G3YNH info: Analysis of Douma's generator PF correction scheme.

Analysis of Douma's generator PF correction scheme.
by David Knight

A drawback associated with having a reactive voltage-sampling network at the transmitter side of an impedance monitoring bridge is that it leaves the generator with a reactive load when the bridge is balanced. In US Patent No. 2734169, Douma offers a solution to this problem which involves producing a correction voltage to be added to the sum of the bridge current and voltage analogs . The situation which Douma considered is illustrated below (note that current-transformer secondary stray capacitance and propagation delay are neglected in the derivation which follows):

In this case there is a capacitance Cα permanently connected across the generator, causing a phase-shift which cannot be taken into account when ZL is adjusted to balance the bridge. We can however envisage a situation in which the system is somehow adjusted to make the generator see a pure resistance R0; in which case, presuming that the voltage-sampling impedance Z1+Z2 is sufficiently large to be neglected (or is actually responsible for the capacitance Cα), and that the current transformer insertion impedance is sufficiently small to be neglected, R0 will be the parallel combination of ZL and the impedance of the capacitor Cα, i.e.:
R0 = ZL // jXCα
which can be written:
1/R0 = (1/ZL) + (1/jXCα)
and so the load admittance will be:
1/ZL = (1/R0) - (1/jXCα)

The output of the bridge will be:

Vdet =

V Z1

Z1 + Z2

I (Ri // jXLi)

Where
I = V' / ZL = V' [ (1/R0) - (1/jXCα) ]
hence:

Vdet =

V Z1

Z1 + Z2

V' (Ri // jXLi)

Ni R0

V' (Ri // jXLi)

jXCα Ni

The first two terms of this expression are the normal balance condition for the bridge, and have parameters which can be chosen so that the terms add to zero when ZL=R0. The third term is the error in the balance condition due to the fact that ZL ≠ R0 when the load seen by the generator is equal to R0. This shows that we can alter the balance condition of the bridge in such a way that the adjustment of ZL will compensate for the presence Cα by adding a correction voltage Vcorr to Vdet such that Vdet+Vcorr=0. The required correction voltage is the negative of the last term in the expression above, i.e.:
Vcorr = -V' (Ri // jXLi) / ( jXCα Ni )
It will take quite an elaborate network to obtain exactly this voltage, but Douma made the sensible observation that the capacitance Cα will generally be small, since the main use of the correction is to cancel the effect of using a capacitive voltage-sampling network at the generator side of the bridge. Hence Cα will only cause a serious shift in load phase angle at high frequencies, and the correction voltage at low frequencies will be negligible. It is therefore legitimate to assume that significant correction will only occur when XLi >> Ri, and hence Ri//jXLi ≈ Ri , and so the simplified correction voltage becomes:

Vcorr = -V' Ri / ( jXCα Ni )

Now consider the voltage divider circuit on the right. Its output is:
Vcorr = V' Rβ / (Rβ + jXCβ)
but if XCβ >> Rβ, this expression becomes to a good approximation:
Vcorr = V' Rβ / (jXCβ)
If we now choose Cβ to be the same as Cα, then setting Rβ=Ri/Ni gives the required correction voltage, except that the sign should be negative (phase reversal is required).

That the approximation XCβ >> Rβ can be made to hold good can be seen by considering that Ri is typically about 50Ω and Ni is usually in a range from 10 to 40 turns. Therefore Rβ will never be greater than about 5Ω, whereas the input-side capacitive voltage-sampling network for which Cα is a model will typically have a reactance of around -400Ω at the highest frequency of operation, and this will only increase in magnitude at lower frequencies. The only remaining problem is that the correction voltage derived by the RC network is of the wrong polarity, and needs to be floating with respect to ground so that it can be added into Vdet. This calls for the use of a 1:1 isolation transformer, which is shown in Douma's final circuit below:

Note that the resistor Rβ is connected to the secondary side of the transformer. This is done so that the transformer magnetising current (i.e., the current which flows because the inductance of the transformer is not infinite) is small in comparison to the current which flows due to the presence of the resistor. The resistor in this position will also damp any transformer resonances. This ensures that the phase of the current flowing out of the secondary is very close to 180° relative to the phase of the current in the primary, thereby establishing the required 90° difference between Vcorr and V' to the best possible approximation.
If we assume that RV >> XC1 at high frequencies, the capacitance at the generator side of the bridge will be, to a good approximation, C1C2/(C1+C2). If we set Rβ=Ri/Ni, and adjust ZL (using an auxiliary bridge) so that the generator sees R0, adjusting Cβ will bring the bridge to balance at high frequencies when Cβ=C1C2/(C1+C2).

The importance of Douma's correction method for generator-side reactance lies in the fact that it introduces the idea of obtaining bridge-balance by performing summations other than the generic Vv-Vi=0. This is the key to making bridges which monitor only a single scalar attribute of a load impedance, such as its resistance or conductance. Douma's method also instructive in its use of a quadrature network, and of a transformer as a voltage sampling network. What is extraordinary however, is that it represents several missed opportunities. Douma used a transformer as a voltage sampling device, but did not invent the transformer voltage-sample bridge. Douma was fully conversant with the problem of generator power-factor error and displayed great ingenuity in overcoming it; but it did not occur to him that the solution might be simply to move the voltage sampling network to the other side of the current transformer, or to include a small compensating inductance in series with the generator.

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