## Frequency Response Analyzer

The Moku Frequency Response Analyzer (FRA) drives a swept sine wave on the Moku outputs and simultaneously measures the received signal amplitude (or power) on the Moku inputs. The FRA can measure the transfer function of a system or device under test (DUT) and thus create a plot of amplitude and phase vs. frequency, commonly referred to as a Bode plot.

## Power units

In order to measure impedance of a device under test (Z_{dut}), we need to understand the power plots of the FRA. The FRA plots use units of dBm, or decibels relative to one milliwatt (1 mW); a convenient unit of measure in this situation. Defined as:

The Moku FRA swept sine output can be set in volts (peak-to-peak) or dBm. We will use volts for the output. For a sinusoid,

substituting for * V_{rms}* in

**(2)**gives

Expressed in dBm, scaling for mW, and knowing the Moku input impedance is 50 Ω gives:

Figure 1 shows use of the FRA to generate a * 1 V_{pp}* sine wave with Moku output 1 connected via direct coaxial to input 1. The resulting amplitude is of course flat across frequency range (0-1 kHz) and at 4.050 dBm, very close to the calculated 3.979 dBm. The discrepancy equates to 1.7 mV (or 0.17%).

**Figure 1:** FRA plot of 1 V_{pp} driven directly in the Moku input

## Impedance

#### Single-port measurement:

Now that the FRA power units are clear, we can measure an impedance.

In this first example we will measure * R_{dut}* of a simple 10 kΩ, 10% tolerance resistor. The effective circuit is:

**Figure 2:** Equivalent circuit

Note, * V_{out}* is 2 V, since this results in 1 V across load of 50 Ω.

*.*

**V**_{in}**Figure 3:** FRA of 10 kΩ DUT, single port

Rearranging power equation **(1)** and substituting **P** from **(4)** we can state:

With measured * P_{db}* of -35.821 dBm, we calculate

*= 10.23 mV.*

**V**_{in}The resistor divider of * R_{dut}* and the Moku 50 Ω inputs and output in Figure 2 gives us:

Thus

Solving gives * R_{dut}* = 9675 Ω.

A digital voltmeter (DVM) reading of this resistor showed 9750 Ω.

From this simple, one-resistor measurement we can conclude the Moku is accurate within 77 Ω (< 1%).

#### Low impedance measurement:

The example above used a standard 10% tolerance resistor. We can also measure a lower impedance to a high level of accuracy. To do this, we will use a 100 Ω, 0.005% tolerance high-precision resistor. Using the above method, we obtain a power magnitude plot (Figure 4)

**Figure 4:** FRA screen capture of 100 Ω, 0.005%, single port

Applying the measured power of -1.972 dBm to equations **(5)** & **(7)** we calculate * R_{dut}* to be 98.41 Ω. This agrees with the known value, but we can do better with a two-port measurement.

#### Two-port measurement:

In order to improve our measurement, we need to account for the loading of the DUT on the Moku 50 Ω output.

We can accomplish this with a two-port measurement whereby we utilize the second input port of the Moku to observe the actual applied signal level. Figure 5 shows an example hardware setup using Moku:Lab.

**Figure 5:** Two-port configuration using Moku:Lab

We can derive * R_{dut}* in Figure 6, from Ohm’s law:

Substitute **(9)** into **(8)**

or

**Figure 6:** Two-port equivalent circuit

We set up this two-port measurement with our tight tolerance 100 Ω, 0.005% resistor and captured the Moku FRA plot in Figure 7.

**Figure 7:** 100 Ω, two-port FRA capture

Note, we have used the FRA math channel to produce V2/V1. This is very quick and simple to configure on the iPad interface.

From **(10)** we see that we can calculate * R_{dut}* simply from the V2:V1 voltage ratio.

The FRA math channel has calculated the power ratio as 9.505 dBm and thus the voltage ratio is:

So we can apply **(10)** to obtain * R_{dut}* = 99.36 Ω.

We can now apply this 2 port method to the original 10 kΩ / 10% resistor; Figure 8 shows the FRA response.

**Figure 8:** 10 kΩ, two-port measurement

Using our established formula, the power ratio of 45.856 dBm gives and improved result of * R_{dut}* = 9762 Ω.

## Summary

The Moku FRA can be used to make impedance measurements and determine the value of a resistance to a <1% accuracy.

The two-port method allows for the loading of the DUT and appears more accurate.

In part two, we move on to using the FRA to measure complex impedance, capacitance & inductance, and more complex systems and across frequency.

## Moku demo mode

You can download the Moku app for macOS and Windows here. The demo mode operates without any hardware and provides an introduction to using Moku:Go or Moku:Pro.

## Questions or comments?

Contact us at [email protected].