Controlling Torque
The controlling torque is provided by two control springs. These springs act as leads to the moving coil.
Damping
Air-friction damping is employed for these instruments and is provided by a pair of aluminium vanes, attached to the spindle at the bottom. These vanes move in a sector-shaped chamber.
In this section, you will how the following parts of an Electrodynamometer Instruments are made.
1. Fixed Coils
2. Moving Coil
3. Control
4. Moving System
5. Damping
6. Shielding
7. Cases and Scales
Let,
i1 = instantaneous value of current in the fixed coils, (A)
i2 = instantaneous value of current in the moving coils, (A)
L1 = self-inductance of fixed coils, (H)
L2 = self-inductance of moving coil, (H)
M = mutual inductance between fixed and moving coils (H)
Flux linkage of Coil 1, ψ1 = L1 i1 + Mi2
Flux linkage of Coil 2, ψ2 = L2 i2 + Mi1
Electrical input energy = e1 i1 dt + e2 i2 dt = i1 dψ1 + i2 dψ2
As e1 = dψ1/dt and e1 = dψ2/dt
Electrical input energy = i1 d(L1 i1 + Mi2)+ i2 d(L2 i2 + Mi1)
Energy stored in the magnetic field = ½ i12L1 + ½ i22L2 + i1 i2M
Change in energy stored = d(½ i12L1 + ½ i22L2 + i1 i2M)
From the principle of conservation of energy,
Total electrical input energy = Change in energy in energy stored + mechanical energy
The mechanical energy can be obtained by subtracting above equations
Therefore, mechanical energy =½ i12dL1 + ½ i22dL2 + i1 i2dM
Now, the self-inductances L1 and L2 are constants and, therefore, dL1 and dL2 both are equal to zero.
Hence, mechanical energy = i1 i2 dM
Suppose Ti is the instantaneous deflecting torque and dθ is the change in deflection, then,
Mechanical energy = work done = Ti dθ
Thus we have,
Ti dθ = i1 i2dM
Ti = i1 i2 (dM/dθ)
1. Operation with DC
Let, I1 = current in the fixed coils, I2 = current in the moving coil
So deflecting torque Td = I1 I2 (dM/dθ)
This shows that the deflecting torque depends in general on the product of current I1 and I2 and the rate of change of mutual inductance.
This deflecting torque deflects the moving coil to such a position where the controlling torque of the spring is equal to the deflecting torque. Suppose θ be the final steady deflection.
Therefore controlling torque Tc = kθ where k = spring constant (N-m/rad)
At final steady position Td = Tc
I1 I2 (dM/dθ) = kθ
Deflection, θ = (I1 I2/k)(dM/dθ)
If the two coils are connected in series for measurement of current, the two currents I1 and I2 are equal.
Say, I1 = I2 = I
Thus, deflection of the pointer is θ = (I2/k)(dM/dθ)
For DC use, the deflection is thus proportional to the square of the current and hence the scale non-uniform and crowded at the ends.
2. Operation with AC
Let, i1 and i2 be the instantaneous values of the current carried by the coils. Therefore, the instantaneous deflecting torque is:
Ti = i1 i2 (dM/dθ)
If the two coils are connected in series for measurement of current, the two instantaneous currents i1 and i2 are equal.
Say, i1 = i2 = i
Thus, instantaneous torque on the pointer is Ti = i2 (dM/dθ)
Thus, for ac use, the instantaneous torque is proportional to the square of the instantaneous current. As the quantity i2 is always positive, the current varies and the instantaneous torque also varies. But the moving system due to its inertia cannot follow such rapid variations in the instantaneous torque and responds only to the average torque.
The average deflecting torque over a complete cycle is given by:
where T is the time period for one complete cycle.
At final steady position Td = TC
Thus, deflection of the pointer is
Deflection is thus a function of the mean of the square of the current.
If the pointer scale is calibrated in terms of the square root of this value, i.e. square root of the mean of the square of current value, then RMS value of the AC quantity can be directly measured by this instrument.
3. Sinusoidal Current
If currents i1 and i2 are sinusoidal and are displaced by a phase angle j, i.e.
i1 = im1 sin ωt and i2 = Im1 sin(wt – j)
The average deflecting torque
where I1 and I2 are the RMS values of the currents flowing through the coils.
At equilibrium, Td = Tc
As was in the case with ac measurement, with sinusoidal current also the deflection is a function of the mean of the square of the current.
If the pointer scale is calibrated in terms of the square root of this value, i.e. square root of the mean of the square of current value, then the RMS value of the ac quantity can be directly measured by this instrument.
In this section, you will learn three types of electrodynamometer instruments.
1. Electrodynamic Ammeter
In an electrodynamic ammeter, the fixed and moving coils are connected in series as shown in the figure. A shunt is connected across the moving coil for limiting the current.
Electrodynamometer ammeter
The reactance–resistance ratio of the shunt and the moving coil is kept nearly the same for the independence of the meter reading with the supply frequency.
Since the coil currents are the same, the deflecting torque is proportional to the mean square value of the current. Thus, the scale is calibrated to read the RMS value.
2. Electrodynamic Voltmeter
The electrodynamic instrument can be used as a voltmeter by connecting a large noninductive resistance (R) of low-temperature coefficient in series with the instrument coil.
Electrodynamometer voltmeter
3. Electrodynamic Wattmeter
The electrodynamic wattmeter consists of two fixed coils ‘a’ and ‘b’ placed symmetrical to each other and producing a uniform magnetic field. They are connected in series with the load and are called the Current Coils (CC).
Electrodynamometer wattmeter
The two fixed coils can be connected in series or parallel to give two different current ratings. The current coils carry the full-load current or a fraction of full load current. Thus the current in the current coils is proportional to the load current.
Basic Arrangement of an Electrodynamometer Wattmeter
The moving coil ‘c’, in series with a high non-inductive resistance Rv is connected across the supply. Thus the current flowing in the moving coil is proportional to, and practically in phase with the supply voltage. The moving coil is also called the voltage coil or Pressure Coil (PC).
The voltage coil is carried on a pivoted spindle which carries the pointer, the pointer moved over a calibrated scale.
Two hairsprings are used for providing the controlling torque and for leading current into and out of the moving coil. Damping is provided by air friction.
4. Torque Equation
Let’s derive the torque equation of an electrodynamometer instrument.
Let, if = current in the fixed coil
im = current in the moving coil
i = load current
v = load voltage
Tin = instantaneous value of the deflecting torque
p = instantaneous power
Tin α if im
Since if α i and im α v
Tin α vi α p
Thus, the instantaneous value of the deflecting torque is proportional to the instantaneous power. Owing to the inertia of the moving system, the pointer reads the average power.
In dc circuits, the power is given by the product of voltage and current, and hence the torque is directly proportional to the power. Thus, the instrument indicates the power.
For AC, the instrument indicates the average power. This can be proved as follows:
Tin α vi
Average deflecting torque × average power
Let, v = V, sin d
I = I m sin (θ – Ф)
Average deflecting torque α average value of Vm sin d × I m sin (θ – Ф) α VI cos θ
If Td be the average torque, then
Td α VI cos θ α true power = kP
where P is the true power and k is the constant.
For spring control Tc = ks θ1
where Tc is the control torque, ks is the spring constant and θ1 is the angle of deflection of the pointer.
For steady deflection,
Tc = Td
ks θ1 = kP
θ1 = (k/ks)P
θ1 α P
Hence, in the case of ac also the deflection is proportional to the true power in the circuit. The scale of the electrodynamometer wattmeter is therefore uniform.
The advantages of electrodynamometer type instruments are
The advantages of electrodynamometer type instruments are
The various errors in electrodynamometer instruments are,
1. Torque to weight ratio
2. Frequency errors
3. Eddy current errors
4. Stray magnetic field error
5. Temperature error
The range of electrodynamometer ammeter and voltmeter are given below.
Ammeter Range
Voltmeter Range – up to 750 V
The comparison of PMMC, moving iron and electrodynamometer type instruments is summarized in the Table
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