OHM'S LAW

GEORGE SIMON OHM

In the year 1827, a German Physicist, George Simon Ohm, published a pamphlet entitled "Die galvanische kette mathematisch bearbeitet" meaning "The Galvanic circuit investigated mathematically".

This pamphlet contained the results of one of the first efforts to measure current and voltage and to describe and relate them mathematically. One result was a statement of the fundamental relationship we now call 'Ohm's law'. Ohm's law is the most fundamental law in Electrical engineering. It states that

" The potential difference between the two ends of a conductor is directly proportional to the current flowing through it, provided its temperature and other physical parameters remain unchanged". That is,

\[V\alpha I\] or \[V=RI\]

The constant of proportionality R is called the resistance of the conductor. Its unit is ohm($\Omega $).

Another way of stating ohm's law is $I\alpha V$ or $I=GV$.

The constant of proportionality G is called the conductance of the conductor. It is the reciprocal of resistance and its SI unit is Siemens(S).

OHM'S LAW IN GRAPHICAL FORM - 

OHM'S LAW IN GRAPHICAL FORM

Ohm's law in graphical form is shown in the adjacent figure. The voltage is shown as independent variable(cause) and current as dependent variable(effect). The slope of the line is the reciprocal of resistance (1/R), and is called conductance.

A conductor showing straight line V-I characteristics is said to have a linear resistance.

SHORT CIRCUIT AND OPEN CIRCUIT - 

If the V-I characteristic graph reads,

Then V=0, and R=0 with I$\ne $0 . It represents a short circuit and is represented by the circuit as shown in the figure given below.

If the V-I characteristic graph reads,

Then V$\ne $0, I=0 and R= $\infty $ . It represents an open circuit and is represented by the circuit as shown in the figure given below.

    Ohm's law alone is not enough to analyze all the complex circuits in electrical. There are some more laws such as Kirchhoff's law which help in better analyzing the circuits. These laws will be discussed in the further posts. 

INDUCTOR

An inductor is a passive element designed to store energy in its magnetic field. They find numerous applications in electronics and power systems. For example, they are used in transformers, radios, TVs, radars and electric motors.

Any conductor of electric current has inductive properties and can be thought of as an inductor. But in order to enhance the inductive effect, a practical inductor is usually formed into a cylindrical coil with many turns of conducting wire as shown in the figure.

It was in the early 1800s, that English experimentalist Michael Faraday and American inventor Joseph Henry discovered almost simultaneously that a changing magnetic field could produce a voltage in a neighboring circuit. They showed that this voltage was proportional to the time rate of change of current which produced the magnetic field. That constant of proportionality is called inductance and is denoted by L.

\[v=L\frac{di}{dt}\]

The unit of inductance is Henry (H).

Now, what is Inductance?

Inductance is the property of the coil by virtue of which it opposes any change in the current flowing through it.

The inductance of an inductor depends on its physical dimensions and construction. Inductance of a solenoid (inductor) is given by

\[L=\frac{\mu {{N}^{2}}A}{l}\]

Where,    N = number of turns

               l = length

               A = cross-sectional area

               $\mu $= permeability of the core

Inductance can be increased by increasing N,$\mu $, and A or by reducing l.

TYPES OF INDUCTORS-

Inductors are available in different types ranging from large high current iron- cored chokes to tiny low current coils.

Air-cored coils are wound on a tubular insulating material such as cardboard, fiber, hard rubber, Bakelite etc. Such coils find use in electronic circuits working at high frequencies. Their inductance values are in millihenry (mH) and microhenry ($\mu $H) range.

Solenoidal wound inductor

Toroidal inductor

Chip inductor

CAPACITOR

A capacitor is a passive element designed to store energy in its electric field. It generally consists of two conducting plates separated by an insulator (or dielectric).

WHAT HAPPENS IN A CAPACITOR?

When a voltage source ‘v’ is connected to a capacitor, the source deposits positive charge q on one plate and a negative charge –q on the other plate. The capacitor is said to store the electric charge. The amount of charge stored is given by

                                                                      \[q=Cv\]

Where,    C – constant of proportionality

               v – Voltage applied

The unit of capacitance is farad (F), in honor of English physicist Michael Faraday (1791-1867). For a parallel plate capacitor, capacitance is given by

                                                           \[C=\in \frac{A}{d}\]

Where,    A- surface area of the plates

               d- Distance between the plates

               $\in $- Permittivity of the material

TYPES OF CAPACITORS –

Capacitors are of two types based on their functioning:-

1. Fixed
2. Variable

The symbols for fixed and variable type capacitors are shown in the figure below.

CIRCUIT SYMBOLS FOR FIXED AND VARIABLE CAPACITOR

The figure below shows common types of fixed value capacitors. Polyester capacitors are light in weight, stable and their change with temperature is predictable. Other dielectrics such as mica or polystyrene may be used. Film capacitors are rolled and housed in metal or plastic films. Electric capacitors produce very high capacitances.

FIXED CAPACITORS:- (a) Polyester (b) Ceramic (c) Electrolytic

The figure below shows some common types of variable capacitors. The capacitance of a trimmer capacitor is varied by turning the screw. It is often placed in parallel with another capacitor so that equivalent capacitance can be varied slightly. The capacitance of the variable capacitor is varied by turning the shaft.

VARIABLE CAPACITORS:- (a) Trimmer (b) Filmtrim

USES OF CAPACITOR –

Besides resistors, capacitors are the most common electrical components. They are used extensively in electronics, communication, computers, and power systems. They are used in tuning circuits and as dynamic memory elements in computer.

Capacitors are also used to block dc, pass ac, shift phase, start motors and suppress noise.

ENERGY STORED IN A CAPACITOR –

We know that,   $q=Cv$

Differentiating this, we get  $\frac{dq}{dt}=C\frac{dv}{dt}$      or      $i=C\frac{dv}{dt}$     (Since, $i=\frac{dq}{dt}$ )

Integrating this, $v=\frac{1}{C}\int\limits_{-\infty }^{t}{idt}$

Or 

                        \[v=\frac{1}{C}\int\limits_{{{t}_{0}}}^{t}{idt+v({{t}_{0}})}\]

V(t₀) is the voltage across the capacitor at time t₀. This shows that the capacitor voltage depends on the past history of the capacitor current.

Instantaneous power delivered to the capacitor is $p=vi=Cv\frac{dv}{dt}$

Therefore, energy stored is

\[w=\int\limits_{-\infty }^{t}{pdt}=C\int\limits_{-\infty }^{t}{v\frac{dv}{dt}dt}=C\int\limits_{-\infty }^{t}{vdv}=\frac{1}{2}C{{v}^{2}}\]

 V ($-\infty $) =0, because the capacitor was not charged at t=$-\infty $

Thus,

                      \[w=\frac{1}{2}C{{v}^{2}}\text{ or }w=\frac{{{q}^{2}}}{2C}\]

PROPERTIES OF A CAPACITOR –

1. When the voltage across a capacitor is not changing with time (i.e. dc voltage), the current through the capacitor is zero.A capacitor is an open circuit to dc. With dc voltage connected across a capacitor, it charges.
2. The voltage on the capacitor must be continuous. The voltage on the capacitor cannot change abruptly because an abrupt change would mean an infinite current, which is physically impossible. Conversely, the current through a capacitor can change instantaneously.

   

   VOLTAGE ACROSS A CAPACITOR (a) possible (b) not possible

3. An ideal capacitor does not dissipate energy.
4. A real, non-ideal capacitor has a parallel-model leakage resistance as shown in figure. The leakage resistance may be as high as 100M$\Omega $  and can be neglected for most practical applications.

   

   CIRCUIT MODEL OF A NON-IDEAL CAPACITOR

MEASURING INSTRUMENTS - AMMETER AND VOLTMETER

Measuring instruments are an important part of electrical engineering and they are essential for analyzing the electrical circuits. Some of the important instruments which are used occasionally are ammeter, voltmeter, wattmeter, energy meter etc. But in this article we are about to discuss the two most fundamental of all - Ammeter and Voltmeter. Ammeters and voltmeters are generally classed together because of the similarities in their operating principles, with some exceptions of course!

WHAT IS AN AMMETER?

An ammeter is an electrical measurement instrument which is used to measure current through a point or wire in a circuit. It is connected in series with the circuit whose current is to be measured. Therefore, they should have a low electrical resistance. This is essential in order that they cause a small voltage drop and consequently absorb small power.

Ammeter can also be made from a galvanometer by placing a shunt resistance in parallel with the galvanometer.

WHAT IS A VOLTMETER?

A voltmeter is an electrical measurement instrument which is used to measure voltage across two given points in a circuit. It is connected in parallel with the circuit or the part of the circuit whose voltage is to be measured. They should have high electrical resistance so that the current drawn by them is small and ultimately the power absorbed is small.

voltmeter can also be made from a galvanometer by placing a high resistance in series with the galvanometer.

Ammeters and voltmeters can be of different types based on their construction and working.

TYPES OF AMMETERS AND VOLTMETERS –

  1. Moving iron type (both for AC/DC)

      (a)  Attraction type

      (b) Repulsion type

  2. Moving coil type

      (a)  Permanent magnet type (for DC only)

      (b) Electrodynamic or dynamometer type (for DC/AC)

  3. Hot wire type (both for DC/AC)

  4. Induction type (for AC only)

      (a)  Split phase type

      (b) Shaded pole type

  5. Electrostatic type (for voltmeters only) – Both DC/AC

MOVING IRON TYPE –

In moving iron type, there are two basic types, which are:

(a)  Attraction type

(b) Repulsion type

For both type of these instruments, the necessary magnetic field is produced by the ampere turns of a current carrying coil. As we are talking about ammeter; the coil has comparatively fewer turns of thick wire so that the ammeter has low resistance because it is to be connected in series with the circuit.

ATTRACTION TYPE INSTRUMENT –

The figure below shows the constructional details of an attraction type moving iron instrument.

ATTRACTION TYPE MOVING IRON INSTRUMENT

The coil is flat disc or a sector eccentrically mounted. When the current flows through the coil, a magnetic field is set up and the moving iron moves from a region of weaker field to region of stronger magnetic field, thereby deflecting the pointer over a calibrated scale.

The controlling torque is provided by springs but can also be gravity controlled. Damping is provided by air friction usually by a vane moving in a sector shaped chamber.

REPULSION TYPE INSTRUMENT –

REPULSION TYPE MOVING IRON ONSTRUMENT

In repulsion type, there are two iron vanes inside the coil; one fixed and other movable. When the current flows through the coil, these vanes get similarly magnetized and there is force of repulsion between the two vanes resulting in the movement of moving vane, and therefore the pointer.

Why moving iron can be used in both AC and DC?

The function of moving iron instruments depends on the attraction or repulsion of iron vanes. Thus, they are unpolarised i.e. they are independent of the direction in which the current flows. Therefore, they can be used both in AC as well as DC.

DEFLECTION PRODUCED –

The deflection in a moving iron instrument is given by

                        \[\theta =\frac{1}{2}\frac{{{I}^{2}}}{K}\frac{dl}{d\theta }\]

Thus, $\theta \propto {{I}^{2}}$

As, the deflection is proportional to the square of the current, it is evident that the scale of such an instrument is non-uniform.

If there is no saturation, the change of inductance with the angle of deflection is uniform i.e. $\frac{dl}{d\theta }$ is constant. Thus, the scale can be easily laid as the measured quantity is proportional to the square root of deflection.

SOURCES OF ERROR –                                                                      

     (a)  Errors with both AC and DC work –

          (i)  Errors due to hysteresis -  

             Because of hysteresis in the iron parts of the moving system, readings are higher            for descending values and lower for ascending values. This can be completely              eliminated by using Mu metal or Perm-alloy, which has negligible hysteresis loss. 

         (ii)  Errors due to stray fields –

               Unless shielding is not done from external fields, the results obtained may be                  wrong. Thus, the instrument is shielded with cast iron.                              

     (b) Errors with AC work –

     Changes of frequency produce change in the impedance of the coil and change in          the magnitude of AC currents.

ADVANTAGES - 

• Cheap and robust.
• Can be used both in AC and DC.

DISADVANTAGE –

• They cannot be calibrated with DC because of the effect of hysteresis in the iron      vanes. Hence, they are usually calibrated by comparison with AC standard.

MOVING COIL INSTRUMENTS

Moving coil instruments can be classified into two types:

       1. Permanent magnet moving coil instrument (for DC)

       2. Dynamometer type (for both AC and DC)

PERMANENT MAGNET MOVING COIL –

PRINCIPLE OF OPERATION –

The operation of a permanent magnet moving coil instrument is based upon the principle that when a current carrying conductor is placed in a magnetic field, it experiences a force or torque which moves it.

CONSTRUCTION –

The instrument consists of a permanent magnet and a rectangular coil of many turns wound on a light aluminium frame inside which is an iron core. A soft iron cylinder is fixed between the magnetic poles which help to

(i) Make the field radial.

(ii) Decrease the reluctance of air path between the poles.

The rectangular coil on the iron core is attached to a light pointer. The sides of the coil are free to move in the air gaps between the poles.

CONTROL –

The controlling torque is provided by two phosphor bronze hair springs. These springs also lead the current in and out of the coil.

The control torque can also be provided by using a ribbon suspension and it is considered to be an advantageous method as it eliminates bearing friction.

DAMPING –

Damping torque is produced by the movement of the aluminium frame moving in magnetic field of the magnet.

DEFLECTING TORQUE AND DEFLECTION –

The deflecting torque produced in the coil is given by

${{T}_{d}}$ = Force * perpendicular distance

     $=NBIl\times b$$=NBIA$ N-m.

Where, N = no. of turns in coil

            B = flux density in Wb/m²

            I = current in the coil

            A = area of the coil

$\Rightarrow {{T}_{d}}\propto I$

Also, controlling torque ${{T}_{c}}\propto \theta $ (Because such instruments are spring controlled)

As T_d = T_c  $\Rightarrow \theta \propto I$

Therefore, such instruments have uniform scales.

Damping is electromagnetic i.e. by eddy currents induced in metal frame over which coil is wound.

ADVANTAGES –

• They have low power consumption.
• Their scales are uniform.
• They have high torque-to-weight ratio.
• No hysteresis loss.
• Effective and efficient eddy current damping.
• They are not much affected by stray magnetic fields, due to the presence of strong permanent magnets.

DISADVANTAGES –

•      Costlier as compared to moving iron instruments.
•      Errors occur with the ageing of instrument.

ELECTRODYNAMOMETER TYPE INSTRUMENTS

An electrodynamometer type instrument is a moving coil instrument in which the operating field is produced by another coil which is fixed. This type of instrument can be used either as an ammeter or as a voltmeter, but is generally used as a wattmeter.

ELECTRODYNAMIC INSTRUMENT AS AN AMMETER

ELECTRODYNAMIC INSTRUMENT AS A VOLTMETER

Electrodynamic instruments are also capable of functioning as transfer instruments. Besides, their use as an ammeter, voltmeter, and wattmeter; they are also used to transfer calibration of working instruments.

CONSTRUCTION –

The instrument consists of a fixed coil and a moving coil. The fixed coil is usually air-cored to avoid hysteresis effects when used on AC circuits. Fixed coils are wound with fine wire for use as a voltmeter.

But, if the instrument is to be used as an ammeter or wattmeter, then the fixed coils are wound with heavy wire carrying the main current.

 The moving coil is mounted on an aluminium spindle. It is wound either as self sustaining coil or else on a non-metallic former so as to prevent eddy currents. Moving coils are also air-cored.

CONTROLLING TORQUE –

Controlling torque is provided by two control springs. These springs act as leads to the moving coil.

DAMPING –

Air friction damping is provided by aluminium vanes attached to the spindle at the bottom.

DEFLECTING TORQUE –

Let the currents passing through fixed and moving coils be I₁ and I₂ respectively. Since, there is no iron, field strength and hence flux density is proportional to I₁.

                \[\therefore B=K{{I}_{1}}\] (K is constant)

Force on each side of the moving coil$=NB{{I}_{2}}l$

Torque produced on whole of the coil$=NB{{I}_{2}}l\times b=NB{{I}_{2}}A$

Which can be written as, ${{T}_{d}}=NK{{I}_{1}}{{I}_{2}}A$ N-m                                 (By putting B=KI₁ )

$\Rightarrow {{T}_{d}}\propto {{I}_{1}}{{I}_{2}}$

It shows that the deflecting torque is proportional to the product of currents flowing in the fixed and the moving coil.

Since, the instrument is spring controlled; the restoring or controlling torque is proportional to angular deflection$\theta $.

                                                                              

                                            \[{{T}_{c}}\propto {{K}_{2}}\theta \]

As, the deflecting torque is equal to the controlling torque, $\Rightarrow \theta \propto {{I}_{1}}{{I}_{2}}$ . The deflection produced in the instrument is proportional to the product of currents flowing in the fixed and the moving coil.

ADVANTAGES –

• These instruments are free from hysteresis and eddy current losses.
• They can be used on both AC and DC.
• They are used as transfer instruments.

DISADVANTAGES –

• Low torque/weight ratio and hence low sensitivity.
• Costlier than PMMC and moving iron type.
• Non-uniform scale.