INPUT IMPEDANCE OF THE LINE

The input impedance of the line depends on features like the ohmic resistance, the conductance, the inductance and the capacitance. It is also related to the resistance that loads the line at the opposite end and to both the frequency and the frequency and the voltage of the input signal.

The purpose of the first part of the test is to measure the modulus of the input impedance of the line under different load conditions: open line, line terminated on a matched load, short-circuited line. In the second part of the test we will measure the phase displacement between the input voltage and current, under the same conditions of line operation.

When the modulus and the phase displacement are known, the Impedance vector is fully identified.

Required components, instruments and accessories

1. Cable with intermediate sockets
2. Line termination resistors
3. Function generator
4. Oscilloscope

F10-5 shows the connections to be realized to perform the measurement of the input impedance modulus.

The signal generator with the line matching resistance supplies the transmission line at one end. The load at the opposite end is composed either by an infinite value resistance (open line), a matched load (68 ohms) or a null resistance (short-circuited line).

The resistance Rm with 1 ohm value in series connected between the generator and the transmission line allows to measure the value of the input current of the line for each value of the voltage connected t the line.

It is suggested that the measurements are repeated in more points within the frequency range 10 kHz to 1 MHz. Moreover, during each measurement the amplitude value of the input signal should be kept constant and of significant amplitude (for instance 10 V rms). This allows a comfortable measurement of the voltage drop value across Rs, which is necessarily small.

The results, gathered in tabular form, are subsequently processed calculating the modulus of the input impedance according to the following formula:

|Zin| = (Vin/I) = (Vin/Vm) . 1 Ω

The measurements have to be repeated in the three cases of open line, short-circuited line and closed on matches load.

The second part of the test consists in measuring the phase displacement between the current and the voltage at the input of the line, under the different load conditions stated above and for different frequency values.

It is possible to use either the measuring setup shown in F10-6 or the arrangement shown in the next F10-7, which allows to obtain the same measuring result with two different methods.

With the first method the duel-trace oscilloscope is connected with both channels for displaying the voltage waveform and the current waveform at the input of the line, respectively.

By using as trigger the input voltage signal, the phase displacement of the current against this signal can be evaluated by measuring the time delay after which the corresponding waveform is displayed against the former one.

With the second measuring method, the oscilloscope is connected with the X-axis to the input voltage signal and with the Y-axis to the current. On the screen the so-called Lissajous display is then determined, which allows the phase displacement between the two signals to be evaluated through the ratio of the semi-axis of the ellipse.

Again the results, gathered in a table, are subsequently translated into graphical form. This provides, together with the graph relevant to the input impedance modulus, the full information of modulus and phase of the input signal for different frequencies and for different load conditions of the line.

F10-5: Measuring the input impedance.

F10-6: Measuring the phase displacement between the input current and voltage.

F10-7: Measuring the phase displacement between the input current and voltage.

FREQUENCY CHARACTERISTICS OF LINE

When the frequency of the input signals increases, the line attenuation due to both the ohmic resistance R′ and the conductance C′ progressively increases because of the so-called “skin effect”.

Starting from a given frequency onward, the line attenuation quickly increases. The “cut-off frequency” of the line is defined as the frequency value at which the attenuation reaches the value of 3 dB compared to the low-frequency attenuation value.
The purpse of this test is to measure the cut-off frequency for the coaxial line provided on the DL2597. This measurement is performed in conditions of line closed on a load equal to the characteristic impedance.
Required components, instruments and accessories

1. Cable with intermediate sockets
2. Variable line termination resistors
3. Connecting cables and chords
4. Function generator
5. Oscilloscope

Procedure

F10-4 shows the circuit used to perform the test. Substantially, the line is closed on the load Rt and the generator is connected to the input of the line. The internal impedance of the generator is matched to the line impedance through the resistor Ri = Rt. As already said in the previous test, this is true when a generator  is used whose impedance is neglectable when compared to 68 ohms. Otherwise, for Ri a value has to be selected equal to the required resistance (68 ohms) and the output impedance of the generator.
The test consists in scanning a wide frequency range, for instance from 100 kHz to 4 MHz, with suitable amplitude steps, measuring the attenuation in the different measurement points selected.
It is suggested to use for the input signal a peak-to-peak amplitude of 2 V, which is kept constant in the whole measuring range. The results can be gathered in tabular form.
From the table data, which are in the following translated in graphical form, the cut-off frequency of the line can be inferred. As stated above, this is the frequency at which the attenuation is reduced f 3 dB compared to the value measured at low frequency (10 kHz).
We recall that a voltage attenuation of 3 dB corresponds to a signal attenuated to one half compared to the input signal.

F10-4: Measuring the frequency characteristic of the line.

MEASURING THE ATTENUATION OF A LINE

The ohmic resistance R′ and the conductance G′ are responsible within the line of the energy dissipation in the form of heat. This occurs within both the conductor and the dielectric material. These losses, which determine the attenuation characteristics of the line, are expressed in terms of “attenuation constant”, represented with the symbol “a”, and can be calculated through the following formula:

av = 20 . Log (V1/V2)

Where:

V1 = amplitude of the signal at the input of the line.

V2 = amplitude of the signal at the output of the line.

av = voltage attenuation in dB.

The purpose of this test is to measure the attenuation for the different trunks of the transmission line.

Procedure

F10-3 shows the method for the measurement to be performed. One end of the line is closed on the rated load Rt, while on the other end the sine-wave signal generator is connected.

The purpose of the resistance Ri is to match the generator to the line.

The value of Ri is equal to the load resistance connected to the output of the line, i.e. 68 ohms. This is true when the signal generator used has a neglectable internal impedance when compared to this ohmic value. In the opposite case it will be necessary to externally provide a resistance Ri whose value is such that, when added to the internal impedance of the generator, the result is 68 ohms.

The test is performed by setting the generator for an output signal of 3 V rms and a frequency of 50 kHz.

By using either the oscilloscope or the multimeter, the amplitude are measured that can be detected at the input of the line and at 25, 50, 75 and 100 meters.

The results are gathered in tabular form t be processed in the following.

The attenuations in the points where the measure has been performed are calculated through the following formula:

av = 20 . Log (V1/Vn)

Where:

Vn = amplitude of the signal measured at the distances of 25, 50, 75 and 100 m.

F10-3: Measuring the attenuation.

MEASURING THE CHARACTERISTICS OF A LINE

The first test proposed consists in experimentally measuring the characteristics (R′, L′, G′, C′, Z₀ and γ) for the transmission line included in DL 2597.

Required components, instruments and accessories

1. Coaxial cable (100 meters) with intermediate sockets.
2. Connecting cables and chords.
3. Universal measuring bridge.
4. Digital multimeter.

Procedure

F10-2 synthetically shows the modalities for the measurement to be performed. Both the inductance and the ohmic resistance of the line are measured by short-circuiting an end of the line and connecting the measuring instruments to the other end. Instead, the capacitance and the conductance are measured by operating on the open line.

The resistance R and the conductance G are measured with an ohmmeter. For the conductance to be measured an ohmmeter is required which is able to perform resistance measurements with a range greater than 100 Mhoms.

For the measurement of both the inductance L and the capacitance C a measuring bridge is required.

The results of these measurements will be values of R, L, C and G referred to the cable length that, in out case, is of 100 meters.

In accordance with the knowledge acquired from the theoretical course, the characteristic impedance Z₀ can be calculated from the measured values through the following formula:

In the great majority of applications the lines are used in working frequency ranges whose terms of series resistance R′ and parallel conductance G′ normally produce a neglectable effect on the characteristic impedance Z₀. The latter, therefore, is essentially determined only by the capacitive and inductive terms. Therefore, assuming ωL′ to be much smaller than R′ and ωC′ to be much smaller than G′, the previous formula is reduced to:

Use the above formula to calculate, from the measured values, the characteristic impedance of the line.

F10-1: Equivalent circuit of a coaxial line.

F10-2a: Measuring the inductance (L) and resistance (R).

F10-2b: Measuring the capacitance (C) and the conductance (G)