# Dielectric Loss-Frequency Relation and Principle for Variable Frequency Measurement

HIMALAYAL - SHANGHAI - CHINA

1. Principle for variable frequency measurement
Variable frequency measurement manages to obtain accurate and reliable outcome under severe interference. For example during 55Hz measurement, measurement system only delivers 55Hz signal, 50Hz interference signal is effectively suppressed, the reason for that is measurement system can easily distinguish different frequencies, calculation below illustrates effect for frequency selective measurement:
Two overlaid sin wave with one time frequency difference, higher frequency is considered interference, with 10 time amplitude to low frequency:

Substitute x=0/90/180/270°come out four measured value Y0=12.4517，Y1= -11.1017，Y2=12.2075，Y3= -13.5576, calculate A=Y1 - Y3=2.4559，B=Y0 - Y2=0.2442, thus:

The outcome coincides with low frequency section phase position and amplitude, interference is suppressed. Measuring point for actual waveform amount to tens of thousands, leading to loads of calculation, the outcome shows overall waveform characteristics.

2.       Dielectric loss-frequency connection
Any capacitor with dielectric loss can be stimulated into two ideal models of RC cascade and RC parallel:
(1)    Parallel model
Dielectric loss is generated from resistance connected to capacitor in parallel. Voltage at two ends of RC circuit is equal under such circumstance:
Active power P=U2/R
Reactive power Q=U2/(Capacitice reactance1/ωC)= ωCU2
Thus , tgδ=P/Q=1/ωRC

Parallel Model
Among which ω＝2πf, f represents power supply frequency. It is thus clear that if dielectric loss is actually stimulated with one pure resistance and one pure capacitor, it will be inversely proportional to frequency. On condition of R=∞, comes out no active power and zero dielectric loss.
This method is commonly applied to laboratorial stimulation for dielectric loss above 10%, or used to produce standard dielectric loss tester.
Dielectric loss is generated from resistance connected to capacitor in cascade. Current in this circuit remains the same:
Active power P=I2R       ,
Reactive power Q=I2 x (Capacitive Reactance1/ωC)=I2/ωC
Thus, tgδ= P/Q=ωRC

Conclusion can be drawn based on analysis above that cascade model tgδ=2πfRC, parallel model tgδ=1/(2πfRC), R and C basically stay the same, f is variation. Substitute 45Hz, 50Hz and 55Hz into the equation, it is observable that tgδ is respectively in direct and inverse proportion to f. As shown on the diagram below, f has more influence on fully proportional and fully inversed proportional model, yet actual capacitor applies hybrid model and minorly influenced by f.

3.       Actual capacitor test object:
(1)    Measurement under fixed frequency
Actual capacitor test object can be signified with both cascade model and parallel model under a fixed frequency. For instance if frequency is fixed at 50Hz, the two circuits below appears the same characteristics:

Dielectric loss remains 31.4% when measuring two test objects with different tan delta bridge, yet capacitance measured with Schering bridge (2801 or OTHER AUTOMATIC BRIDGE) is 10000pF, capacitance measured with current comparator bridge (for instance WS30) is 9101.7pF. The reason for that is 2801 tan delta bridge assumes test object loss to be cascade model, yet HCL2876 assumes test object to be parallel model.
Parallel model is normally considered to be closer to practical situation, this is due to active current flow through insulating layer between electrodes resembles more of loss resistance connected between electrodes in parallel, whereas electrode resistance is zero, without dielectric loss.
In fact when dielectric loss is below 10%, capacitance difference like this is minor.

(2)    Variable frequency measurement
Experts engaged in on-site test must learn from experience that: traditional instrument, for instance OTHER AUTOMATIC BRIDGE, is incapable of balancing tan delta bridge when measuring dielectric loss with repeated measurement applying phase shift and phase inversion under severe on-site interference.
Along with voltage level increase, interference will get severer. In this case variable frequency measurement appears to be a good or even the sole solution. Anti interference capacity of variable frequency measurement improved over one order of magnitude compared to phase shift and phase inversion method. It’s like when two radio station at one frequency, it’s hardly possible to suppress the other signal, whereas it’d be far easier to distinguish when the two radio station are at different frequency.

4.       Automatic frequency variation and 50Hz equivalent
The only uncertainty about variable frequency measurement is frequency equivalence. Based on model above, dielectric loss varies with frequency. For instance 1% dielectric loss at 50Hz applies 55Hz measurement, with outcome for cascade model turns into 1.1% (direct proportion), outcome for parallel model turns into 0.91% (inverse proportion). Even though error like this may be tolerable for on-site measurement, error is still large.
To solve this problem, firstly we come up with the principle of double variable frequency measurement: measure once respectively at symmetric positions of 45Hz and 55Hz to 50Hz, then average test data to minimize error. Theoretical analysis result as shown in the form below:

 Model 50Hz actual   dielectric loss 45Hz measured   dielectric loss 55Hz measured   dielectric loss Average Cascade 1% 0.9% 1.1% 1% Parallel 1% 1.111% 0.909% 1.010%

Seen from above maximum error appears in parallel model, relative error 1%.
Analysis above shows that double variable frequency measurement exerts strong anti interference capacity of frequency measurement, with theoretical maximum relative error less than 1%, meets the requirement for on-site measurement. It is also recommended to apply 47.5Hz, 52.5Hz double variable frequency measurement, theoretical error will be reduced to 0.25%, yet anti-interference at such frequency is inferior than at 45Hz, 55Hz.
Practical measurement shows that variable frequency measurement has stable data, excellent repeatability. Laboratorial calibration also shows great precision indicator. Principle for variable frequency measurement is now widely accepted.