In the previous article in this series, we looked at bridge circuits used to measure impedance. Now, let's look at some more (simple) math.

Z-meter performance capability is expressed mainly by measurable impedance and frequency ranges. The Z-meter voltage and current of the unknown impedance, Z_{x} , have measurement ranges that determine the overall resistance (R), inductance (L), and capacitance (C) ranges of the meter. The relationship is rather simple. If the voltage, v_{x} , across Z_{x} is measured over V decades and the current through it, i_{x} , is measured over I decades, then the measurement range of Z_{x} is Z = V + I decades, where Z, V, and I are ranges corresponding to the indicated variables and are on a log_{10} scale of decades. Therefore, the impedance measurement range of the meter extends from the minimum impedance magnitude of

to a maximum impedance magnitude of

The range of frequencies over which impedance is measured extends the measurement capability, as shown on the following reactance plot for Z meters.

The frequency range, F, extends the reactance measurement range to

L = C = X = Z + F

The scaled bridge voltage and current, v_{vx} and v_{ix} (discussed in my previous article), are converted to digital form and displayed by a digital voltmeter (DVM). DVMs have the quasistatic (low-frequency) transfer function of

where V_{O} is a digital displayed quantity with full-scale value of 2^{n} counts, V_{X} is the converted input voltage, and V_{REF} , the reference voltage, is the full-scale value of V_{X} . A ratiometric DVM converts the analog voltage ratio to a digital count ratio, and the division in the ratioing divides v_{vx} by v_{ix} to give X_{X} or R_{X} when v_{vx} and v_{ix} are correctly scaled.

The voltmeter range, including that of the DVM and its display, is an additional consideration. A DVM with more digits has more range, of course, though over that range, resolution (measured in counts) and accuracy decrease proportionally from the full-scale (FS) value. A meter with four digits has effectively one more decade of range than one with three digits for the same resolution. For the above range equations, the resulting values are relative to FS, whatever that might be for the DVM. The DVM itself can add additional range, depending on how the instrument accuracy is specified.

A Z meter with a three-digit DVM range and a 1 percent (two-decade) inaccuracy specification needs two digits of resolution to show the specified accuracy over the range, and then it has an additional decade that can be used for ranging. The range increments of R_{R} and R_{g} are typically chosen to be in decades, so that the DVM never needs to exceed one additional decade in range to maintain the spec down to zero scale, a decade lower than FS. Usually, inaccuracy or resolution is specified relative to the FS reading. Below FS, the specified performance also degrades. Consequently, a Z meter specified for two digits of FS accuracy and having a three-digit DVM has an additional decade of range that can be used to extend the impedance measurement range, Z.

As an example, a B&K 875A handheld, battery-operated RLC meter has no range switching of A_{VX} for its voltage measurement; V = 0 decades. The current amplifier range resistors, R_{R} (from the previous article), span six decades from 10Ω to 1MΩ. Consequently,

Z = V + I; Z = 0 + 6; Z = 6 decades

This defines the range on the vertical (log ‖Z‖) axis of the above reactance plot. It corresponds to the full-scale resistance range of the meter: R_{X} ϵ [20Ω, 20MΩ]. This meter has two excitation frequencies — 120Hz and 1kHz — giving it nearly 1 decade of F. Then

X = Z + F ≈ 6 + 1 ≈ 7

The B&K 875A does not make use of the highest L range (2kH); thus L = 6 decades FS (200μH to 200H). The lower frequency is included to extend the high end of the C range, and it is seven decades: 200pF to 2mF. The DVM is a 3.3-decade (3.5-digit) or 2,000-count meter. This results in a measurement range (with decreasing accuracy below FS) of 6 + 3.3 = 9.3 decades of total range for L and 7 + 3.3 = 10.3 for C.

In the next blog, we'll take a close look at the frequency sources used for excitation voltage of the impedance being measured.

**Related posts:**

- 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

Thanks Dennis for the great tutorial and in-depth mathematical analysis.

I just wanted to comment on a neat use of a Z-Meter: Characterizing a thermoelectric element over a large temperature—The Z-meter can heat the sample to a very high temperature in a vacuum. The Z-meter can then simultaneously measure all three thermoelectric parameters (Seebeck coefficient, thermal conductivity, and electrical conductivity), as well as measure the generated power and the efficiency for a single TE leg.

@Steve – I would not have thought that the excitation voltage source on most Z-meters would have sufficient current drive capability to heat up much of anything – but perhaps some do….

Hi Brad,

For very high temperatures, the power elements would probably have to be off on a heatsink to drive hefty current—although, class D drivers can run relatively cool and efficient

Hi Steve & Brad,

Steve, your notion of significant power delivery by Z meters can be inferred from the Z-meter range article. To measure low Z (which is where thermocouples are), a high current from the Z-meter is needed to develop a measurable voltage. 100 mA is not uncommon, and I am working on some Z meter designs that use up to 5 A. So Z meters do have some capability for heating a load.

The impedance meter test supports the confirmation of distributed amplifier installations and displays detailed info of the associated load. The test consequence displays if the associated speaker system is in decent order in addition to the needed amplifier power-ratio.