Electrodynamometer Instrument

ELECTRODYNAMOMETER INSTRUMENT

This instrument can be used for the measurement of voltage current and with some modification it can be used for the measurement of power factor and frequency.

This instrument serves as a transfer instrument and provide same accuracy for both AC and DC.

Instrument is calibrated with a de source and then the same instrument is used without modification to measure AC. This type of instruments are called as transfer instruments.

It consist of two coils. They are

1. Fixed coil and

2. Moving coil.

Current flows through the fixed coil and produces magnetic field.

In order to give more magnetic field near centre, fixed coil is divided into two sections.

Fixed coils are wound with five wire for voltage measurement and wound with heavy wire for its usage as ammeter and wattmeter.

To reduce change in dimension, coils are varnished.

Moving coil is wound on a non-metallic former and mounted on an Aluminium spindle. In this instrument operating field is weak, so eddy current damping is not used. For damping torque air friction damping is used. Control torque is provided by two springs. Both fixed coil and moving coil are air cored this eliminates hysteresis error. Supports are made up of ceramics, moving system carries counter weight and pointer.

Even earthis magnetic field may affect the readings in DC measurement. In order to protect from external magnetic field, a casing made up to high permeability alloy is used.

Principle:

Current through the fixed coil produces a magnetic field and the moving coil also produces a magnetic field when current flous through it moving coil is placed in between the two sections of fixed coil. These two fields creates a force between the two coils. This force causes the moving system to deflect.

Torque Equation:

Flux linkages of two coils

Ψ₁ = L₁i₁ + Mi₂

Ψ₂ = L₂ i₂+ Mi₁ ……….(1)

where L₁ is the self inductance of Fixed coil..

L₂ is the self inductance of moving coil.

M is the mutual inductance between fixed coil and moving coil.

i₁ is the current through the Fixed coil.

i₂ is the current through the moving coil.

Electrical input energy, = e₁ i₁ dt + e₂ i₂ dt

= i₁ dΨ₁ + i₂ dΨ₂

where e₁ dt = dΨ₁ and e₂ dt = dΨ₂ use equation (1) here,

= i1 d (L₁i₁ + Mi₂) + i₂ d (L₂ i₂ + Mi₁)

= i₁ L₁ di₁ + i₁² dL₁ + i₁ , i₂ dM + i₁ M di₂ + i₂ L₂ di₂ + i₂² dL₂ +  i₁, i₂ dM + i₂ M di₁ ……….(2)

Energy stored in the magnetic field = ½ i₁² L₁ + ½ i₂² L₂ +  i₁ i₂ M

Change in stored energy = d ﴾½ i₁² L₁ + ½ i₂² L₂ +  i₁ i₂ M﴿

= i₁ L₁ di₁ + ﴾i₁²/2﴿ dL₁ + i₂ L₂ di₂ + ﴾i₂²/2﴿ dL₂ + i₁ M di₂ + i₂ M di₁ + i₁ i₂ dM ……..(3)

Where d ﴾½ i₁² L₁﴿ = L₁ 2i₁/2 di₁ + i₁²/2 dL₁

From the principle of conservation of energy,

Electrical energy

Supplied i.e. input = change in stored energy + mechanical work done

Mechanical work done = Electrical input energy - Change in stored energy.

Mechanical work done is T_d .dƟ; use equation (2) and (3)

T_d  dƟ = [i₁ L₁ di₁ + i₁² dL₁ + i₁ i₂ dM + i₁ M di₂ + i₂ L₂ di₂ + i₂² dL₂ +  i₁, i₂ dM + i₂ M di₁] – [i₁ L₁ di₁ + ﴾i₁²/2﴿ dL₁ + i₂ L₂ di₂ + ﴾i₂²/2﴿ dL₂ + i₁ M di₂ + i₂ M di₁ + i₁ i₂ dM]

T_d  dƟ = i₁ i₂ dM

Self inductance L₁ and L₂ are constant Therefore dL₁ and dL₂ are zero's (underlined terms are cancelled)

T_d = i₁ i₂ dM/dƟ ……….(4)

For DC Measurement:

T_d = I₁ I₂ dM/dƟ and T_c = kƟ (spring control)

For final steady deflection,

T_c = T_d

KƟ = I₁ I₂ dM/dƟ

Ɵ = I₁I₂/K . dM/dƟ

For AC measurement:

i₁ & i₂ are the instantaneous values of current carried by the two coils.

Average deflecting torque over a complete cycle, T_d = 1/T ₀∫^T T_i dt

Where 'T' is the time period for one complete cycle

T_i is the instantaneous deflecting torque

= i₁ i₂ dM/dƟ

T_d for the currents with phase angle difference o is given by,

T_d = 1/2π ₀∫^{2 π} i₁ i₂ dM/dƟ dt

= 1/2π ₀∫^{2 π} I_m1 sinωt . I_m2 sin(ωt – ϕ) d(ωt) dM/dƟ

= I_m1 I_m2 dM/dƟ

T_d = I_m1 I_m2 cosϕ dM/dƟ

T_d = I₁I₂/K cosϕ dM/dƟ

where I₁ & I₂ are the RMS value of currents in the fixed coil and moving coil respectively.

ϕ is the phase angle between the two coil currents.

T_c = kƟ

T_c for spring control.

For final steady deflection, T_c = T_d

KƟ = I₁ I₂ cosϕ dM/dƟ

Ɵ = I₁I₂/K cosϕ dM/dƟ

When AC voltage is applied, alternating current flows through the fixed coil and moving coil. When the current in the fixed coil reverses, the current in the moving coil also reverses and the Torque remains in the same direction. Scale is non uniform, in nature.

Errors:

1. To get reasonable torque, no of turns of the coil should be increased more no of turns increases the weight of the moving system, if weight increases, torque weight ratio becomes low and cause frictional error.

2. Variation of self-reactance of coil with frequency cause frequency error. To eliminate frequency error volt meter coil is made very small part of the circuit and in shunted ammeter this error is eliminated by fixed coil and moving coil with same time constant.

3. eddy current produced due to moving coil and adjacent metal parts. and causes error. This is eliminated by replacing metal parts or made them highly resistive.

4. External magnetic field cause error, this is reduced by using metal shields metal shield cause eddy current to flow. Therefore Astatic system is used in precision type instrument for eliminating magnetic field error. Astatic system consist of two moving coil, so additional field influences one moving coil in one direction and the second moving coil in opposite direction. Thus net effect is zero.

5. Temperature changes causes error. This is eliminated by temperature compensating resistor.

Advantages:

1. Accurate for frequency upto 500Hz, if astatic system is used accurate for frequency upto 10kHz.

2. Act as transfer instrument.

3. Deflecting torque is proportional to square of rms value of applied voltage and does not depends on the waveform.

4. Hysteresis error and eddy current error are absent.

Dis-advantage:

1. Low torque weight ratio

2. Cost is high

3. Non uniform scale

4. Power consumption is high

5. Frequency error occurs.

6. Temperature and stray field error occurs