if the deflecting torque is directly proportional to the current, I i.e.,
Td = kI ...(12.18)
We have at the equilibrium position
Td = Tc
or kI = kg sin θ
or I = g k
k
sin θ (12.19)
This relation shows that current I is proportional to sin θ and not θ. Hence in gravity controlled
instruments the scale is not uniform. It is cramped for the lower readings, instead of being uniformly
divided, for the deflecting torque assumed to be directly proportional to the quantity being measured.
Advantanges of Gravity Control
1. It is cheap and not affected by temperature variations.
2. It does not deteriorate with time.
3. It is not subject to fatigue.

Disadvantages of Gravity Control

1. Since the controlling torque is proportional to the sine of the angle of deflection, the scale is
not uniformly divided but cramped at its lower end.
2. It is not suitable for use in portable instruments (in which spring control is always preferred).
3. Gravity control instruments must be used in vertical position so that the control weight may
operate and also must be leveled otherwise they will give zero error.
In view of these reasons, gravity control is not used for indicating instruments in general and
portable instruments in particular.


Damping Torque

We have already seen that the moving system of the instrument will tend to move under the action of the
deflecting torque. But on account of the control torque, it will try to occupy a position of rest when the
two torques are equal and opposite. However, due to inertia
of the moving system, the pointer will not come to rest
immediately but oscillate about its final deflected position
as shown in Fig. 12.3 and takes appreciable time to come
to steady state.
To overcome this difficulty a damping torque is to be
developed by using a damping device attached to the moving
system. The damping torque is proportional to the speed of
rotation of the moving system, that is
Tv = kv d
dt
θ
where kv = damping torque constant
d
dt
θ = speed of rotation of the moving system
Depending upon the degree of damping introduced in the moving system, the instrument may have
any one of the following conditions as depicted in Fig.12.3.

1. Under damped condition: The response is oscillatory
2. Over damped condition: The response is sluggish and it rises very slowly from its zero
position to final position.
3. Critically damped condition: When the response settles quickly without any oscillation, the
system is said to be critically damped.

In practice, the best response is slightly obtained when the damping is below the critical value i.e.,
the instrument is slightly under damped.
The damping torque is produced by the following methods:

Air Friction Damping

In this type of damping a light vane or vanes having considerable area is attached to the moving system
to develop a frictional force opposing the motion by reason of the air they displace. Two methods of
damping by air friction are depicted
• The arrangement shown in Fig. 12.4(a) consists of a light aluminium vane which moves in a
quadrant (sector) shaped air chamber. The chamber also carries a cover plate at the top. The
vane is mounted on the spindle of the moving system. The aluminium vane should not touch
the air-chamber walls otherwise a serious error in the deflection of the instrument will be
introduced. Now, with the motion, the vane displaces air and thereby a damping force is
created on the vane that produces a torque (damping) on the spindle. When the movement is
quicker the damping force is greater; when the spindle is at rest, the damping force is zero.
• The arrangement of Fig.12.4 (b) consists of a light aluminium piston which is attached to the
moving system. This piston moves in a fixed chamber which is closed at one end. Either
circular or rectangular chamber may be used. The clearance (or gap) between the piston and
chamber walls should be uniform thorughout and as small as possible. When the piston
moves rapidly into the chamber the air in the closed space is compressed and the pressure of
air thus developed opposes the motion of the piston and thereby the whole moving system. If
the piston is moving out of the chamber, rapidly, the pressure in the closed space falls and the
pressure on the open side of the piston is greater than that on the opposite side. Motion is thus
again opposed. With this damping system care must be taken to ensure that the arm carrying
the piston should not touch the sides of the chamber during its movement. The friction which
otherwise would occur may introduce a serious error in the deflection.
The air friction damping is very simple and cheap. But care must be taken to ensure that the piston
is not bent or twisted. This method is used in moving iron and hot wire instruments.

Fluid Friction Damping

• This form is damping is similar to air friction damping. The action is the same as in the air
friction damping. Mineral oil is used in place of air and as the viscosity of oil is greater, the
damping force is also much greater. The vane attached to the spindle is arranged to move in
the damping oil.
• It is rarely used in commercial type instruments.
• The oil used must fulfill the following requirements.
* It should not evaporate quickly
* It should not have any corrosive effect on metals.
* Its viscosity should not change appreciably with temperature.
* It should be good insulator.

Advantages of Fluid Friction Damping

1. The oil used for damping can also be used for insulation purpose in some forms of instruments
which are submerged in oil.
2. The clearance between the vanes and oil chamber is not as critical as with the air friction
clamping system.
3. This method is suitable for use with instruments such as electrostatic type where the movement
is suspended rather than pivoted.
4. Due to the up thrust of oil, the loads on bearings or suspension system is reduced thereby the
reducing the frictional errors.
Disadvantages of Fluid Friction Damping
1. The instruments with this type of damping must be kept always in a vertical position.
2. It is difficult to keep the instrument clean due to leakage of oil.
3. It is not suitable for portable instruments.
The fluid friction damping can be used for laboratory type electrostatic instruments.

Eddy Current Damping

Eddy current damping is the most efficient form of damping. The essential components in this type of
damping are a permanent magnet; and a light conducting disc usually of alumninum.
When a sheet of conducting material moves in a magnetic field so as to cut through lines of force,
eddy currents are set up in it and a force exists between these currents and the magnetic field, which is
always in the direction opposing the motion. This force is proportional to the magnitude of the current,
and to the strength of field. The former is proportional to the velocity of movement of the conductor,
and thus, if the magnetic field is constant, the damping force is proportional to the velocity of the
moving system and is zero when there is no movement of the system.
Figure 12.6 shows two methods of applying this method of damping. In Fig. 12.6(a) a thin disc of
conducting, but non-magnetic material-usually copper of aluminium is mounted on the spindle which
carries the pointer of the instrument. When the spindle rotates, the edge of the disc cuts through the
lines of force in the gap of a permanent magnet, and eddy currents, with consequent damping, are
produced. An arrangement similar to this is often used in hotwire instruments.

Figure 12.6(b) shows the essential parts of a permanent-magnet, moving coil, instrument. The
coil is wound on a light metal former in which eddy currents are induced when the coil moves in the
permanent-magnet field. The directions of the eddy-current which in turn produce the damping torque
due to the motion of the coil (clockwise) are as shown in Fig.12.6(b) and this will produce damping
forces as indicated in the figure.

Electromagnetic Damping

• The movement of a coil in a magnetic field produces a current in the coil which interacts with
the magnetic field to produce a torque. This torque opposes the movement of the coil and
slows the response.
• The magnitude of the current and hence the damping torque is dependent upon the resistance
of the circuit which the instrument is connected.
• This damping method is used in galvanometers.

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CLASSIFICATION OF MEASURING INSTRUMENTS

Electrical measuring instruments may be classified into two groups:
(a) Absolute (or primary) instruments.
(b) Secondary instruments.

Absolute Instruments

• These instruments give the value of the electrical quantity in terms of absolute quantities (or
some constants) of the instruments and their deflections.
• In this type of instruments no calibration or comparison with other instruments is necessary.
• They are generally not used in laboratories and are seldom used in practice by electricians and
engineers. They are mostly used as means of standard measurements and are maintained lay
national laboratories and similar institutions.
• Some of the examples of absolute instruments are:
* Tangent galvanometer
* Raleigh current balance
* Absolute electrometer.


 Secondary Instruments

• They are direct reading instruments. The quantity to be measured by these instruments can be
determined from the deflection of the instruments.
• They are often calibrated by comparing them with either some absolute instruments or with
those which have already been calibrated.
• The deflections obtained with secondary instruments will be meaningless untill it is not calibrated.
• These instruments are used in general for all laboratory purposes.
• Some of the very widely used secondary instruments are: ammeters, voltmeter, wattmeter,
energy meter (watt-hour meter), ampere-hour meters etc.
Classification of Secondary Instruments

(a) Classification based on the various effects of electric current (or voltage) upon which their
operation depend. They are:

• Magnetic effect: Used in ammeters, voltmeters, watt-meters, integrating meters etc.
• Heating effect: Used in ammeters and voltmeters.
• Chemical effect: Used in dc ampere hour meters.
• Electrostatic effect: Used in voltmeters.
• Electromagnetic induction effect: Used in ac ammeters, voltmeters, watt meters and
integrating meters.
Generally the magnetic effect and the electromagnetic induction effect are utilized for the construction
of the commercial instruments. Some of the instruments are also named based on the above effect such
as electrostatic voltmeter, induction instruments, etc.

(b) Classification based on the Nature of their Operations

We have the following instruments.
• Indicating instruments: Indicating instruments indicate, generally the quantity to be
measured by means of a pointer which moves on a scale. Examples are ammeter, voltmeter,
wattmeter etc.
• Recording instruments: These instruments record continuously the variation of any
electrical quantity with respect to time. In principle, these are indicating instruments but so
arranged that a permanent continuous record of the indication is made on a chart or dial.
The recording is generally made by a pen on a graph paper which is rotated on a dice or
drum at a uniform speed. The amount of the quantity at any time (instant) may be read
from the traced chart. Any variation in the quantity with time is recorded by these instruments.
Any electrical quantity like current, voltage, power etc., (which may be measured lay the
indicating instruments) may be arranged to be recorded by a suitable recording mechanism.
• Integrating instruments: These instruments record the consumption of the total quantity
of electricity, energy etc., during a particular period of time. That is, these instruments
totalize events over a specified period of time. No indication of the rate or variation or the
amount at a particular instant are available from them. Some widely used integrating
instruments are: Ampere-hour meter: kilowatthour (kWh) meter, kilovolt-ampere-hour
(kVARh) meter.

(c) Classification based on the Kind of Current that can be Measurand.

Under this heading, we have:
• Direct current (dc) instruments
• Alternating current (ac) instruments
• Both direct current and alternating current instruments (dc/ac instruments).

(d) Classification based on the method used.

Under this category, we have:
• Direct measuring instruments: These instruments converts the energy of the measured
quantity directly into energy that actuates the instrument and the value of the unknown
quantity is measured or displayed or recorded directly. These instruments are most widely
used in engineering practice because they are simple and inexpensive. Also, time involved in
the measurement is shortest. Examples are Ammeter, Voltmeter, Watt meter etc.
• Comparison instruments: These instruments measure the unknown quantity by comparison
with a standard. Examples are dc and ac bridges and potentiometers. They are used when
a higher accuracy of measurements is desired.

(e) Classification based on the Accuracy Class of Instruments.

Groups of error in the measured quantity for instruments of various class of accuracy are
listed below:

Class of accuracy   0.2    0.5     1.0      1.5      2.5     5

Limit of error %   ± 0.2  ± 0.5  ± 1.0  ± 1.5   ± 2.5  ± 5

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DEFINITIONS OF IMPORTANT TERMS

Measurement work employs a number of terms which are defined below:
Measurand: The quantity or variable being measured is called measurand or measurement variable.
Accuracy: It is defined in terms of the closeness with which an instrument reading approaches
the true or expected (desired) value of the variable being measured.
Precision: It is measure of the consistency of reproducibility (repeatability) of the measurement
(i.e., the successive reading do not differ). For a given fixed value of an input variable, precision is a
measure of the degree to which successive measurement differ from one another.
Sensitivity: It is defined by the change in the output or response of the instrument for a unit
change of input or measured variable.
Resolution: Resolution is the smallest change in a measured variable (or measurand) to which the
instrument will respond.
True or Expected Value: The true or expected value of a quantity to be measured may be defined
as the average of an infinite number of measured values when the average deviation due to the various
contributing factors tends to zero. It also refers to a value of the quanity under consideration that would
be obtained by a method (known as exemplar method) agreed upon by experts. In other words, it is the
most probable value that calculations indicate and one should expect to measure.
Note that the value of the unknown obtained by making use of primary standards and measuring
instruments is considered to be its ture value.
Error: It is the deviation of the measured (or indicated) value from the true (or expected) value of
a quantity. In other words, error is the difference between the measured value and the true value of the
unknown quantity. It is also called absolute error are maximum possible error. Then error of measurements
is given by
∈A = Am – At ...(12.1)
where Am = measured value of the quantity
At = true value of the quantity
Absolute error, ∈0, is the limit of error in measurement. In other words, ∈A must not be higher
than ∈0.
Thus, |∈0| = max |Am – At| ...(12.2)
Note that the absolute error does not give any information about accuracy. For example, an error
of (–1) volt in measurement of 1000 volt is negligible, but the same error in measurement of 10 volts is
never acceptable. Thus, error is expressed in terms of another term called the relative error which is the
ratio of absolute error of the ture value of the quantity being measured. Therefore, the relative error, ∈R
is given by
∈R = Absolute error
True value
= 0
At
∈
= (Am – At)/ At ...(12.3)
The percentage relative error % ∈R = ∈R × 100.
Also, from Eqn. (12.3), we have
(1 + ∈r ) At = Am
or At =
1 [ ]
m
R
A
+ ∈
...(12.4)
If the absolute error ∈A is sufficiently small, then Eqn. (12.1) shows that
∈A = Am – At ≈ 0
or At ≅ Am ...(12.5)
That is, Am may be substituted for At in Eqn. (12.3) for practical purpose. Now in view Eqn. (12.3)
becomes ∈R = 0
Am
∈
...(12.6)
Correction: The difference between the true value and the measured value of the sought quantity
is defined as the reading correction or simply correction. That is, correction is negative or error. Thus,
δC = – ∈A ...(12.7)
= At – Am ...(12.8)
or At = expected value = Am + δC ...(12.9)
Therefore addition of correction in measured value gives the true (or accurate or expected) value.
Bandwidth: The bandwidth of an instrument relates to the maximum range of frequency over
which it is suitable for use. It is normally quoted in terms of 3 dB (dB = decibel) point. For an amplifier,
it is the range of frequencies between which the gain or amplitude ratio is constant to within 3 dB (this
corresponds 30% reduction in gain).
Significant Figures: An indication of the precision of the measurement is obtained from the
number of significant figures in which it is expressed.
Significant figures convey actual information regarding the magnitude and the measurement precision
of a quantity. The more is the significant figures, the greater will be the precision of measurement. For
example, if a resistor is specified as having a resistance of 105 Ω and 105.3 Ω, than in 105 Ω there are
three significant figures whereas in 105.3 Ω there are four. The later with more significant figures,
expresses a measurement of greater precision than the former.

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Introduction to Measurements

Measurements are the basic means of acquiring knowledge about the parameters and variables involved
in the operation of a physical system. Measurement generally involves using an instrument as a physical
means of determining a quantity or variable. An instrument or a measuring instrument is, therefore,
defined as a device for determining the value or magnitude of a quantity or variable. The electrical
measuring insturment, as its name implies, is based on electrical principles for its measurement function.
These days a number of measuring instruments, both analog as well as digital ones, are available
for the measurement of electrical quantities like voltage, current, power energy, frequency, power
factor, etc. The instruments considered in this book are analog devices in which the output or display is
a continuous-time signal and bears a fixed relationship to the input.
Analog instruments may be divided into three groups:
(a) Electromechanical instruments;
(b) Electronic instruments which are often constructed by the addition of electronic circuits to
electromechanical indicators thus increasing their sensitivity and input impedances; and
(c) Graphical instruments which are electromechanical and electronic instruments having a modified
display arrangement so that a graphical trace, that is, a display of instantaneous values against
time is obtained.
This chapter presents general concepts related to the working principles and construction and
certain features common to many electrical indicating instruments particularly of electromechanical
types. Important definitions relevant to instruments will be discussed first.

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Clippers

A circuit which removes the peak of a waveform is known as a clipper. A negative clipper is shown in Figure below. This schematic diagram was produced with Xcircuit schematic capture program. Xcircuit produced the SPICE net list Figure below, except for the second, and next to last pair of lines which were inserted with a text editor.

*SPICE 03437.eps * A K ModelName D1 0 2 diode R1 2 1 1.0k V1 1 0 SIN(0 5 1k) .model diode d .tran .05m 3m .end

Clipper: clips negative peak at -0.7 V.

During the positive half cycle of the 5 V peak input, the diode is reversed biased. The diode does not conduct. It is as if the diode were not there. The positive half cycle is unchanged at the output V(2) in Figure below. Since the output positive peaks actually overlays the input sinewave V(1), the input has been shifted upward in the plot for clarity. In Nutmeg, the SPICE display module, the command “plot v(1)+1)” accomplishes this.

V(1)+1 is actually V(1), a 10 Vptp sinewave, offset by 1 V for display clarity. V(2) output is clipped at -0.7 V, by diode D1.

During the negative half cycle of sinewave input of Figure above, the diode is forward biased, that is, conducting. The negative half cycle of the sinewave is shorted out. The negative half cycle of V(2) would be clipped at 0 V for an ideal diode. The waveform is clipped at -0.7 V due to the forward voltage drop of the silicon diode. The spice model defaults to 0.7 V unless parameters in the model statement specify otherwise. Germanium or Schottky diodes clip at lower voltages.

Closer examination of the negative clipped peak (Figure above) reveals that it follows the input for a slight period of time while the sinewave is moving toward -0.7 V. The clipping action is only effective after the input sinewave exceeds -0.7 V. The diode is not conducting for the complete half cycle, though, during most of it.

The addition of an anti-parallel diode to the existing diode in Figure above yields the symmetrical clipper in Figure below.

*SPICE 03438.eps D1 0 2 diode D2 2 0 diode R1 2 1 1.0k V1 1 0 SIN(0 5 1k) .model diode d .tran 0.05m 3m .end

Symmetrical clipper: Anti-parallel diodes clip both positive and negative peak, leaving a ± 0.7 V output.

Diode D1 clips the negative peak at -0.7 V as before. The additional diode D2 conducts for positive half cycles of the sine wave as it exceeds 0.7 V, the forward diode drop. The remainder of the voltage drops across the series resistor. Thus, both peaks of the input sinewave are clipped in Figure below. The net list is in Figure above

Diode D1 clips at -0.7 V as it conducts during negative peaks. D2 conducts for positive peaks, clipping at 0.7V.

The most general form of the diode clipper is shown in Figure below. For an ideal diode, the clipping occurs at the level of the clipping voltage, V1 and V2. However, the voltage sources have been adjusted to account for the 0.7 V forward drop of the real silicon diodes. D1 clips at 1.3V +0.7V=2.0V when the diode begins to conduct. D2 clips at -2.3V -0.7V=-3.0V when D2 conducts.

*SPICE 03439.eps V1 3 0 1.3 V2 4 0 -2.3 D1 2 3 diode D2 4 2 diode R1 2 1 1.0k V3 1 0 SIN(0 5 1k) .model diode d .tran 0.05m 3m .end

D1 clips the input sinewave at 2V. D2 clips at -3V.

The clipper in Figure above does not have to clip both levels. To clip at one level with one diode and one voltage source, remove the other diode and source.

The net list is in Figure above. The waveforms in Figure below show the clipping of v(1) at output v(2).

D1 clips the sinewave at 2V. D2 clips at -3V.

There is also a zener diode clipper circuit in the “Zener diode” section. A zener diode replaces both the diode and the DC voltage source.

A practical application of a clipper is to prevent an amplified speech signal from overdriving a radio transmitter in Figure below. Over driving the transmitter generates spurious radio signals which causes interference with other stations. The clipper is a protective measure.

Clipper prevents over driving radio transmitter by voice peaks.

A sinewave may be squared up by overdriving a clipper. Another clipper application is the protection of exposed inputs of integrated circuits. The input of the IC is connected to a pair of diodes as at node “2” of Figure above . The voltage sources are replaced by the power supply rails of the IC. For example, CMOS IC's use 0V and +5 V. Analog amplifiers might use ±12V for the V1 and V2 sources.

• REVIEW
• A resistor and diode driven by an AC voltage source clips the signal observed across the diode.
• A pair of anti-parallel Si diodes clip symmetrically at ±0.7V
• The grounded end of a clipper diode(s) can be disconnected and wired to a DC voltage to clip at an arbitrary level.
• A clipper can serve as a protective measure, preventing a signal from exceeding the clip limits

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Classification of Clippers

Classification[edit source | editbeta]

Clippers may be classified into two types based on the positioning of the diode. ^[1]

• Series Clippers, where the diode is in series with the load resistance, and
• Shunt Clippers, where the diode in shunted across the load resistance.

The diode capacitance affects the operation of the clipper at high frequency and influences the choice between the above two types. High frequency signals are attenuated in the shunt clipper as the diode capacitance provides an alternative path to output current. In the series clipper, clipping effectiveness is reduced for the same reason as the high frequency current passes through without being sufficiently blocked.

Clippers may be classified based on the orientation(s) of the diode. The orientation decides which half cycle is affected by the clipping action.

The clipping action can be made to happen at an arbitrary level by using a biasing elements (potential sources) in series with the diode.

• Positively Biased Diode Clipper
• Negatively Biased Diode Clipper

The signal can be clipped to between two levels by using both types of diode clippers in combination. ^[2] This clipper is referred to as

• Combinational Diode Clipper or Two-Level Clippers

The clamping network is the one that will "clamp" a signal to a different dc level. The network must have capacitor, a diode, and a resistive element, but it also employs an independent dc supply to introduce an additional shift.

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Intro-clipper

In electronics, a clipper is a device designed to prevent the output of a circuit from exceeding a predetermined voltage level without distorting the remaining part of the applied waveform.

A clipping circuit consists of linear elements like resistors and non-linear elements like junction diodes or transistors, but it does not contain energy-storage elements like capacitors. Clipping circuits are used to select for purposes of transmission, that part of a signal wave form which lies above or below a certain reference voltage level.

Thus a clipper circuit can remove certain portions of an arbitrary waveform near the positive or negative peaks. Clipping may be achieved either at one level or two levels. Usually under the section of clipping, there is a change brought about in the wave shape of the signal.

Clipping Circuits are also called as Slicers, amplitude selectors or limiters. Using square waveform it is easier to analyze the clipper network than sinusoidal waveform,because in square waveform only two level (i.e.two DC level) have to be considered

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Second Order Transient Analysis

Second Order Transient Analysis

How do recognize it?

A transient problem is one that asks you to find voltage (or current) vs. time. It also includes some instant event: a switch opens or closes, the power supply turns on, a part is suddenly pulled out the circuit, a fuse blows. Your answer can either be a formula for v(t) or i(t) or it might be a graph of v or i on the vertical axis and time on the horizontal axis.
Second-order problems have two independent energy storage elements (capacitors and/or inductors). Note that two capacitors in parallel or series only count as one capacitor because you can find the equivalent capacitance (same with inductors).

How do you do it?

1. For first-order, the only form possible was an exponential response. For second-order there are three possible forms. You can compute which one based on the following three constants that are coefficients of the differential equation that describes these types of circuits:

   Constant	Series RLC	Parallel RLC
   a	1	1
   b	R/L	1/(RC)
   c	1/(LC)	1/(LC)

   Note that in each case, R is the Thevenin resistance seen by the LC pair. If the LC pair is in parallel but the Thevenin R is in series, you may need to convert to a Norton so that all the elements are in parallel.
2. Now compare three constants to determine which of the three response forms you have:
   • if b^2 > 4*a*c then you have overdamped.
   • if b^2 = 4*a*c then you have critically damped.
   • if b^2 < 4*a*c then you have underdamped.

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First-Order Transient Analysis

First-Order Transient Analysis

How do recognize it?

A transient problem is one that asks you to find voltage (or current) vs. time. It also includes some instant event: a switch opens or closes, the power supply turns on, a part is suddenly pulled out the circuit, a fuse blows. Your answer can either be a formula for v(t) or i(t) or it might be a graph of v or i on the vertical axis and time on the horizontal axis.
First-order problems have only one energy storage element (either a capacitor or inductor, but not both). Note that two capacitors in parallel or series only count as one capacitor because you can find the equivalent capacitance (same with inductors).

How do you do it?

1. Find the inital condition (t=0+)
   1. Find the current for each inductor just before the switch is flipped (t=0-). Current through an inductor will remain the same for the instant just after the switch is flipped.
   2. Find the voltage over each capacitor just before the switch is flipped (t=0-). Voltage over a capacitor will remain the same the instant just after the switch is flipped.
   3. In order to find the currents and voltages suggested above, first determine if you are given the values in the problem or you have a formula for the current/voltage from a previous step. If you are not given any other method for finding the current/voltage, then you can usually assume that steady-state conditions exist up to the time the switch is flipped. This will simply the circuit considerably, since inductors look like shorts and capacitors look like opens in steady-state.
   4. Once you have the currents through each inductor and the voltage over any capacitors for the time just before the switch is flipped, then draw the circuit for just after the switch is flipped (t=0+). Each inductor will look like a current source (at its initial current), and each capacitor will look like a voltage source (at its initial current). Note that if the initial current on an inductor is 0 amps, then it will look like an open for that instant, and if the initial voltage on a capacitor is 0 volts, then it will look like a short for that instant. If your variable of interest is not the cap voltage or the inductor current, then use your normal circuit analysis techniques to find the variable of interest at the initial instant after the switch is flipped.
2. Find the final condition (t=infinity)
   1. A long time after the switch is flipped, the inductors will look like shorts and the capacitors will look like opens. Find the variable of interest (voltage/current) for that simplified circuit.
3. Find the time constant for the time between initial and final conditions ( 0 <= t <= infinity)
   1. The time constant for a circuit with a single capacitor is RC, where R is the Thevenin Resistance seen by the capacitor after the switch has flipped. The time constant for a circuit with a single inductor is L/R where R is the Thevenin Resistance seen by the inductor after the switch has flipped.
   2. For a circuit with more than one (equivalent) capacitor and/or inductor does not have a simple exponential response, but the time constant is still part of the generalized second order response. For a circuit with energy storage elements in series, the time constant is L/R, and for a parallel combination the time constant is RC. In both cases, R is the Thevenin Resistance seen by the energy storage elements.
4. Write down the final answer.
   1. For a circuit with a single energy storage element (first order transient), the formula is:
      
      where the "unknown" is the desired voltage or current (as a function of time), "final" is the final value of the unknown, "initial" is the initial value of the unknown, "t" is the time variable, and tau is the time constant. Note that for t=0, unknown = intial and for t=infinity, unknown = final.

What are some difficulties I might run into?

1. If the desired value is not the current through an inductor or the voltage over a cap, then you can still use this procedure, but the initial and final values must be derived based on what the inductor and capacitors do under instantaneous change and under steady-state. That is, you must "back up" to your desired value after seeing what the caps and inductors do in your circuit.
2. If there is more than one switch, treat each switch one at a time in order of the time they change. You can probably assume steady-state conditions before the first switch flip (unless you are told otherwise). Ignore the following switch flips when computing the final value (since the circuit doesn't know another switch flip is coming). For all the other switch flips, you must use your formula to compute what the initial conditions will be. The time constant will be different for each segment of time in between switch flips, since the cap and/or inductor will see a different Thevenin resistance depending on the switch positions.

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Continuity condition of Inductors

Continuity condition of Inductors
The current that flows through a linear inductor must always be a continuous. , the voltage across the inductor is not proportional to the current flowing through it but to the rate of change of the current with respect to time,()ditdt. The voltage across the inductor () is zero when the current flowing through an inductor does not change with time. This observation implies that the inductor acts as a short circuit under steady state dc current. In other words, under the steady state condition, the inductor terminals are shorted through a conducting wire. Alternating current (ac), on the other hand, is constantly changing; therefore, an inductor will create an opposition voltage polarity that tends to limit the changing current. If current changes very rapidly with time, then inductor causes a large opposition voltage across its terminals. If current changes through the inductor from one level to another level instantaneously i.e. in sec., then the voltage across it would become infinite and this would require infinite power at the terminals of the inductor. Thus, instantaneous changes in the current through an inductor are not possible at all in practice. Lv0dt􀀑
Remark-2: (i) The current flowing through the inductor cannot change instantaneously (i.e. just right before the change of current = (0)i−(0)i+ just right after the change of current). However, the voltage across an inductor can change abruptly. (ii) The inductor acts as a short circuit (i.e. inductor terminals are shorted with a conducting wire) when the current flowing through the inductor does not change (constant). (iii) These properties of inductor are important since they will be used to determine “boundary conditions”.

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Introduction to Transient Analysis

So far we have considered dc resistive network in which currents and voltages were independent of time. More specifically, Voltage (cause input) and current (effect output) responses displayed simultaneously except for a constant multiplicative factor (VR). Two basic passive elements namely, inductor →→RI=×()L and capacitor () are introduced in the dc network. Automatically, the question will arise whether or not the methods developed in lesson-3 to lesson-8 for resistive circuit analysis are still valid. The voltage/current relationship for these two passive elements are defined by the derivative (voltage across the inductor C()()LLditvtLdt=, where =current flowing through the inductor ; current through the capacitor ()Lit()()CCdvtitCdt=, = voltage across the capacitor) or in integral form as ()Cvt0011()()(0)()()(0)ttLLLCitvtdtiorvtitdtvLC=+=+∫∫ rather than the algebraic equation (VIR=) for all resistors. One can still apply the KCL, KVL, Mesh-current method, Node-voltage method and all network theorems but they result in differential equations rather than the algebraic equations that we have considered in resistive networks
An electric switch is turned on or off in some circuit (for example in a circuit consisting of resistance and inductance), transient currents or voltages (quickly changing current or voltage) will occur for a short period after these switching actions. After the transient has ended, the current or voltage in question returns to its steady state situation (or normal steady value). Duration of transient phenomena are over after only a few micro or milliseconds, or few seconds or more depending on the values of circuit parameters (like ,,RLandC).The situation relating to the sudden application of dc voltage to circuits possessing resistance (R), inductance (L), and capacitance (C) will

now be investigated in this lesson. We will continue our discussion on transients occurring in a dc circuit. It is needless to mention that transients also occur in ac circuit but they are not included in this lesson.

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Network Theorms to AC

SUPERPOSITION THEOREM

A circuit is linear when superposition theorem can be used to obtain its currents and voltages. When this theorem is applied to an ac circuit, it has to be remembered that the voltage and current sources are in the phasor form and the passive elements are impedances.

WORKED EXAMPLE 1:

A circuit is presented in Fig. 30. The task is to find the current through the load current, using superposition theorem.

The values of components are specified by equation (99). We can find the contribution due to one voltage source, by replacing the other voltage by a short circuit. Only the ideal part of the source is to be replaced by a short-circuit, and its internal impedance should be left in the circuit. For example, we can view Z₁ and Z₂ as the internal impedance of sources, V_A and V_B respectively. The application of superposition theorem is illustrated by the sketch in Fig. 31.

The equations obtained are shown below. The current supplied source can be obtained after determining the impedance seen by the source. The impedance seen by source V_A is the sum of impedance Z₁ and the parallel value of Z₂ and Z_L. Once the source current is determined, current I_L1 through impedance Z_L can be found out using the current division rule. In the same way, we can find current I_L2 through impedance Z_L due to source V_B acting alone. Using superposition theorem, we get the load current as the phasor sum of I_L1 and I_L2.

The answer can be obtained much more easily by using nodal analysis. The voltage across the load Z_L can be found out by solving a single KCL equation. The purpose here is to illustrate the application of superposition theorem. Its direct application is not always handy, bit its importance lies in that the entire analysis of linear circuits rests on it.

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MAXIMUM POWER TRANSFER THEOREM

In some low power circuits, the need for maximum power transfer tends to be more important than the need for efficiency. For example, extraction of maximum signal from a weak source such as a transducer may be necessary. Another example is the case of impedance matching, where the output impedance of an amplifier is to be matched with that of the load such as a loudspeaker. On the other hand, efficiency is the important criterion for a voltage source with low output impedance. For example, efficiency is of utmost importance in electrical power generation and transmission, and it is over 80% for electrical power generation units. Under the maximum power transfer condition, efficiency is no more than 50 %.

When maximum power transfer is to be brought about in ac circuits, the result depends upon type of load. At first, a circuit with resistive load is taken up for study. In the circuit in Fig 32a, the load is purely resistive, designated as R . The source has source impedance containing source resistance, R_G, and positive source reactance X_G. A typical ac voltage source has this type of internal impedance.

When the load is purely resistive, maximum power transfer to the load resistance occurs when the load resistance equals the magnitude of the source impedance. This point is proved as follows.

We consider the case when the load is purely resistive. From Fig. 32a, current supplied by the source can be obtained as shown by equation (105). Then power delivered to the load is obtained as shown by equation (106). In equation (106), the current is expressed as the ratio of source voltage over impedance. A factor of half is present, because the source voltage is represented by its peak value. To find the value of load resistance at which maximum power transfer occurs, equation (106) is differentiated with respect to load resistance R_L, and the derivative is equated to zero. Then the value of load resistance R_L at maximum power transfer can be obtained.

From equation (106), the derivative obtained is shown by equation (107). The formula used for obtaining the derivative is shown below equation (108). On setting the derivative to zero, we obtain the value of load resistance R_L as shown by equation (108). When the load resistance equals the magnitude of source impedance, the power transferred to the load is maximum. That the power transferred is the maximum and not the minimum can be verified by evaluating the second derivative with respect to load resistanceR_L and finding it to be negative. Equation (108) states the condition for maximum power transfer when the load is purely resistive.

When the load impedance is complex, the condition for maximum power transfer is obtained as follows.

From Fig. 32b, the load current can be obtained as shown by equation (109). Then power delivered to the load is obtained as shown by equation (1110). First let us consider the case when the load resistance R_L is fixed and the load reactance X_L is variable.

When the load resistance R_L is fixed and the load reactance X_L is variable, maximum power transfer occurs when equation (111) is satisfied. In this case, the maximum power transferred is obtained from equation (112). If the source impedance has inductive reactance, then the load impedance should contain capacitive reactance. In this case, suffix L is a pointer to load reactance. It is not a pointer to inductive reactance.

If both the load resistance R_L and the load reactance X_L are variable, one of the conditions for maximum power transfer is specified by equation (113). Then the power delivered to the load is obtained as shown by equation (114). When load resistance R_L is variable, equation (114) is differentiated with respect to load resistance R_L. The derivative obtained is shown next.

The derivative obtained is expressed by equation (115). As shown by equation (116), the derivative is zero, when the load resistance equals the source resistance. The maximum power transfer to load occurs when equation (116) is satisfied. In this case, the maximum power transferred to the load is obtained from equation (117).

The condition for maximum power transfer is specified by equation (118), when both the load resistance R_L and the load reactanceX_L are variable . That is, the load impedance is the conjugate of the source impedance and the power factor of the circuit is unity. Since the source impedance normally has inductive reactance, the load impedance should contain capacitive reactance. In this case, suffix L is a pointer to load reactance. It is not a pointer to inductive reactance.

WORKED EXAMPLE 2:

An example is presented now to illustrate the application of maximum power transfer theorem.

For the circuit in Fig. 33, find the value of load resistance at which maximum power transfer occurs and find the maximum power transferred to the load. The values of components used are specified below.

The solution is as follows:

At first, the circuit containing the source, the resistors, R₁ and R₂ and the capacitance should be replaced by its Thevenin's equivalent circuit. Then the results of maximum power transfer theorem can be applied. In short, the procedure is as follows. Get Thevenin's Equivalent Circuit. Then load resistance R_L equals the magnitude of Thevenin's impedance.

From the circuit in Fig. 33, the open-circuit voltage is the voltage across the capacitor, as shown by equation (120). Next we find the Thevenin's impedance by replacing source voltage V_S by a short-circuit. Then the Thevenin's impedance is obtained as shown by equation (121). The load resistance equals the magnitude of Thevenin's impedance and its value is shown by equation (122).

The Thevenin's equivalent circuit is shown by the circuit in Fig. 34. Note that Thevenin's equivalent circuit does not contain the load resistance. Some worked examples are presented next, to illustrate the use of network theorems and the use of mesh and nodal analysis.

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reciprocity theorem.

In many electrical network it is found that if positions ofvoltage source and ammeter are interchanged, the reading of ammeter remains same. It is not clear to you. Let’s explain in details. Suppose avoltage source is connected to a passive network and an ammeter is connected to other part of the network to indicate the response. Now any one interchanges the positions of ammeter and voltage source that means he or she connects the voltage source at the part of the network where the ammeter was connected and connects ammeter to that part of the network where the voltage source was connected. The response of the ammeter means current through the ammeter would be same in both cases. This is where the property of reciprocity comes in circuit. The particular circuit which has this reciprocal property is called reciprocal circuit. This type of circuit perfectly obeys reciprocity theorem.

The voltage source and the ammeter used in this theorem must be ideal. That means the internal resistance of both voltage source and ammeter must be zero. The reciprocal circuit may be a simple or complex network. But every complex reciprocal passive network can be simplified to a simple network. As per reciprocity theorem in a linear passive network, supply voltage V and output current I are mutually transferable. The ratio of V and I is called the transfer resistance. The theorem can easily be understood by this following example

Reciprocity Theorem

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Tellegen theorem

This theorem has been introduced in the year of 1952 by Dutch Electrical Engineer Bernard D.H. Tellegen. This is very useful theorem in network analysis. According to Tellegen theorem the summation of instantaneous powers for the n number of branches in an electrical network is zero. Are you confused ? Let’s explain. Suppose n number of branches in an electrical network have i1, i2, i3, ………….inrespective instantaneous currents through them. These currents satisfyKirchhoff current law. Again, suppose these branches have instantaneous voltages across them are v1, v2, v3, ……….. vn respectively. If these voltages across these elements satisfy Kirchhoff Voltage law then,

Where vk is the instantaneous voltage across the kth branch and ik is the instantaneous current flowing through this branch. Tellegen theorem is applicable mainly general class of lumped networks consists of linear, non-linear, active, passive, time variant and time variant elements. This theorem can easily be explained by the following example.

In the network shown, arbitrary reference directions have been selected for all of the branch currents, and the corresponding branch voltages have been indicated, with positive reference direction at the tail of the current arrow. For this network, we will assume a set of branch voltages satisfy the Kirchhoff voltage law and a set of branch current satisfy Kirchhoff current law at each node. We will then show that these arbitrary assumed voltage and currents satisfy the equation

and it is the condition of Tellegen theorem,

In the network shown in the figure, let v1, v2 and v3 be 7, 2 and 3 volts respectively. Applying Kirchhoff voltage law around loop ABCDEA. We see that v4 = 2 volt is required. Around loop CDFC, v5 is required to be 3 volt and around loop DFED, v6 is required to be 2. We next apply Kirchhoff current law successively to nodes B, C and D.

At node B let ii = 5 A, then it is required that i2 = − 5 A. At node C let i3 = 3 A and then i5 is required to be − 8. At node D assume i4 to be 4 then i6 is required to be − 9. Carrying out the operation of equation,

we get, 7 X 5 + 2 X ( − 5) + 3 X 3 + 2 X 4 + 3 X ( − 8) + 2 X ( − 9) = 0
Hence Tellegen theorem is verified.

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