To extract the resistive (real) and reactive (imaginary) components of Zx, two multipliers are used, one for each component. The resistive component is extracted by multiplying vx by a sine-wave in-phase (I) with the generator (θg) and the reactive component with a sine-wave in quadrature (Q):
The results are voltages scaled by Io which directly affect the accuracy. Circuit realization of the multiplier with divider scaling is directly performed by transconductance multipliers such as the LM13700, the multiplier used previously for oscillator amplitude control.
The ESI253, B&K875A, and Elektor meters do not use translinear multipliers; analog switches are used instead. The reference input of the phase comparator from the generator is converted to a square-wave by an analog comparator. The resulting square-wave is in-phase with the generator. The Elektor phase detector is shown below, based on a precision rectifier or absolute-value circuit with synchronous gain switching.
The B&K 875A has a full-wave phase detector that outputs twice the voltage, shown below.
Both the I and Q waveforms and their complements (inverted 180°) are used followed by simple RC integrators. (The I component, or 0° and 180°, produces the reactive output because the phase of the square-waves has been shifted by 90°.) The gating effect of the square-waves is a multiplication having only two values, zero and one. The resulting frequency spectrum is not the usual sum and difference frequency convolutions of the analog sine multiplier but includes the additional sinx/x Fourier distribution of frequencies around the sum and difference frequency components. The low-pass filter is designed to be sufficient for removing these extra components as well as the usual modulation products of sine-wave multiplication. This is not difficult to design because usually meter readings are slow (one to a few per second) and a low break frequency for the filter can remove the higher square-wave frequencies.
Different circuit schemes can produce the Q square-wave for the phase detector. In the ESI 253, B&K 875A, and Elektor meters, the oscillator sine-wave is shifted 90° by an op-amp all-pass filter. The Elektor circuit is shown below.
For Rf = Ri, the all-pass filter transfer function is
The magnitude is flat with frequency and the phase decreases linearly so that when the pole and RHP zero frequency magnitudes are set to the input frequency (of vi), the phase of vo is -90 ° relative to that of vi, with -45° from both RHP zero and pole. This Q generation is simple and effective but for an accurate 90° phase shift, must be adjusted for the input frequency so that R•C = 1/2•π•fi. Hence the adjustment of R is needed in the above circuit.
The quadrature oscillator compensates somewhat for its disadvantage of 3 switched resistances for frequency ranging by outputting both I and Q sine waveforms. They can be input to a translinear multiplier or converted to square-wave transitions at their zero-crossings and used in a square-wave-driven phase comparator.