Let's continue our discussion of the subsystems in the impedance meter that we've been discussing in the previous posts (see “Related Posts” at the end of page 3). We now move on to the next to last major functional block: the phase comparator or detector.

This block also offers a variety of implementation choices, and all of them can be integrated monolithically. A simple reactance-measuring meter would not need a phase detector, and the bridge equations presented in the first article (Z Meter on a Chip? Impedance Meter Bridge Circuits), which are based on a simple reactance model of the unknown to be measured, Z_{x} , would suffice.

While reactance measurement (or measurement of L, C, and R) is the main objective, Z meters are expected to also acquire values of circuit elements of more complicated models for Z_{x} based on the decomposition of impedance into its real (or resistive) and imaginary (reactive) components, as shown in the impedance diagram below.

Z_{L} is the impedance of an inductor with parasitic series resistance, R_{L} . Z_{C} is the impedance of a capacitor with series resistance R_{C} . Both of these parasitic values are of interest and can be acquired by decomposing Z_{x} into these components.

The magnitude of Z_{x} is derived directly from the bridge variables by Ohm’s Law. The phase angle can be measured in the time domain by measuring the time intervals between zero-crossings of the two sine waveforms. This is an attractive method when a μC with a high-resolution counting capability is involved.

Otherwise, the phase is determined in the frequency domain using a phase detector: a phase comparator followed by a low-pass filter. The comparator multiplies v_{vx} for L or v_{ix} for C by the generator sine-waves that are 0° (I, or in-phase) and 90° (Q, or quadrature) relative to the v_{g} – stabilized quantity (v_{ix} for L, v_{vx} for C). The conceptually simplest form of detector multiplies sine-wave by sine-wave and then integrates the result over the cycle. The integration or low-pass filtering can be implemented multiple ways. The simplest is a passive RC integrator. A more elegant choice is a synchronous integrator.

The multiplier as a phase comparator is expressed in its equations. Let the bridge quantity to be compared be v_{x} and the reference sine-wave derived from the generator waveform(s) be v_{g} . Then the bridge current through measured impedance Z_{x} has an amplitude of I_{g} and

θ_{x} is the phase of v_{x} relative to v_{g} . The cosine with a difference of angles is expanded so that

The generator sine-wave to be multiplied by v_{x} is scaled for a desired output current, I_{o} , to be multiplied by the impedance component being measured. The generator input amplitude to the multiplier is removed by dividing it out using the factor, I_{g} /2. Then what is multiplied by v_{x} is a unitless quantity,

and results in

Following the multiplier is an integrator that synchronously integrates over the cycle. The integrals are cycle averages;

What these equations tell us is that in-phase waveforms (sin and sin, or cos and cos) result in a cycle-averaged output of ½ while sin and cos, having a 90° phase difference (in quadrature), average to zero.

To extract the resistive (real) and reactive (imaginary) components of Z_{x} , two multipliers are used, one for each component. The resistive component is extracted by multiplying v_{x} by a sine-wave in-phase (I) with the generator (θ_{g} ) and the reactive component with a sine-wave in quadrature (Q):

The results are voltages scaled by I_{o} 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 R_{f} = R_{i} , 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 v_{i} ), the phase of v_{o} is -90 ° relative to that of v_{i} , 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•f_{i} . 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.

The B&K 875A and HP 4261A generate the phase waveforms for the detector with a PLL and modulus 4 counter consisting of two flip-flops (FF). The B&K scheme is shown below.

A CMOS 4046 PLL IC is driven by a square-wave derived from the oscillator loop waveform, then phase-shifted (with an all-pass filter circuit, as shown previously) for phase compensation before driving the PLL as the reference phase. The CMOS 4027 dual JK FF outputs the four phases. PLL locking over the limited frequency range of Z meters, which do not sweep frequency as Z analyzers do, can be made very accurate and no analog adjustments are required. It is a preferred analog-digital scheme.

Having the ability to separate real and reactive impedances, the measurement circuits in the functional form of blocks look like this.

The voltage and current amplifiers must be phase-matched at the bridge frequencies. Amplitudes must also roll off together. For single-pole rolloff, amplitude error for a pole at f_{bw} and generator frequency at a lower f_{g} has a magnitude error of

and is tabulated below. The influence of phase extends more broadly in frequency range from f_{bw} than magnitude and its error is

and is also tabulated.

What the two right columns show are the required frequency ratios for a given error. It is quite evident that phase error dominates magnitude error in mismatched waveform paths to the phase detector. For a 0.2% accuracy goal from the phase detector output, the bridge amplifiers must have a bandwidth of 500 times f_{g} . For a 1kHz bridge excitation, the amplifiers must have a bandwidth of 500kHz. Usually, these op-amp-based amplifiers have low gain to maximize bandwidth, or else all-pass filters are used for phase adjustment at each frequency, as in the B&K 875A.

The phase detector can be auto-zeroed by selecting the same inputs from either amplifier and acquiring the phase offset errors as nonzero V_{R} and V_{x} . Consequently, if the two amplifiers are designed to be similar, using dual op-amps in a single IC and precision resistors, then even with significant phase shift, phase offset compensation can be performed in a μC or with analog tomfoolery. The point is that the tabulated ratios of frequency are what a simple, textbook design would require and an actual design should not be far from these ratios. With auto-zero and full-scale phase calibration using the amplifier input switches, two-point calibration of the phase detector is possible, leaving only the requirement that it be linear.

Next, we will look at the digital voltmeter and its ADC. This will finish our discussion of the Z meter.

**Related posts:**

- Z Meter on a Chip? Impedance Meter Oscillators
- Z Meter on a Chip? Impedance Meter Range Capabilities
- Z Meter on a Chip? Impedance Meter Bridge Circuits
- An Instrument on a Chip? The Minimum-Subsystem Instrument
- An Instrument on a Chip? The Configuration Problem
- An Instrument on a Chip? Some Emerging Instruments & the China Factor
- An Instrument on a Chip? A Look Back
- Thermocouple Nodules, Cold Junctions & Integration Opportunities

I wonder whether there is a calibration to get correct measurement in the circuit as shown in the BK 875A. In most case, device measurement tool would be adjusted every year based on warranty description.

I am little confused, but please correct me if i am wrong, expression suggests a quadrature phase detector can be made by summing the outputs of two multipliers.

The B&K875A has a preponderance of trimpots – over 20 as I recall my count of them – and this is not unique to it. The ESI 253 has plenty of them too (only 10-turn pots instead). This ponderous amount of adjusting is caused by the lack of a uC in these products (too early) to do the calibrating. So yes, everything you can imagine that might need adjusting is included.

As for the two multipliers, the B&K875A had a clever phase detector that was full-wave: it used both half-cycles of the sine waveforms to contribute to the phase detector output. Each half-cycle was multiplied using an analog switch with either an in-line or quadrature reference sine from the generator. This scheme results in twice the PD gain of the more common half-wave design.

Hi Dennis,

Great article, I love these kind of explorations into the Analog Domain 😉

I have a couple of things to mention, but I am sure you already know them:

4046PLL is a beast. The phase comparator is excellent. But there is a little dead band in it, so Philips (now NxP) did a newer design, the 7046 to avoid this dead band

Further I found a mixer that is working from DC to 6GHZ with just 4 FET's inside. Basically this is for an analog frond end (of course) but I found out that for audio this one can be used as well. I build a chopper amplifier with it, gain of 50 and a 7nV/SQR(Hz) result, mainly due to this mixer. Normally this cannot be done with 'nuts and bolds' components and it should be integrated, nevertheless, it works fine here on my desk, as long as there is no wind blowing passby 😉

These new parts with theur enormous nice specs can help you and also break your design in 2, but Analog folks like us will find their way around.

“Mistakes were made.” Donald Rumsfeld, former Sec. of Defense, USA

In this article, it is stated that

“The B&K 875A has a full-wave phase detector that outputs twice the voltage…”

Also in one of my follow-up comments, I wrote

“This scheme results in twice the PD gain of the more common half-wave design….”

Don't believe this. It is true of peak detectors but not these phase detectors. The gain of the half-wave and full-wave synchronous detectors is the same. What is different is that the full-wave dectector outputs at twice the sample rate, not voltage. This has the advantage of making the low-pass filtering ripple less, or else the time constant less for the same ripple.

In the impedance diagram of inductance/capacitance with series resistor is supposed that parallel capacitance is negligible?

To measure the phase angle by utilizing an high resolution μController is a good idea , provided that the bandwidth of the whole measurement system is enough to include the range of frequencies of interest.