SSZTBQ2 january   2016

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Manjing Xie

Bode plot measurements are essential for system stability and transient response optimization.  This post describes how the connection wires affect bode plot measurements and the methods to improve measurement accuracy.

Figure 1 shows a typical Bode plot measurement setup with a broken control loop and a small resistor inserted. The frequency analyzer has a source that injects an AC disturbance, ṽds, across this small resistor. This setup also usually includes an isolation adapter.

Figure 1 Typical Bode Plot Measurement Setup

AC fluctuation occurs at the two nodes, A and B, across the breaking point. The frequency analyzer has two receivers to measure signals at nodes A and B, ṽA and ṽB. You can calculate the system loop gain, TV, with Equation 1:

Equation 1.

Equation 2 and Equation 3 calculate the voltage at nodes A and B, respectively:

Equation 2.
Equation 3.

## How and Why Connection Wires Affect Bode Plot Measurements

In an actual setup, you would have to solder connection wires to the two ends of the small resistor and connect both the disturbance-injection and signal-measurement cables to the other ends of the wires. Figure 2 shows a more accurate block diagram of the actual test setup.

Figure 2 Bode Plot Setup with a Single Pair of Connection Wires

The signal receivers A and B measure not only ṽA and ṽB, but also the voltage drop across the wires, ṽdrop. Equation 4 and Equation 5 calculate the measured signals, ṽ’A and ṽ’B:

Equation 4.
Equation 5.

Where Xds is the impedance between the break points and XL is the impedance of the shared wires. Xds is usually dominated by the resistor inserted between point A and point B.

Thus, the measured loop gain, TV_measure, is affected by the impedance of the wires’ XL, as shown in Equation 6:

Equation 6.

At low frequency, when |TV| >> |Xds/XL|, Equation 7 can approximate |TV_measure|:

Equation 7.

At high frequency, when |TV| << 1, Equation 8 can approximate |TV_measure|:

Equation 8.

When |XL| is comparable to |Xds|, TV_measure is much different than the TV. Because designers rely on Bode plots for stability and load-transient response optimization, this measurement error greatly affects the system’s ultimate performance.

Figure 3 shows |XL| (the impedance of wires of different lengths and gauges), with the two wires at a distance of 1 inch.

Figure 3 Wire Impedance of Different Lengths and Gauges

## Isolation Toolbox Limitations

Certain frequency analyzers might come with an isolation toolbox that pre-connects the receivers A and B to the output of the isolator, as shown in Figure 8. The toolbox is designed to lower the connection wire count. Doing so, however, forces the two receivers to include the voltage drop across the connection wires. Figure 3 shows the impedance of one 3-foot AWG 16 wire. Around 1 MHz, the impedance rapidly increases to over 10Ω.

Figure 8 Example Toolbox Connection

Figure 9 compares a Bode plot measured with a toolbox and two 3-foot cables of AWG 16 measured with two pairs of wires. The Bode plot measured with the toolbox has a gain margin less than 6dB, which falsely indicates a marginal stability.

Figure 9 Comparison of Bode Plots Measured with a Toolbox and the Setup Shown in Figure 4

The isolation toolbox is a convenient tool. However, the impedance of the connection wires at high frequency affects the Bode plot measurement too much. This misleading Bode plot would hinder the process to improve transient response. When using the toolbox, you should always use cables of minimal length. For systems that have high control bandwidth, I do not recommend the toolbox.

In this post, I showed how and why connection wires could affect Bode plot measurements. The best approach is to add a separate pair of connection wires to the breaking points for measurement only. In applications where adding a separate pair of connection wires is not feasible, keep the connection wires as short as possible.