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# Equating VNA delay to Equivalent Insertion Phase

How can I calculate the equivalent phase delay, as a function of frequency, for a given electrical delay as set on an Agilent Vector Network Analyzer?

Equivalent phase delay, based upon electrical delay is computed as:
θ˚ = Ftest * Delay * 360

Where:

• θ˚ = Equivalent Phase (degrees)
• Ftest = Frequency (Hz)
• Delay = Delay (seconds)
• 360 = Conversion factor, radians to degree.

### Example:

Figure 1 is a PNG image acquisition from an Agilent PNA network analyzer.

• Start Frequency 1.0GHz, Stop Frequency 2.0GHz, 201 Points
• Full 2-Port Calibration via an Agilent electronic calibration module (Ecal)
• Single channel, two traces, one trace in each of two windows
• Device Under Test – Agilent N4419AK20, 3’ 3.5mm male to 3.5mm female cable assembly.
• Upper Window, ‘Tr1’, S21, Format Phase, Delay = 0S (no electrical delay)
• Lower Window, ‘Tr2’, S21, Format Phase, Delay = 3.623045-9S (delay applied automatically via the Marker Function - Delay selection on the PNA network analyzer).

Figure 1 – PNA Screen Capture, Single Channel, S21 Phase, Dual Window, Tr1 No Delay, Tr2 Auto Delay.

A partial sample of the acquired trace data / discrete data points for Tr1 and Tr2, as imported and processed within Microsoft Excel, are noted in Figure 2 below.

Figure 2 – S21 Phase, Trace 1 (Tr1) & Trace 2 (Tr2), Partial Data, Points 1..28/201 as Imported to Excel.

Sample Calculation:

Equivalent Phase Delay, Frequency = 1.0 GHz.
θ˚ = Ftest * Delay * 360
θ˚ = 1E9 * 3.623045E-9S * 360˚
θ˚ = 1304.2962˚

The Agilent vector network analyzer default phase format wraps the phase at the limits of ±180˚. The calculated equivalent phase shift must be manipulated for phase rotations in excess of 360˚ (which is equivalent to one wavelength) . Thus the modulo* function is applied to the resultant data. Additionally, if the modulo function result is greater than 180˚ then an additional 360˚ is subtracted from the result (i.e. from the remainder). If the result of the modulo function is less than 180˚ then the remainder is the result.

*Modulo function calculates the remainder of division of one number, ‘X’, by another, ‘Y’. Modulo is applied here as Modulo(X,Y) where X = computed phase and Y = 360˚.

Based upon the notes above, the equivalent insertion phasem (θ˚) @ 1.0 GHz = 1304.2962˚ is calculated as follows:

Applying the modulo function, modulo (1304.2962, 360) = 224.2962˚.
Applying the rule:
IF
(Modulo (1304.2962, 360)) > 180
THEN
(Modulo (1304.2962, 360)-360)
ELSE
(Modulo (1304.2962, 360))

The THEN condition is true since modulo (1304.2962, 360) = 224.2962 (i.e. > 180˚). Subtract 360˚ or,
Modified equivalent insertion phase = 224.2962˚ - 360˚
Modified equivalent insertion phase = -135.7038˚

The modified equivalent insertion phase at 1.0GHz with 3.623045nS delay = -135.7038˚.

The original S21 phase parameter of Tr1 @ 1GHz with 0 nS delay = 134.5573˚ (from ‘Exported PNA PRN S21 Phase Trace Data Tr1 & Tr2’ in Figure 2 above). The Insertion Phase at 1.0 GHz with 3.623045nS delay (Tr2) = -1.1465˚ (again, from ‘Exported PNA PRN S21 Phase Trace Data Tr1 & Tr2’ in Figure 2 above).

The delayed phase (‘Tr2’ phase) is calculated as the sum of the Original phase (i.e. ‘Tr1’ phase, 0s delay, at a specified frequency) + Modified Equivalent Insertion Phase;
Final Phase with delay = Original phase + Modified Equivalent Insertion Phase
Final Phase = 134.5573˚ + -135.7038˚
Final Phase = -1.1465˚.
Final Phase, absolute, Tr2, based upon 3.623045nS @ 1GHz = -1.1465˚. Thus, correlation with the Phase data for 1GHz, Tr2, 3.623045nS delay.