_24-Transmission Line

Transmission Lines 24-1 TTTTTransmissionransmissionransmissionransmissionransmission LinesLinesLinesLinesLines Chapter 24 Basic Theory of Transmission Lines The desirability of installing an antenna in a clear space, not too near buildings or power and telephone lines, cannot be stressed too strongly. On the other hand, the transmitter that generates the RF power for driving the antenna is usually, as a matter of necessity, located some distance from the antenna terminals. The connecting link between the two is the RF transmission line, feeder or feed line. Its sole purpose is to carry RF power from one place to another, and to do it as efficiently as possible. That is, the ratio of the power transferred by the line to the power lost in it should be as large as the circumstances permit. At radio frequencies, every conductor that has appreciable length compared with the wavelength in use radiates powerevery conductor is an antenna. Special care must be used, therefore, to minimize radiation from the conductors used in RF transmission lines. Without such care, the power radiated by the line may be much larger than that which is lost in the resistance of conduc- tors and dielectrics (insulating materials). Power loss in resistance is inescapable, at least to a degree, but loss by radiation is largely avoidable. Radiation loss from transmission lines can be pre- vented by using two conductors arranged and operated so the electromagnetic field from one is balanced every- where by an equal and opposite field from the other. In such a case, the resultant field is zero everywhere in spacethere is no radiation from the line. For example, Fig 1A shows two parallel conductors having currents I1 and I2 flowing in opposite directions. If the current I1 at point Y on the upper conductor has the same amplitude as the current I2 at the correspond- ing point X on the lower conductor, the fields set up by the two currents are equal in magnitude. Because the two currents are flowing in opposite directions, the field from I1 at Y is 180° out of phase with the field from I2 at X. However, it takes a measurable interval of time for the field from X to travel to Y. If I1 and I2 are alternating currents, the phase of the field from I1 at Y changes in such a time interval, so at the instant the field from X reaches Y, the two fields at Y are not exactly 180° out of phase. The two fields are exactly 180° out of phase at every point in space only when the two conductors occupy the same spacean obviously impossible condition if they are to remain separate conductors. The best that can be done is to make the two fields cancel each other as completely as possible. This can be achieved by keeping the distance d between the two con- ductors small enough so the time interval during which the field from X is moving to Y is a very small part of a cycle. When this is the case, the phase difference between the two fields at any given point is so close to 180° that cancellation is nearly complete. Fig 1Two basic types of transmission lines. Ch24.pmd 9/2/2003, 3:09 PM1 24-2 Chapter 24 Practical values of d (the separation between the two conductors) are determined by the physical limitations of line construction. A separation that meets the condi- tion of being “very small” at one frequency may be quite large at another. For example, if d is 6 inches, the phase difference between the two fields at Y is only a fraction of a degree if the frequency is 3.5 MHz. This is because a distance of 6 inches is such a small fraction of a wave- length (1 λ = 281 feet) at 3.5 MHz. But at 144 MHz, the phase difference is 26°, and at 420 MHz, it is 77°. In neither of these cases could the two fields be considered to “cancel” each other. Conductor separation must be very small in comparison with the wavelength used; it should never exceed 1% of the wavelength, and smaller separa- tions are desirable. Transmission lines consisting of two parallel conductors as in Fig 1A are called open-wire lines, parallel-conductor lines or two-wire lines. A second general type of line construction is shown in Fig 1B. In this case, one of the conductors is tube- shaped and encloses the other conductor. This is called a coaxial line (coax, pronounced “co-ax”) or concentric line. The current flowing on the inner conductor is bal- anced by an equal current flowing in the opposite direc- tion on the inside surface of the outer conductor. Because of skin effect, the current on the inner surface of the outer conductor does not penetrate far enough to appear on the outside surface. In fact, the total electromagnetic field outside the coaxial line (as a result of currents flowing on the conductors inside) is always zero, because the outer conductor acts as a shield at radio frequencies. The sepa- ration between the inner conductor and the outer con- ductor is therefore unimportant from the standpoint of reducing radiation. A third general type of transmission line is the waveguide. Waveguides are discussed in detail in Chap- ter 18, VHF and UHF Antenna Systems. CURRENT FLOW IN LONG LINES In Fig 2, imagine that the connection between the battery and the two wires is made instantaneously and then broken. During the time the wires are in contact with the battery terminals, electrons in wire 1 will be attracted to the positive battery terminal and an equal number of electrons in wire 2 will be repelled from the negative ter- minal. This happens only near the battery terminals at first, because electromagnetic waves do not travel at infi- nite speed. Some time does elapse before the currents flow at the more extreme parts of the wires. By ordinary stan- dards, the elapsed time is very short. Because the speed of wave travel along the wires may approach the speed of light at 300,000,000 meters per second, it becomes nec- essary to measure time in millionths of a second (micro- seconds). For example, suppose that the contact with the bat- tery is so short that it can be measured in a very small fraction of a microsecond. Then the “pulse” of current Fig 2A representation of current flow on a long transmission line. Fig 3A current pulse traveling along a transmission line at the speed of light would reach the successive positions shown at intervals of 0.1 microsecond. that flows at the battery terminals during this time can be represented by the vertical line in Fig 3. At the speed of light this pulse travels 30 meters along the line in 0.1 microsecond, 60 meters in 0.2 microsecond, 90 meters in 0.3 microsecond, and so on, as far as the line reaches. The current does not exist all along the wires; it is only present at the point that the pulse has reached in its travel. At this point it is present in both wires, with the electrons moving in one direction in one wire and in the other direction in the other wire. If the line is infinitely long and has no resistance (or other cause of energy loss), the pulse will travel undiminished forever. By extending the example of Fig 3, it is not hard to see that if, instead of one pulse, a whole series of them were started on the line at equal time intervals, the pulses Ch24.pmd 9/2/2003, 3:09 PM2 Transmission Lines 24-3 conductors occur one cycle later in time than the cur- rents at A and C. Put another way, the currents initiated at A and C do not appear at B and D, one wavelength away, until the applied voltage has gone through a com- plete cycle. Because the applied voltage is always changing, the currents at A and C change in proportion. The current a short distance away from A and C—for instance, at X and Y—is not the same as the current at A and C. This is be- cause the current at X and Y was caused by a value of voltage that occurred slightly earlier in the cycle. This situ- ation holds true all along the line; at any instant the cur- rent anywhere along the line from A to B and C to D is different from the current at any other point on that sec- tion of the line. The remaining series of drawings in Fig 4 shows how the instantaneous currents might be distributed if we could take snapshots of them at intervals of 1/4 cycle. The cur- rent travels out from the input end of the line in waves. At any given point on the line, the current goes through its complete range of ac values in one cycle, just as it does at the input end. Therefore (if there are no losses) an ammeter inserted in either conductor reads exactly the same current at any point along the line, because the ammeter averages the current over a whole cycle. (The phases of the currents at any two separate points are dif- ferent, but the ammeter cannot show phase.) VELOCITY OF PROPAGATION In the example above it was assumed that energy travels along the line at the velocity of light. The actual velocity is very close to that of light only in lines in which the insulation between conductors is air. The presence of dielectrics other than air reduces the velocity. Current flows at the speed of light in any medium only in a vacuum, although the speed in air is close to that in a vacuum. Therefore, the time required for a sig- nal of a given frequency to travel down a length of prac- tical transmission line is longer than the time required for the same signal to travel the same distance in free space. Because of this propagation delay, 360º of a given wave exists in a physically shorter distance on a given transmission line than in free space. The exact delay for a given transmission line is a function of the properties of the line, mainly the dielectric constant of the insulat- ing material between the conductors. This delay is expressed in terms of the speed of light (either as a per- centage or a decimal fraction), and is referred to as velocity factor (VF). The velocity factor is related to the dielectric constant (ε) by ε 1 =VF (Eq 1) The wavelength in a practical line is always shorter than the wavelength in free space, which has a dielectric constant ε = 1.0. Whenever reference is made to a line as Fig 4Instantaneous current along a transmission line at successive time intervals. The frequency is 10 MHz; the time for each complete cycle is 0.1 microsecond. would travel along the line with the same time and dis- tance spacing between them, each pulse independent of the others. In fact, each pulse could even have a different amplitude if the battery voltage were varied between pulses. Furthermore, the pulses could be so closely spaced that they touched each other, in which case current would be present everywhere along the line simultaneously. It follows from this that an alternating voltage applied to the line would give rise to the sort of current flow shown in Fig 4. If the frequency of the ac voltage is 10,000,000 hertz or 10 MHz, each cycle occupies 0.1 µsecond, so a complete cycle of current will be present along each 30 meters of line. This is a distance of one wavelength. Any currents at points B and D on the two Ch24.pmd 9/2/2003, 3:09 PM3 24-4 Chapter 24 being a half wavelength or quarter wavelength long (λ/2 or λ/4), it is understood that what is meant by this is the electrical length of the line. The physical length corre- sponding to an electrical wavelength on a given line is given by VF f 6.983 =)feet( ×λ (Eq 2) where f = frequency in MHz VF = velocity factor Values of VF for several common types of lines are given later in this chapter. The actual VF of a given cable varies slightly from one production run or manufacturer to another, even though the cables may have exactly the same specifications. As we shall see later, a quarter-wavelength line is frequently used as an impedance transformer, and so it is convenient to calculate the length of a quarter-wave line directly by λ/4 = VFf 9.245 × (Eq 2A) CHARACTERISTIC IMPEDANCE If the line could be perfect—having no resistive losses—a question might arise: What is the amplitude of the current in a pulse applied to this line? Will a larger voltage result in a larger current, or is the current theo- retically infinite for an applied voltage, as we would expect from applying Ohm’s Law to a circuit without resistance? The answer is that the current does depend directly on the voltage, just as though resistance were present. The reason for this is that the current flowing in the line is something like the charging current that flows when a battery is connected to a capacitor. That is, the line has capacitance. However, it also has inductance. Both of these are “distributed” properties. We may think of the line as being composed of a whole series of small induc- tors and capacitors, connected as in Fig 5, where each coil is the inductance of an extremely small section of wire, and the capacitance is that existing between the same two sections. Each series inductor acts to limit the rate at which current can charge the following shunt capacitor, and in so doing establishes a very important property of a transmission line: its surge impedance, more commonly known as its characteristic impedance. This is abbrevi- ated by convention as Z0. TERMINATED LINES The value of the characteristic impedance is equal to C/L in a perfect line—that is, one in which the con- ductors have no resistance and there is no leakage between them—where L and C are the inductance and capacitance, respectively, per unit length of line. The inductance decreases with increasing conductor diameter, and the capacitance decreases with increasing spacing between the conductors. Hence a line with closely spaced large conductors has a relatively low characteristic impedance, while one with widely spaced thin conductors has a high impedance. Practical values of Z0 for parallel-conductor lines range from about 200 to 800 Ω. Typical coaxial lines have characteristic impedances from 30 to 100 Ω. Physi- cal constraints on practical wire diameters and spacings limit Z0 values to these ranges. In the earlier discussion of current traveling along a transmission line, we assumed that the line was infinitely long. Practical lines have a definite length, and they are terminated in a load at the output or load end (the end to which the power is delivered). In Fig 6, if the load is a pure resistance of a value equal to the characteristic impedance of a perfect, lossless line, the current travel- ing along the line to the load finds that the load simply “looks like” more transmission line of the same charac- teristic impedance. The reason for this can be more easily understood by considering it from another viewpoint. Along a trans- mission line, power is transferred successively from one elementary section in Fig 5 to the next. When the line is infinitely long, this power transfer goes on in one directionaway from the source of power. From the standpoint of Section B, Fig 5, for instance, the power transferred to section C has simply disappeared in C. As far as section B is concerned, it makes no differ- ence whether C has absorbed the power itself or has trans- ferred it along to more transmission line. Consequently, if we substitute a load for section C that has the same Fig 5Equivalent of an ideal (lossless) transmission line in terms of ordinary circuit elements (lumped constants). The values of inductance and capacitance depend on the line construction. Fig 6A transmission line terminated in a resistive load equal to the characteristic impedance of the line. Ch24.pmd 9/2/2003, 3:09 PM4 Transmission Lines 24-5 electrical characteristics as the transmission line, section B will transfer power into it just as if it were more trans- mission line. A pure resistance equal to the characteristic impedance of C, which is also the characteristic imped- ance of the line, meets this condition. It absorbs all the power just as the infinitely long line absorbs all the power transferred by section B. Matched Lines A line terminated in a load equal to the complex char- acteristic line impedance is said to be matched. In a matched transmission line, power is transferred outward along the line from the source until it reaches the load, where it is completely absorbed. Thus with either the infinitely long line or its matched counterpart, the impedance presented to the source of power (the line- input impedance) is the same regardless of the line length. It is simply equal to the characteristic impedance of the line. The current in such a line is equal to the applied voltage divided by the characteristic impedance, and the power put into it is E2/Z0 or I2Z0, by Ohm’s Law. Mismatched Lines Now take the case where the terminating load is not equal to Z0, as in Fig 7. The load no longer looks like more line to the section of line immediately adjacent. Such a line is said to be mismatched. The more that the load impedance differs from Z0, the greater the mismatch. The power reaching the load is not totally absorbed, as it was when the load was equal to Z0, because the load requires a voltage to current ratio that is different from the one traveling along the line. The result is that the load absorbs only part of the power reaching it (the incident power); the remainder acts as though it had bounced off a wall and starts back along the line toward the source. This is known as reflected power, and the greater the mismatch, the larger is the percentage of the incident power that is reflected. In the extreme case where the load is zero (a short circuit) or infinity (an open circuit), all of the power reaching the end of the line is reflected back toward the source. Whenever there is a mismatch, power is transferred in both directions along the line. The voltage to current ratio is the same for the reflected power as for the inci- dent power, because this ratio is determined by the Z0 of the line. The voltage and current travel along the line in both directions in the same wave motion shown in Fig 4. If the source of power is an ac generator, the incident (outgoing) voltage and the reflected (returning) voltage are simultaneously present all along the line. The actual voltage at any point along the line is the vector sum of the two components, taking into account the phases of each component. The same is true of the current. The effect of the incident and reflected components on the behavior of the line can be understood more readily by considering first the two limiting casesthe short- circuited line and the open-circuited line. If the line is short-circuited as in Fig 7B, the voltage at the end must be zero. Thus the incident voltage must disappear sud- denly at the short. It can do this only if the reflected volt- age is opposite in phase and of the same amplitude. This is shown by the vectors in Fig 8. The current, however, does not disappear in the short circuit; in fact, the inci- dent current flows through the short and there is in addi- tion the reflected component in phase with it and of the same amplitude. The reflected voltage and current must have the same amplitudes as the incident voltage and current, because no power is dissipated in the short circuit; all the power starts back toward the source. Reversing the phase of either the current or voltage (but not both) reverses the Fig 7Mismatched lines; extreme cases. At A, termination not equal to Z0; at B, short-circuited line; At C, open-circuited line. Fig 8Voltage and current at the short circuit on a short-circuited line. These vectors show how the outgoing voltage and current (A) combine with the reflected voltage and current (B) to result in high current and very low voltage in the short circuit (C). Ch24.pm