General principles of measuring current and voltage

General principles of measuring

Current and voltage

Contents:

1. General principles of measuring current and voltage
1.1 Instrument transformers
1.2 Current transformers operating principles
1.2.1 Measuring errors
1.2.2 Calculation of errors
1.2.3 Variation of errors with current
1.2.4 Saturation factor
1.2.5 Core dimensions
1.3 Voltage transformers operating principles
1.3.1 Measuring errors
1.3.2 Determination of errors
1.3.3 Calculation of the short-circuit impedance Zk
1.3.4 Variation of errors with voltage
1.3.5 Winding dimensions
1.3.6 Accuracy and burden capability

1.1 Instrument Transformers
The main tasks of instrument transformers are:
• To transform currents or voltages from a usually high value to a value easy to handle for relays and instruments.
• To insulate the metering circuit from the primary high voltage system.
• To provide possibilities of standardizing the instruments and relays to a few rated currents and voltages.

Instrument transformers are special types of transformers intended to measure currents and voltages. The common laws for transformers are valid.

Current transformers
For a short-circuited transformer the following is valid:

I₁/I₂=N₂/N₁

This equation gives current transformation in proportion to the primary and secondary turns.
A current transformer is ideally a short-circuited transformer where the secondary terminal voltage is zero and the magnetizing current is negligible.


Voltage transformers
For a transformer in no load the following is valid:

E₁/E₂=N₁/N₂

This equation gives voltage transformation in proportion to the primary and secondary turns.
A voltage transformer is ideally a transformer under no-load conditions where the load current is zero and the voltage drop is only caused by the magnetizing current and is thus negligible.            


1.2 Current transformers operating principles
A current transformer is, in many respects, different from other transformers. The primary is connected in series with the network, which means that the primary and secondary currents are stiff and completely unaffected by the secondary burden. The currents are the prime quantities and the voltage drops are only of interest regarding exciting current and measuring cores.

1.2.1 Measuring errors

If the exciting current could be neglected the transformer should reproduce the primary current without errors and the following equation should apply to the primary and secondary currents:

I_s = (N_p/N_s). I_p

In reality, however, it is not possible to neglect the exciting current.
Figure 1.2 shows a simplified equivalent current transformer diagram converted to the secondary side.

The diagram shows that not all the primary current passes through the secondary circuit. Part of it is consumed by the core, which means that the primary current is not reproduced exactly. The relation between the currents will in this case be:

I_s = (N_p/N_s). I_p - I_e

The error in the reproduction will appear both in amplitude and phase. The error in amplitude is called current or ratio error and the error in phase is called phase error or phase displacement.

Figure 1.3 shows a vector representation of the three currents in the equivalent diagram.
Figure 1.4 shows the area within the dashed lines on an enlarged scale.

In Figure 1.4 the secondary current has been chosen as the reference vector and given the dimension of 100%. Moreover, a system of coordinates with the axles divided into percent has been constructed with the origin of coordinates on the top of the reference vector. Since δ is a very small angle, the current error ε and the phase error δ could be directly read in percent on the axis ( δ = 1% = 1 centiradian = 34.4 minutes).

According to the definition, the current error is positive if the secondary current is too high, and the phase error is positive if the secondary current is leading the primary. Consequently, in Figure 1.4, the positive direction will be downwards on the ε axis and to the right on the δ axis.


1.2.2 Calculation of errors

The equivalent diagram in Figure 1.5 comprises all quantities necessary for error calculations. The primary internal voltage drop does not affect the exciting current, and the errors - and therefore the primary internal impedance - are not indicated in the diagram. The secondary internal impedance, however, must be taken into account, but only the winding resistance Ri. The leakage reactance is negligible where continuous ring cores and uniformly distributed secondary windings are concerned. The exiting impedance is represented by an inductive reactance in parallel with a resistance. Iµ and If are the reactive and loss components of the exiting current.

The error calculation is performed in the following four steps:
1. The secondary induced voltage Esi can be calculated from

Esi = Is * Z [V]

where
Z = the total secondary impedance

Z = sqr [(R_i + R_b)² +Xb²]

2. The inductive flux density necessary for inducing the voltage Esi can be calculated from

B = Esi / (pi * sqr (2) * f * A_j * N_s)

3. The exciting current, Iµ and If, necessary for producing the magnetic flux B. The magnetic data for the core material in question must be known. This information is obtained from an exciting curve showing the flux density in Gauss versus the magnetizing force H in ampere-turns/cm core length.

Both the reactive component Hµ and the loss component Hf must be given.
When H_µ and H_f are obtained from the curve, Iµ and If can be calculated from:

I_µ = H_µ * (L_j/N_s) [A]

I_f = H_f * (L_j/N_s) [A]

where magnetic path
L_j = length in cm
Ns = number of secondary turns
4. The vector diagram in Figure 1.4 is used for determining the errors. The vectors I_m and I_f, expressed as a percent of the secondary current Is, are constructed in the diagram shown in Figure 1.6. The directions of the two vectors are given by the phase angle between the induced voltage vector Esi and the reference vector Is.

The reactive component Iµ is 90 degrees out of phase with Esi and the loss component I_f is in phase with E_si.

1.2.3 Variation of errors with current
If the errors are calculated at two different currents and with the same burden it will appear that the errors are different for the two currents. The reason for this is the non-linear characteristic of the exciting curve. If a linear characteristic had been supposed, the errors would have remained constant. This is illustrated in Figure 1.7 and Figure 1.8. The dashed lines apply to the linear case.

Figure 1.8 shows that the error decreases when the current increases. This goes on until the current and the flux have reached a value (point 3) where the core starts to saturate. A further increase of current will result in a rapid increase of the error. At a certain current Ips (4) the error reaches a limit stated in the current transformer standards.

1.2.4 Saturation factor
Ips is called the instrument security current for a measuring transformer and accuracy limit current for a protective transformer. The ratio of Ips to the rated primary current Ipn is called the Instrument Security Factor (FS) and Accuracy Limit Factor (ALF) for the measuring transformer and the protective transformer respectively. These two saturation factors are practically the same, even if they are determined with different error limits.

If the primary current increases from Ipn to Ips, the induced voltage and the fluxm increase at
approximately the same proportion.

Because of the flat shape of the excitation curve in the saturated region, Bs could be looked upon as approximately constant and independent of the burden magnitude. Bn, however, is directly proportional to the burden impedance, which means that the formula above could be written

(FS)ALF ~ 1/B_n ~1/Z

The formula states that the saturation factor depends on the magnitude of the burden. This factor must therefore always be related to a certain burden. If the rated saturation factor (the saturation factor at rated burden) is given, the saturation factor for other burdens can be roughly estimated from:

ALF ~ ALF_n * (Z_n/Z)

where
ALFn = rated saturation factor
Zn = rated burden including secondary winding resistance
Z = actual burden including secondary winding resistance


1.2.5 Core dimensions
Designing a core for certain requirements is always a matter of determining the core area. Factors, which must be taken into account in this respect, are:
• Rated primary current (number of ampere-turns)
• Rated burden
• Secondary winding resistance
• Accuracy class
• Rated saturation factor
• Magnetic path length
The procedure when calculating a core with respect to accuracy is in principle as follows:

A core area is chosen. The errors are calculated within the relevant burden and current ranges. If the calculated errors are too big, the core area must be increased and a new calculation must be performed. This continues until the errors are within the limits. If the errors in the first calculation had been too small the core area would have had to be decreased.

The procedure when calculating a core with respect to a certain saturation factor, ALF, is much simpler:
The core area can be estimated from the following formula:

Aj ~ (K*ALF * I_sn*Z_n)/N_s

where
K = constant which depends on the core material (for cold rolled oriented steel K~25)
Isn = rated secondary current
Zn = rated burden including the secondary winding resistance.

NOTE! It is important for low ampere turns that the accuracy is controlled according to the class.

1.3 Voltage transformers operating principles
The following short introduction to voltage transformers concerns magnetic (inductive) voltage transformers. The content is, however, in general also applicable to capacitor voltage transformers as far as accuracy and measuring errors are concerned.

1.3.1 Measuring errors

If the voltage drops could be neglected, the transformer should reproduce the primary voltage without errors and the following equation should apply to the primary and secondary voltages:

Us = (Ns/Np) *Up

In reality, however, it is not possible to neglect the voltage drops in the winding resistances and the leakage reactances. The primary voltage is therefore not reproduced exactly. The equation between the voltages will in this case be:

Us = (Ns/Np)*Up –∆U

where
∆U = voltage drop

The error in the reproduction will appear both in amplitude and phase. The error in amplitude is called voltage error or ratio error, and the error in phase is called phase error or phase displacement.

Figure 1.10 shows a vector representation of the three voltages. Figure 1.11 shows the area within the dashed lines on an enlarged scale. In Figure 1.11 the secondary voltage has been chosen as the reference vector and given the dimension of 100%. Moreover a system of coordinates with the axis divided into percent has been created with origin of coordinates on the top of the reference vector. Since δ is a very small angle the voltage error ε and the phase error δ could be directly read in percent on the axis (ε = 1% = 1 centiradian = 34.4 minutes).
According to the definition, the voltage error is positive if the secondary voltage is too high, and the phase error is positive if the secondary voltage is leading the primary. Consequently, the positive direction will be downwards on the e axis and to the right on the δ axis.

1.3.2 Determination of errors

Figure 1.12 shows an equivalent voltage transformer diagram converted to the secondary side.
The impedance Zp represents the resistance and leakage reactance of the primary, Zs represents the corresponding quantities of the secondary. It is practical to look upon the total voltage drop as the sum of a no-load voltage drop caused by Is. The diagram in Figure 1.12 is therefore divided into a no-load diagram shown by Figure 1.13 and a load diagram shown in Figure 1.14.

The no-load voltage drop is, in general, very small and moreover it is always of the same magnitude for a certain design. For these reasons, the no-load voltage drop will be given little attention in the future. The attention will be turned to Figure 1.14 and the load voltage drop ∆Ub

The vector diagram in Figure 1.11 is used for determining the errors. The two vectors ∆Ur and ∆Ux are constructed in the diagram shown by Figure 1.15.
The direction of the two vectors is given by the phase angle between the load current vector Is and the reference vector Us

The resistive component ∆Ur is in phase with Is and the reactive component ∆Ux is 90º out of phase with Is.

1.3.3 Calculation of the short-circuit impedance Zk
Figure 1.16 shows, in principle, how the windings are built up. All quantities, which are of interest concerning Zk, are given in the figure.

The two components Rk and Xk composing Zk are calculated in the following way.

 1.3.4 Variation of errors with voltage
The errors vary if the voltage is changed. This variation depends on the non-linear characteristic of the exciting curve which means that the variation will appear in the no-load errors. The error contribution from the load current will not be affected at all by a voltage change.

The variation of errors is small even if the voltage varies with wide limits. Typical error curves are shown in Figure 1.17.

 1.3.5 Winding dimensions

Designing a transformer for certain requirements is always a matter of determining the cross-sectional area of the winding conductors. Factors, which must be taken into account in this respect, are:
• Rated primary and secondary voltages
• Number of secondary windings
• Rated burden on each winding
• Accuracy class on each winding
• Rated frequency
• Rated voltage factor

The procedure is in principle as follows:
1. The number of turns are determined from
where

N = number of turns (primary or secondary)
U_n = rated voltage (primary or secondary)
f = rated frequency in Hz
A_j = core area in m2
B_n = flux density at rated voltage (Tesla)

The value of Bn depends on the rated voltage factor.
2. Determination of the short-circuit resistance R_k
The highest percentage resistive voltage drop ∆U_r permissible for the approximate accuracy class is estimated. R_k is determined from ∆U_r and the rated burden impedance Z_b
3. The cross-sectional areas of the primary and secondary winding conductors are chosen with respect to the calculated value of R_k.
4. The short-circuit reactance X_k is calculated when the dimensions of the windings are determined.
5. The errors are calculated. If the errors are too high the area of the winding conductors must be increased.

If a transformer is provided with two measuring windings it is often prescribed that each of these windings shall maintain the accuracy, when the other winding is simultaneously loaded. The load current from the other winding passes through the primary winding and gives rise to a primary voltage drop, which is introduced into the first winding. This influence must be taken into account when designing the windings.

1.3.6 Accuracy and burden capability
For a certain transformer design, the burden capability depends on the value of the short-circuit impedance. A low value for the short-circuit impedance (a high quantity of copper) means a high burden capability and vice versa. The burden capability must always be referred to a certain accuracy class.

If 200 VA, class 1 is performed with a certain quantity of copper, the class 0.5 capability is 100 VA with the same quantity of copper, on condition that the turns correction is given values adequate to the two classes. The ratio between accuracy class and burden capability is approximately constant. This constant may be called the “accuracy quality factor” K of the winding

K=100*A/P

where
A = accuracy class
P = rated burden in VA