The Western Electric 13C Transmitter - Harmonic Power

This
document presents an analysis of the output stage of the Western
Electric 13C transmitter. The analysis, which assumes operation into an
antenna of the dorsal V-type installed on Amelia Earhart's second world
flight Lockheed Electra 10E Special, was carried out primarily to
estimate the power of harmonic radiation that may have resulted when the
transmitter was tuned for operation at 6.21 MHz.

The document is divided into three sections:

I. The 13C Output Stage Plate Current Fourier Components

II. The Dorsal V Transmit Antenna

III. The 13C Output Stage Radiated Power Calculations

Chuck Varney

June 2011

Last edit September 2017

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I. The 13C Output Stage Plate Current Fourier Components

Figure 1. is a schematic diagram of the
Western Electric 13C transmitter. In the diagram, tubes V1 (crystal oscillator)

and V5 (audio amplifier) are Western Electric 205-D triodes. Tubes V2 (1st RF
amplifier) and paralleled tubes V3 and V4 (2nd RF amplifier) are Western
Electric 282-A screen grid tetrodes.

Note: I've used the terms *output stage*
and *2nd RF amplifier* interchangeably in this document. Morgan, in Figure
2. below, uses a third variation, *2nd amplifier*.

The schematic shows that the audio amplifier, through transformer T5 and inductor L6, simultaneously modulated the screen voltage of both the 1st and 2nd RF amplifiers. For a voice transmission, the 2nd RF amplifier was therefore required to amplify a modulated control grid voltage. To preserve the modulation it was required to operate as a linear amplifier, ruling out Class C operation and output stage plate current conduction angles of less than 180°.

Figure 1. Western Electric 13C transmitter schematic diagram, from H. K. Morgan's Aircraft Radio and Electrical Equipment, p146 (1939 and 1941 editions).

Figure 2. tabulates the approximate currents and voltages expected for a properly adjusted 13C transmitter.

Figure 2. Table of operating currents and voltages for the 13C transmitter,

from H. K. Morgan's Aircraft Radio and Electrical Equipment,

p150 (1939 and 1941 editions).

Figure 3. gives the control grid voltage-to-plate current transfer characteristics of the 282-A tetrode for three values of screen grid voltage (100, 150, and 200 volts), each at three plate voltages (500, 750, and 1000 volts). The Figure 3 characterisitc is a static characteristic, but is used as a dynamic characteristic in the development that follows.

Figure 3. Static transfer characteristics from 1936 Western Electric 282-A vacuum tube

documentation.
For use with the 13C 2nd RF amplifier, which used a pair of 282-A

tubes
connected in parallel, the indicated currents must be doubled.

Data
from Figures 2. and 3. may be used to find the 2nd RF amplifier plate
current pulse shape, and from it, the Fourier components. For this, key
entries in Figure 2. are:

* 2nd amplifier plate current, 175 mA

* Plate supply voltage, 1050 V

* Grid bias voltage, -27 V

* Screen voltage, 50 V.

Figure 3-1. The 1000-volt (plate) curves for screen grid voltages of 200 V (green) and 100 V (red) from Figure 3, and the 50 V curve (blue) derived from them. The method for obtaining the 50 V curve is given to the right of the plot.

The six data points for the 50 Vsg curve in Figure 3-1 were used with a cubic spline interpolation program, converted from FORTRAN to BASIC and modified to accept control grid bias voltage, plate current cutoff voltage, grid voltage amplitude, and a sampling interval as inputs.

An outline of the procedure for obtaining the plate current pulse shape, and finding its Fourier components follows. Use the derived 50 V screen grid voltage curve with a sinusoidal grid voltage input centered at the -27 V control grid bias point and sample the resulting plate current over one cycle. Run a Discrete Fourier Transform (DFT) on the sampled data to determine the DC, fundamental, and harmonic current components. Adjust the grid voltage amplitude to obtain a DFT output with a DC component of 175 mA, as given for the 2nd RF amplifier plate current in Figure 2.

The Fourier Analysis tool in Microsoft Excel's Data Analysis
Toolbox was used for the DFT task. A half-wave rectified sine wave was used to
test the method and select an adequate sample size, as its exact Fourier
components are known. A 512-point sample size gave good results to four
decimal places, so it was used for both the cubic spline interpolation
and the DFT. This gave a sampling interval of 360/512, or 0.703°.

Figure
4. summarizes the results of the method just described. The plotted
cycle of plate current, with peak value, Imax, of 0.561 A and plate
current conduction angle,
θp, of 181.2°, was obtained with the control grid
of the 2nd RF amplifier biased at -27 V and driven by a sinusoidal input
of 292 V amplitude.

Figure 4. Plot of the plate current pulse and tabulation of its Fourier components through the sixth harmonic

The Fourier component current identified as
Idc (0.175 A) is the average value of the plate current over one cycle.
Currents I1 through I5 are the amplitudes (peak, or crest values) of the
Fourier sinusoidal components of the plate current pulse, with I1 the
fundamental and I2 through I5 the second through fifth harmonics.

II. The Dorsal V Transmit Antenna

The
input impedance of the dorsal V antenna at 6.21 MHz and its harmonics
is an important element of the analysis to be done in section III. This
section describes the origin of the impedance values used there.

The dorsal V antenna was modeled, using 4NEC2, on a wire grid of the Electra, 9 degrees nose up, 7 inches above 6 inches of sea water. I've assumed that if the aircraft is in sea water deeper than skin depth, it's the equivalent of a sea water ground plane. (Sea water skin depth is 3.6 inches at 6.21 MHz and 1.8 inches at 24.84 MHz.).

Models
were made at each frequency of interest to maintain a fixed segment
length of 0.005 wavelength (200 segments per wavelength).

Dimensions were scaled from TIGHAR site Harney drawings, Port.gif and Planform.gif. A starboard-side photograph from the Purdue collection was used to estimate the location of the antenna feed-through insulator.

Figure 5.
Wire grid model of the Electra and dorsal V transmit antenna

The antenna input impedances and radiation efficiencies for the frequencies of interest are:

Frequency, MHz Impedance, ohms Radiation efficiency, %

6.21 2.18 + j 39.3
45.5

12.42 162 + j 397 90.3

18.63
536 + j 185
96.8

24.84 916 - j 2103
93.5

III. The 13C Output Stage Radiated Power Calculations

Figure 6. provides a diagram of the Western Electric 13C output circuit reduced to analytical form. In the diagram,

* Zload is the load impedance seen at the paired 282-A (V3 and V4) plates

* XC13 is the reactance of 4000 pF capacitor C13 in Figure 1.

* XL is the total reactance of the tuning coil (L7, L8, or L9 in Figure 1.)

* Rc is the total resistance of the tuning coil

* k is a factor (value 0 to 1) that apportions XL and Rc either side of the plate tap

* XC is the reactance of capacitor C (C10, C11, or C12 in Figure 1.)

* Ra is the resistive part of the antenna input (feed-point) impedance

* Xa is the reactive part of the antenna input impedance

The
diagram shows that the 13C transmitter used a tapped parallel RLC
output circuit with the dorsal V antenna impedance
(Ra + Xa) comprising a portion of one leg. The circuit served to
transform the antenna impedance to an appropriate load impedance,
attenuate radiation of harmonic energy, and
maintain a sinusoidal voltage at the plates of V3 and V4.

Figure 6. Diagram
of the 13C transmitter output circuit, at right, and power calculation
tables for the transmitter tuned to 6.21 MHz. The dashed box at left in
the diagram represents the tapped tuning coil. The dashed box at right,
the antenna.

Figure
6. includes two tables. The bottom table provides input data. The top
table is the output table, and gives the following for the 6.21 MHz
fundamental and its first three harmonics:

* Zload, given as the resistive and reactive components of the output circuit impedance

* Pp, the radio frequency power delivered to Zload

* Pa, the power delivered to the antenna. It is given in watts (W) and in decibels referenced to 1 watt (dBW).

Note: Pa is the power to be used with propagation models like VOACAP and ICEPAC.

* Circuit efficiency, equal to the ratio of Pa to Pp

* Radiation efficiency, equal to the antenna radiation resistance divided by total antenna resistance, Ra

*
Radiated power, Prad, equal to the product of Pa and the radiation
efficiency. It is tabulated in both watts and decibels.

Zload
is equal to the parallel combination of [k Rc + k XL + XC13] and [Ra +
(1-k) Rc + Xa + (1-k) XL + XC].

The resisistive (R) and reactive (X) components of Zload are given
in the output table for each frequency considered.

The plate RF power output, Pp, for a fundamental or harmonic current, In, is calculated as:

Pp = | In | ^{2} x Re (Zload) / 2, or | In | ^{2}
x R / 2
watts
[Eq. 1]

The antenna current, Ia, is calculated for each frequency and used to calculate Pa,

Pa = | Ia | ^{2} x Re (Ra + Xa) / 2, or | Ia | ^{2} x Ra / 2 watts [Eq. 2]

Figure
6. shows that at 6.21 MHz, 50 W is delivered to Zload; 44.1 watts is delivered to the antenna
itself, and 20.1 W is radiated.

The inductance calculator (Corum & Corum's sheath helix waveguide formula) at hamwaves.com was used for the coil reactance and resistance calculations. I used a coil form 1.75 inches in diameter wound with AWG 12 copper wire at a pitch of 4.23 mm. Coil turns were adjusted in small increments until the required reactance was reached for operation at 6.21 MHz . The coil resistance, Rc, was then recorded. With the coil dimensions fixed, the calculator was run for the harmonic frequencies and their resistances and reactances recorded.

The tabulation below gives the Figure 6. radiated power, Prad, in watts and in dB relative to the power radiated at 6.21 MHz.

Frequency,
MHz 6.21
12.42
18.63
24.84

Prad,
W 20.1
0.039 0.00031
0.0058

Prad,
dB relative
0 -27 -48
-35

the fundamental

The Prad values at harmonic frequencies are due to the combined effects of harmonic current magnitudes, output circuit harmonic attenuation, circuit efficiency, and antenna radiation efficiency.

The output circuit harmonic attenuation results from unfavorable values of
Zload that the circuit presents to currents at harmonic
frequencies. The attenuation is equal to the plate output power at the
fundamental frequency divided by the plate output power at a given
harmonic frequency. The harmonic attenuation, in dB, is given below.

Frequency,
MHz
12.42 18.63
24.84

Harmonic
attenuation,
dB 31
52
39

13C Transmitter Output Stage Summary of Results

* Plate current conduction angle,
θp: 181°

* Class of operation: AB_{2}

*
Plate current pulse peak value, Imax: 561 mA

* DC plate
current:
175 mA

* Fundamental current amplitude: 277 mA

* Second harmonic current amplitude: 123 mA

* Third harmonic current amplitude: 4 mA

* Fourth harmonic current amplitude: 24 mA

* Radiated power, fundamental: 20 W

* Radiated power, second harmonic: 39 mW (milliwatts)

* Radiated power, third harmonic: 310 uW (microwatts)

* Radiated power, fourth harmonic: 6 mW

* Harmonic attenuation, second harmonic: 31 dB

* Harmonic attenuation, third harmonic: 42 dB

* Harmonic attenuation, fourth harmonic: 39 dB

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