Full version in "Microwave System Design Tools and EW Applications" published by Artech House, Inc.
See web home page www.Y1PWE.co.uk

 

VSWR of Cascaded Mismatches

 Applet Notes

Two plots are available: Plot 1 shows the VSWR for two combinations of components. Plot 2 shows the corresponding power delivered to the load. Shown and tagged in black are the results of using simple linear combination.

The Z Combination drop-down menu allows all combinations of impedance for the selected VSWR's to be explored. The component impedances, based on a system characteristic impedance of 100ohms is sent to the Java console on selection.

Moving the mouse in VSWR/Frequency space displays the plotted values referred to the mouse x-axis position.

Default values may be modified and the display updated using the applet or keyboard 'Enter' Button. The display automatically scales to the geometry specified.

'Out of Range' is indicated when parameter values are selected which are out of the computable range.

The Save Button saves plot data to the Java Console ( slow if the console is open) with a summary of the relevant component parameters. Data may be selected on the Console using the mouse (Shift-click) and copied to the clipboard ( Ctrl C ) and saved in a text file for a permanent record or transferring to Excel for example.

If visual plots are required, Alt-Printscreen, copies the current window to MS Clipboard.

 Background

1) Introduction

A broadband microwave component is normally characterised by its VSWR and insertion loss measured at the design characteristic impedance. The relation between VSWR, impedance and return loss is:

VSWR :

Vs = Zc/Z0, if Zc, the component impedance is greater than Z0, the system characteristic impedance or, Vs = Z0/Zc, if Z0>Zc.

Return Loss,

and VSWR,

The convention for cascaded components is to treat them as being effectively isolated by linearly accumulating the insertion and VSWR losses. In addition the overall VSWR is calculated by accumulating the component powers returned at each junction. The overall VSWR is calculated from the total return loss of the supercomponent thus formed.

This simplification does not take into account either the phasing of the returned vectors or more importantly the real mismatch of impedances between components. For instance, two transmission line components may have identical impedances and hence VSWR's. Cascading these components produces no return component at the junction and therefore no change in the coupled VSWR. In most real situations the combined VSWR will also be a function of frequency and not only be dependant upon the component VSWR's but also on their complex impedances.

The example used in this applet comprises two components separated by an adjustable attenuator.

2) Linear Combination

The example used in this applet comprises two components separated by an adjustable attenuator as shown below.

P1 is the available power from the source and the power delivered to the load is,

An approximation to the load power ripple with frequency and its period can be estimated by vector summing the returned voltages with appropriate component line length phasing.

Using the linear combination method, the total return loss is,

from which, the overall VSWR can be calculated. A better approximation is to use the source and sink impedances to determine the junction VSWR and return losses before applying the above equations. Depending upon the impedance values, the junction mismatch can be either, Va.Vb or, Va/Vb.

3) Vector Analysis

The correct approach to determine the cascaded circuit performance is to iteratively calculate the component input impedances in the complex domain from the component circuit parameters in the conventional way. zilo is the load impedance, knowing this and component 2 characteristics, zi2 can be calculated. Similarly for zia and finally zi1, the loaded input impedance at component 1. The voltages in the cascade can then be found by working forward, starting from the source voltage Vs. In this example, Components 1 and 2 are transmission line devices (coaxial filter for example) having a voltage transfer characteristic,

The 'T' attenuator transfer function is,

The corresponding input impedances are,

and for the 'T' attenuator,

4) Conclusions

The applet enables the two methods to be compared. With five components, specified by just VSWR and insertion loss and measured at the system design characteristic impedance , there are 32 possible combinations of component impedances. These options can also be compared.

Interactively adjusting component VSWR and loss values illustrates how in worst cases even a few moderately mismatched components can give rise to large power variations in the load.

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