Characteristic Impedance of Cables
at High and Low Frequencies

By Marco Pavincich
Ref:- RSI, Times Microwave, Belden Companies

The increasing use of electrical pulses in the transmission of data by cable has resulted in a need for a better understanding of the electrical characteristics of a cable.

Many system specifications state that the cables used shall have a certain characteristic impedance, specified in ohms. Any cable-maker's catalogue will list the characteristic impedance values of most coaxial cables, which usually range from 50 to 95 ohms. But impedance information on the more common types of shielded pairs is not readily available to the cable user. Why? Because there are too many variations of applications involved with these cables.

IMPEDANCE DEFINED

Let us examine characteristic impedance - what it is and what it is not - so that a better understanding of what the numbers mean and how they apply to our application can be derived.

Magnitude of Impedance

Ohm's Law states that if a voltage (E) is applied to a pair of terminals and a current (I) is measured in this circuit, the following equation can be used to determine the magnitude of the impedance (Z).

(Equation 1)

Equation 1: E/I = Z

This relationship holds true whether talking about direct current (DC) or alternating current (AC).

Phase

In the case of alternating current, impedance has two components: magnitude in ohms (discussed earlier) and phase. Phase refers to the instantaneous relationship between voltage and current. Looking at the voltage and current simultaneously on a dual-trace oscilloscope, observe that in the case of a resistor (see Figure 1A), the voltage and current attain a maximum at the same time and are said to be “in phase” with each other.

Figure 1:  Phase Relationships

Figure 1. Phase relationship between voltage and current. (The horizontal axis of the oscilloscope is a linear time base, reading from left to right.)

However, in the case of an inductor (usually a coil, but even a straight wire has some inductance) the voltage goes through a maximum earlier than the current (voltage is said to lead the current) (see Figure 1B). This is because an inductance resists a change in current.

Conversely, in the case of a capacitor, the voltage goes through a maximum later than the current (voltage is said to lag the current) because it takes time to charge the capacitor. (See Figure 1C.)

Phase Angle

If one complete cycle (the distance between two successive voltage or current maxima as shown in Figure 1A) is equivalent to 360o, then the number of degrees between a current maximum and a voltage maximum can be determined. For a perfect inductor, the voltage maximum occurs cycle (or 90o) before the current maximum. (The voltage is 90o ahead of the current, or the phase angle is +90o). (See Figure 1B). In the case of a perfect capacitor, the voltage is 1/4 cycle (or 90o) behind the current, so the phase angle is -90o.

In reality, perfect inductors and capacitors do not exist. They contain some resistance, so the phase angle is always less than 90o, whether it be a positive or negative 90o. (See Figure 1D.)

CABLE CHARACTERISTICS

What does all this have to do with cable? Simply this - the conductors, insulation, and shield (if present), when assembled into a cable, result in resistances, capacitances, and inductances. The equivalent circuit of a cable is shown in Figure 2.

Figure 2:  Equivalent circuit

Figure 2. Equivalent circuit of a cable connected to an AC voltage source.

The circuit of Figure 2 is not meant to imply that a cable has only one capacitance (C), one inductance (L), one conductance (G), and one resistance (R). The cable is equivalent to many of each depending on the length of cable being tested. When calculations are made on a specific length of cable, all four parameters must maintain the same length units throughout the calculations. These parameters are usually given as “per foot” or “per meter.”

Note that the resistance across the line, which represents the losses in the insulation, is shown as a parallel conductance (G), rather than a parallel resistance (R). The reason for this is that when there exists many resistances in parallel, the method of finding the total or equivalent resistance requires tedious mathematics, whereas when dealing with conductances1 in parallel, the calculation required is simple addition.

When an alternating voltage is applied to the cable, with the far end open, a current will flow. With voltage (E) and current (I) measured in this circuit, impedance (Z) can now be calculated (Z = E/I). The impedance will have some magnitude and some phase angle, which can be either positive or negative. However, if a portion of the cable is cut off and the measurement is repeated, a different impedance magnitude and a different phase angle will be observed.

The characteristic impedance (Zo) of a cable is independent of length, so obviously these measurements do not yield the characteristic impedance.

Theoretical Definition of Characteristic Impedance

For the moment, imagine that a cable has infinite length. Application of an alternating voltage to this infinitely long cable would allow a measurement of current and calculation of impedance (both magnitude and phase). Of course, there is no such thing as an infinitely long cable. The concept is introduced to promote the idea of a cable which is so long that the signal never gets to the far end. The impedance measurement in this case would yield the impedance of the cable itself, or its characteristic impedance, and not the combination of the cable impedance plus the effect of conditions at the far end. The phase angle would be zero or negative (between 0 and -45o). Recalling circuit theory, a negative phase angle indicates that at the specific frequency at which the measurements were taken, the cable resembles a capacitor with a resistor in series.

Calculations of an equivalent capacitor and resistor to replace the infinitely long cable could be made. Replacement of the cable in the test circuit by these two components and application of the test voltage would result in the identical current and phase angle as observed when using the cable. As long as the frequency of the applied AC voltage does not change, impedance measurements would be identical for both the capacitor-resistor combination and the infinitely long cable.

If the infinitely long cable is cut to some finite length and the far end of this cable is connected to a capacitor-resistor combination which is assembled and found to be equal to the characteristic impedance, an astounding discovery is made. The impedance measured looking into the cable which is terminated at the far end with its matching characteristic impedance (the capacitor-resistor combination) is still the same as it was for the infinitely long cable! Cut the cable to any length and if the termination at the far end is unchanged and the frequency is unchanged, no difference in the measured impedance will be noticed.

EFFECT OF FREQUENCY

Note that the frequency of the AC signal has not been mentioned; only the characteristic impedance at the specific frequency is defined.

The impedance of inductors and capacitors is dependent on frequency.

Inductive Reactance

The impedance which is due to pure inductance is called inductive reactance (XL) and, in ohms, is equal to 2(3.1416)f L where f is frequency in hertz (cycles per second) and L is inductance in henries. It can be seen by the equation that inductive reactance increases when frequency increases.

Capacitive Reactance

The impedance which is due to pure capacitance, called capacitive reactance (Xc), is equal in ohms to 1/(2(3.1416) f C), where f is frequency in hertz and C is capacitance in farads. It can be seen by this equation that capacitive reactance decreases when frequency increases.

Characteristic Impedance

What effect does the frequency dependence of the inductive and capacitive reactance have on the characteristic impedance of the cable? The answer will depend on whether the discussion is about “low” frequencies or “high” frequencies.

It can be shown that the characteristic impedance (Zo) of a cable is:

(Equation 2)

Equation 2

 

Where: R = The series resistance of the conductor in ohms per unit length.
G = The shunt conductance in mhos per unit length
j = A symbol indicating that the term has a phase angle of +90o
pi = 3.1416
Symbols f, L. and C are as previously identified.

For materials commonly used for cable insulation, G is small enough that it can be neglected when compared with 2(3.1416) f C.

At “low” frequencies, 2(3.1416) f L is so small compared with R that it can be neglected. Therefore, at low frequencies, the following equation can be used:

(Equation 3)

Equation 3

 

Where: R = DC resistance.

Figure 3: Relative dieletric constant vs. frequency

Figure 3. Relative dielectric constant vs. frequency.

If the capacitance does not vary with frequency, the Zo varies inversely with the square root of the frequency and has a phase angle which is -45o near DC and decreases to 0o as frequency increases. Polyvinyl chloride and rubber decrease somewhat in capacitance as frequency increases, while polyethylene, polypropylene, and Teflon* do not vary significantly. (See Figure 3.)

If the magnitude of the impedance (in ohms) is plotted as a function of frequency (in hertz) on log-log graph paper, the result is a straight line having a slope of one decade of ohms for two decades of frequency, extending downward and to the right. (See Figure 4.)

Referring to Equation 2 again, observe what happens at “high” frequencies.

When f becomes large enough, the two terms containing f become so large that R and G may be neglected and the resultant equation is:

(Equation 4)

Equation 4

 

Note that j2(3.1416)f appears in both numerator and denominator of Equation 4 and, therefore, can be cancelled leaving:

(Equation 5)

Equation 5

 

Figure 4: Slope of Low Frequency Impedance curve

Figure 4. Slope of Low Frequency Impedance curve. (Actual data may be above or below the curve but will be parallel to it in the low frequency region.)

If L and C are independent of frequency, which is essentially true in the “high” frequency region, Zo is constant, and since there are no “j” terms, the phase angle is zero, indicating that the “high” frequency characteristic impedance is a pure resistance. This could be plotted on log paper as a straight horizontal line (Figure 5.) The characteristic impedance which is normally listed in a cable catalogue is this constant high frequency impedance.

 

Firgure 5: Slope of High Frequency Impedance curve

Figure 5. Slope of High Frequency Impedance curve. (Actual data may be above or below this curve but will be parallel to it in the high frequency region.)

How does one determine whether a cable is operating in the “low” frequency region or in the “high” frequency region?

The Figure 4 and Figure 5 curves are plotted on the same log-log graph paper in Figure 6. If a particular cable characteristic point falls well to the left of where the high and low frequency curves intersect, the location would be in the low frequency region. If the cable characteristic point falls well to the right of the intersection, it would be considered in the high frequency region. Near the intersection, a transition region appears, where all the terms in Equation 2 have some effect and the true impedance curve will have a smooth transition between the two straight intersecting lines (see Figure 6). At the high frequency end of the transition region the inductance decreases slightly as frequency increases so there is a slight downward slope of the Zo curve as it approaches the final “high” frequency value.

 

Figure 6:  Impedance vs. frequency

Figure 6. Impedance vs. frequency. (Dotted line joining low frequency curve and high frequency curve represents the transition region. Frequency range at which it occurs depends on the position of the low frequency curve.)

Actual test data from two cables having different wire sizes and hence different conductor resistances are plotted together in Figure 7 to illustrate the effect of conductor resistance on the position of the low frequency region and the transition region. These two cables are the largest and the smallest in a series of shielded polyethylene insulted single pair cables which were designed to have essentially the same capacitance regardless of conductor size. Note that neither curve reaches the final “high” frequency value for Zo within the frequencies covered by the data.

How are actual values determined for R, L, C, and G at each frequency in the transition region so that Zo can be determined? It is possible to calculate R, L, C, and G from the dimensions of the cable and from the characteristics of the conductor and insulation but this is very tedious. R varies with frequency in the transition region because as frequency increases, the current tends to travel only near the surface of the conductor (skin effect). Reliable values for C are difficult to obtain if the capacitance of the insulating material varies with frequency.

Fortunately, a relationship exists which makes determination of Zo rather simple with the proper equipment. It can be shown that if, at a given frequency, the impedance of a length of cable is measure with the far end open (Zoc), and the measurement is repeated with the far end shorted (Zsc), the following equation may be used to determine Zo:

(Equation 6)

Equation 6

 

NOTE

The Zoc and Zsc measurements both have magnitude and phase, so the Zo determined by Equation 6 will also have magnitude and phase.

“High” frequency measurements of Zo are made by determining the velocity of propagation and capacitance of the cable or by reflectometry.

Why is characteristic impedance important in data transmission? Remember, if a cable is terminated in its matching characteristic impedance you can't tell from the sending end that the cable is not infinitely long - all the signal that is fed into the cable is taken by the cable and the load much as if evenly spaced waves were being generated from the shore of a very large body of water. The waves are all outward bound and undisturbed.

However, if a barrier is placed a short distance out in the body of water, the waves will be reflected back toward the shore, distorting the outbound waves. When these reflected waves hit the wave generator, they are again reflected and mingle with the outbound waves so that it is difficult to tell which waves are original and which are re-reflections.

 

Figure 7:  Impedance vs. Frequency

Figure 7. Impedance vs. Frequency. (Actual test data from a No. 24 AWG shielded pair and No. 12 AWG shielded pair.)

 

The same thing happens when pulses are sent down the cable - when they encounter an impedance other than the characteristic impedance of the cable, a portion of their energy is reflected back to the sending end. If the pulses encounter an open circuit or a short circuit, all of the energy is reflected (except for losses due to attenuation - another subject). For other terminations, smaller amounts of energy will be reflected.

This reflected energy distorts the pulse, and if the impedance of the pulse generator is not the same as the characteristic impedance of the cable, the energy will be re-reflected back down the cable, appearing as extra pulses. These extra reflected pulses may result in erroneous data being received at the end of the cable, depending on the magnitude and timing of the reflected pulse compared to the original pulse.

In the high frequency region, it is a relatively simple matter to make the load resistive and equal to the cable impedance. However, pulses are a mixture of low and high frequencies, depending on their rise time, duration, and repetition rate. It is up to the system designer to determine whether the rising impedance of the cable at low frequencies is going to cause any difficulty and to take whatever design steps may be necessary to allow for it.

Another problem which must be considered, particularly in the low frequency region, is the fact that a pulse generator which may be matched to the characteristic impedance of the cable at high frequencies may be working into a much different impedance at low frequencies. This may affect the operation of the signal source.

MICROPHONE CABLE

To clarify a point which confuses many people, description of microphone cables as “high impedance” and “low impedance” has nothing to do with actual matching of the characteristic impedance of the cable to that of the microphone. A “high impedance” microphone has an internal impedance of tens of thousands of ohms. To avoid excessive loading, the cable should have low capacitance, while the resistance of the conductor is of little importance. A low impedance microphone on the other hand, has an internal impedance of a few hundred ohms, so capacitive loading is less important but conductor resistance can become important in long runs.

Notes:
1
A conductance is equal to the reciprocal of a resistance. Example: a 20 ohm resistance = 1/20 = .05 mho conductance.

*Trademark of DuPont


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Last revised: Tuesday, 10 November 1998