# Understanding RF Transmission Lines by Measurement and Calculation page 2

In the application of a coaxial transmission line, it is normally necessary to match both the signal source impedance and the load impedance to that of the coaxial cable in order to ensure the maximum power transfer and minimum reflections. In the majority of RF and Microwave systems, there is a single characteristic impedance. The most common industry standard being 50 Ohms and all of the measurements made in this study are assumed to be made using 50 Ohm equipment. The following measurements simply prove that the impedance of a termination, signal generator and coaxial cable are all 50 Ohms.

In paragraph 1.2, the input voltage to the termination with +13 dBm at 50 MHz was measured as 1 Volt. In paragraph 1.3, the input current to the termination with +13 dBm at 50 MHz was measured as 20 mA. Using Ohms law, we can easily calculate the impedance of the termination.

R = V/I

Substituting Z for R  Z = V/I Formula 2.1

Z = 1/.02 = 50 Ohms

This proves that the LOAD termination is 50 Ohms at 50 MHz and demonstrates that we can use simple dc formulas to assist in understanding the operation of a high frequency circuit if we use suitable test equipment.

The theoretical output circuit of a signal generator with an output of 1 Volt rms, or approximately +13 dB, is shown in Figure 2.1. The generator has a source resistance (Rs) of 50 Ohms. This resistance (and it can be an actual resistor at the signal generator output stage) matches the signal generator source to any 50 Ohm cables or equipment at the output.

Figure 2.1 We first measure the Signal Generator output voltage with the LOAD Termination and then again without the LOAD Termination, in order to demonstrate the source impedance of the Signal Generator. The Voltage and Current Detector is working in a linear part of its range, so there is no need to linearise the indicated readings for this measurement.

The Voltage and Current Detector is connected directly to the output of the Signal Generator and is terminated with a 50 Ohm load as shown in Figure 1.1. The Signal Generator output is set to +13 dBm at a frequency of 50 MHz. The DMM is first connected to the Voltage and Current Detector, current output and should indicate approximately 200 mV dc (representing a signal current of 20mA). The DMM is then connected to the Voltage and Current Detector, voltage output and should indicate approximately 1 Volt dc (representing a signal of 1 Volt rms). The measured voltage of 1 Volt rms, represents the Potential Difference (PD) at the output socket of the Signal Generator into the 50 Ohm LOAD Termination ( Rl).

The LOAD Termination is now removed and the output voltage is doubled to 2 Volt rms giving a DMM reading of 2 Volts dc. There is now virtually no current flowing in the source resistance (Rs) of the Signal Generator and therefore no voltage drop across it. The V/C Detector is now measuring the Electromotive Force (EMF) of the voltage source in the Signal Generator.

With the 50 Ohm termination connected, the generator output (PD) was 1 Volt rms at 20 mA.

The EMF was 2 Volts, therefore the voltage across Rs was EMF - PD or 2 - 1 Volts = 1 Volt

So, Rs must be 1 Volt / 0.02 Amps = 50 Ohms.

We have now proven that the signal generator source resistance (Rs), or more correctly the signal generator impedance and the LOAD termination impedance are both 50 Ohms at 50 MHz.

The effect that the coaxial cable has in a circuit is related to the cable length, characteristic impedance in relation to termination impedance and the operating frequency of the system. The longer a cable is, the more it will approach to its characteristic impedance and with a theoretical infinite length cable the input impedance is equal to the characteristic impedance regardless of what terminates the other end. For this reason, when measuring the characteristic impedance of a cable it is advisable to use a reasonably long length and a two metre length is adequate for the frequencies used in the next measurement. The selected coaxial cable for the demonstration is RG223, which is a good quality cable and its characteristic impedance is close to the ideal of 50 Ohms.

The coaxial cable of length two metres is fitted between the Voltage and Current Detector and the LOAD Termination as per Figure 2.2. The Voltage and Current Detector is connected directly to the output of the Signal Generator. The signal generator is set to give an output of +13 dBm ( or 1 Volt rms ) at a frequency of 50 MHz. The indicated voltage and current are recorded.

Figure 2.2 The Voltage and Current Detector is working at its calibration point, so there is no need to linearise the indicated readings.

The measurement results can now be used to calculate the impedance of the Coaxial Cable using:-

Z = V/I   Formula 2.1

As can be seen from the calculation results, the impedance remains at 50 Ohms with the cable fitted.

We have now proven by measurement that the LOAD Termination, Signal Generator and Coaxial Cable all have an impedance of 50 Ohms.

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