KL7AJ on the Conjugate Match Theorem

KL7AJ proposed a little test for his readers on QRZ:

One of the most useful (and sometimes astonishing) principles in radio is the Conjugate Match theorem. In the simplest terms, what this says is that the maximum power will be transferred between a source (like a transmitter) and a load (like an antenna), when the source impedance is the COMPLEX CONJUGATE of the load impedance (or vice versa).
Here’s a neat little experiment to prove the conjugate match theorem. You need four basic ingredients: an antenna analyzer like the MFJ259 (or an actual impedance bridge, if you know how to use one). A good low loss antenna tuner. A good 50 ohm resistor. And a good 200 ohm resistor. And some appropriate connecting hardware, namely some short bits of coax.

Step 1) connect the 50 ohm resistor to the OUTPUT of the antenna tuner. Connect the antenna analyzer to the INPUT of the antenna tuner.

Step 2) Adjust the antenna tuner to get precisely 50 ohms, zero reactance on the antenna analyzer. This step simply confirms everything is working.

Step 3) Replace the 50 ohm resistor with the 200 ohm resistor. Readjust the antenna tuner to get 50 ohms, zero reactance on the antenna analyzer. Do not disturb the antenna tuner adjustments after this point.

Step 4) Remove the 200 ohm resistor and insert the antenna analyzer in its place (at the OUTPUT of the antenna tuner).

Step 5) Insert the 50 ohm resistor at the INPUT of the antenna tuner.

Step 6) Take a careful reading of the antenna analyzer. (What do you think it will say?)

10 points for anyone who will correctly explain why this works.

Some clarifications

Jacobi maximum power transfer theorem

Jacobi published his maximum power transfer theorem in 1840. It states that maximum power is transferred from a (Thevenin) source to a load when the load resistance is equal to the (Thevenin equivalent) source resistance.

It was later adapted to apply to AC circuits with sinusoidal excitation, maximum power is transferred from a (Thevenin) source to a load when the load impedance is the complex conjugate of the (Thevenin equivalent) source impedance.

Walt Maxwell’s Conjugate Mirror

(Maxwell 2001 24.5) states

To expand on this definition, conjugate match means that if in one direction from a junction the impedance has the dimensions R + jX, then in the opposite direction the impedance will have the dimensions R − jX. Further paraphrasing of the theorem, when a conjugate match is accomplished at any of the junctions in the system, any reactance appearing at any junction is canceled by an equal and opposite reactance, which also includes any reactance appearing in the load, such as a non-resonant antenna. This reactance cancellation results in a net system reactance of zero, establishing resonance in the entire system. In this resonant condition the source delivers its maximum available power to the load. …(1)

Note that it states that if a conjugate match is established an any junction, then a conjugate match occurs in any (all) other junctions, simultaneously a conjugate match exists everywhere. Continue reading KL7AJ on the Conjugate Match Theorem

Last update: 2nd November, 2017, 7:58 PM

UHF series coaxial connector characteristic impedance

Measurements of Insertion VSWR of UHF series connectors consistently show increasing Insertion VSWR with frequency, an issue that often impacts measurement accuracy.

My own article Exploiting your antenna analyser #12 is but one of many.

Measurements consistently hint that the defect is that the characteristic impedance is typically somewhere between 30 and 40Ω.

Above is a dimensioned drawing from Amphenol (https://www.amphenolrf.com/connectors/uhf.html). Continue reading UHF series coaxial connector characteristic impedance

Last update: 29th October, 2017, 11:16 AM

Review of the Amidon AB_200_10 balun

The Amidon AB_200_10 2-30MHz, 1KW balun and knock-offs have been around for a very long time, I recall Dick Smith selling them in the early 1970s in Australia.

They were regarded as the epitome of the art… but it was not a very well understood art.

Lets analyse the common implementation as a Ruthroff 4:1 voltage balun in a 50:200Ω scenario.

Ruthroff 4:1 voltage balun

In this implementation, Amidon’s instructions show 16 bifilar turns on a T200-2 core.

A very simple model is to consider the device as an ideal transformer with a shunt magnetising impedance equal to the impedance of the 16t winding that appears across the 50Ω terminals. This has its greatest effect at low frequencies and although it is specified from 2-30MHz, lets analyse it at 3.5MHz.

The powdered iron core has very low loss at 3.5MHz, sufficiently so that we can ignore the imaginary component of µr for this analysis and take µr to be 10+j0.

Above is a calculation of the magnetising impedance and admittance under those assumptions. The magnetising admittance (0.00-j0.0134S) appears in shunt with the transformed load admittance (0.02S) so we can simply add them to find the admittance seen by the transmitter (0.02-j0.0134S). Continue reading Review of the Amidon AB_200_10 balun

Last update: 9th November, 2020, 6:32 AM

Tuned Plate Tuned Grid oscillator – a simple, but complete explanation

A correspondent trying to get his head around old designs was challenged by the Tuned Plate Tuned Grid (TPTG) oscillator in common cathode configuration.

A superficial analysis is that the feedback to the grid from the anode via the anode to grid capacitance (Cag) is in phase with the anode voltage, which because of inversion in the valve means it is negative feed back. How can it cause self oscillation? Continue reading Tuned Plate Tuned Grid oscillator – a simple, but complete explanation

Last update: 28th October, 2017, 7:05 AM

A flexible test panel for microcontroller based power control projects

I do a lot of experiments with microcontrollers switching mains powered equipment, and the test beds have always been improvised. It has always been my intention to formalise something for convenience but mainly for better safety.

The article describes a test panel to fill that need.

The panel is constructed on a piece of 3mm aluminium sheet, drilled and tapped to take two sections of 35mm DIN rail for flexible mounting of accessories.

Above is a pic of the test panel in use to test the generic heating / cooling controller (hcctl), a flexible bang-bang controller based on an ATTiny25. Continue reading A flexible test panel for microcontroller based power control projects

Last update: 26th October, 2017, 10:27 AM

What is a Ground Plane Antenna?

Ask half a dozen hams to define a Ground Plane Antenna and you will probably get half a dozen different answers, yet it is thought of as one of the basic antenna types that newcomers will be introduced to in their education.

There seems credible acceptance by some writers that the Ground Plane Antenna was invented by George Brown (more completely Brown, Lewis and Epstein (BLE)) from RCA, and is described in US Patent 2,234,333 for a Demountable Antenna  filed in 1939. The patent does not call the thing a Ground Plane Antenna, but it does describe what could be naively described as a quarter wave vertical radiator with four equally spaced quarter wave horizontal radials (plus some other embellishments).

BLE gives the dimensions for the antenna at 41.5MHz and offers that the feed point impedance is 21.5Ω, transformed up to (the then popular) 70Ω transmission line by his custom quarter wave transformer (part of the invention).

The naively basic Ground Plane Antenna

Lets look at an NEC model of the vertical quarter wave and four quarter wave radials alone (ie in free space).

An NEC-4.2 model gives the feed point impedance as 23.4+j6.11Ω. The reactance is not surprising since the elements are actually slightly longer than a free space quarter wave, and resonance would occur at a little less length. Importantly, the R value is in the ball park of their estimate, so that reconciles reasonable well. BLE do not give a gain figure, but the gain of a lossless model in NEC is 1.44dB.

Above is the pattern, no surprises there (unless you were expecting it to look like a quarter wave monopole on perfectly conducting earth). Continue reading What is a Ground Plane Antenna?

Last update: 28th October, 2017, 10:13 AM

A low Insertion VSWR high Zcm Guanella 1:1 balun for HF – more detail

This article expands on the detail behind A low Insertion VSWR high Zcm Guanella 1:1 balun for HF.

Choice of core

Online experts all have a preferred core material, but there is a dearth of measurement data to show the difference in actual use. If someone recommends a particular core material and cannot provide measured Zcm data to support the recommendation, regard it as a weak recommendation.

Beware the magic of unobtainium… just because something is hard to get is not an indication that it is desirable.

Above is the complex permeability characteristic of the #43 material used. Inductance calculators that do not take that frequency dependent complex characterisitic into consideration produce invalid results. (Duffy 2015) gives a suitable approximation, and there are links to calculators that do work properly at the bottom of this article. Continue reading A low Insertion VSWR high Zcm Guanella 1:1 balun for HF – more detail

Last update: 17th January, 2019, 4:45 AM

A low Insertion VSWR high Zcm Guanella 1:1 balun for HF

This article describes a Guanella 1:1 current balun which has high common mode impedance (Zcm) and low Insertion VSWR. It is for application on antennas that have low VSWR(50) on at least some bands, especially if they would be used without an ATU on some bands.

The purpose of the balun is to minimise common mode feed line current which may contribute to EMC problems when transmitting, and contribute to increased ambient noise when receiving. Reduction of feed line common mode current also helps in achievement of expected load impedance characteristic, radiation pattern and gain. This article gives measured Zcm, but the definitive test of the effectiveness of such a balun is direct measurement of common mode current Icm… and it is so easy.

Example applications are half wave centre fed dipoles, fan dipoles, trapped dipoles, G5RV with hybrid feed, ZS6BKW, trapped verticals, monopoles, ground planes.

To obtain low Insertion VSWR, the choke will be wound with 50Ω coax, to demonstrate the practicality of the design budget (but good quality) regular (ie solid PE dielectric) RG58C/U will be used. Foam dielectric is NOT recommended. Solid PTFE coax could be used, but avoid coax with steel cored inner conductor, it may be lossier than you think at low frequencies with the silver cladding is relatively thin.

The candidate core is the readily available FT240-43 (Fair-rite 2643803802, 5943003801), it is a low cost NiZn ferrite with medium µr, and its µr and loss characteristic contributes to a broad high impedance choke well suited to this application.

Above is a model of the expected Zcm with 11 turns of RG58C/U coax and an equivalent shunt capacitance of 4.6pF. Continue reading A low Insertion VSWR high Zcm Guanella 1:1 balun for HF

Last update: 24th April, 2019, 10:18 AM

G3LNP balun with symmetric ‘matched’ load

My article G3LNP balun explored the operation of the G3LNP 4:1 balun on a 200Ω asymmetric load and found it exhibited extreme Insertion VSWR on what should have been an ideal impedance transformation but for the asymmetric element.

The balun is in fact a Voltage Balun and cannot be expected to work properly on asymmetric loads.

A correspondent proposes that the balun probably works very well on a nearly symmetric load such as a half wave dipole.

There are two aspectes to this proposition:

  1. the assumption that a common half wave dipole implementation is nearly symmetric; and
  2. the balun works well on a nearly symmetric load.

Continue reading G3LNP balun with symmetric ‘matched’ load

Last update: 14th October, 2017, 6:47 AM

G3LNP balun

G3LNP described a 4:1 balun for HF antennas in Radcom Nov 2017.

Above is the schematic supplied by G3LNP. He describes the dashed link at the bottom as optional, but uses it in his prototype so this analysis is with that link installed. The prototype used equal lengths of coax (1m PF100, an RG-6 like coax), and the toroidal choke appears to be 8t on a T130-2 powdered iron core.

Exploration of behaviour of baluns on extreme asymmetric load often reveals whether they work properly for asymmetric loads.
Continue reading G3LNP balun

Last update: 14th October, 2017, 6:50 AM