SWR, Return Loss, and Reflection Coefficient
This paper is intended to give the newcomer to RF terminology a brief overview of SWR, return loss and reflection coefficient. Instead of concentrating on mathematical derivations or formulas which can make simple ideas seem complicated, this paper will endeavor to explain the fundamental principals and physical relevance of the terms.
SWR, which stands for standing wave ratio, may be illustrated by considering the voltage at various points along a cable driving a poorly matched antenna. A mismatched antenna reflects some of the incident power back toward the transmitter and since this reflected wave is traveling in the opposite direction as the incident wave, there will be some points along the cable where the two waves are in phase and other points where the waves are out of phase (assuming a sufficiently long cable). If one could attach an RF voltmeter at these two points, the two voltages could be measured and their ratio would be the SWR. Identical results would be obtained by measuring currents. By convention, this ratio is calculated with the higher voltage or current in the numerator so that the SWR is one or greater.
Here are two examples to illustrate how the numbers work. Consider a 1 volt source driving a cable with either a short or open on the end such that all of the power is reflected. Since the reflected wave is as big as the incident wave there will be points where the two voltages cancel completely and other places where the voltage will be 2 volts. The ratio of 2/0 is infinity which is as “bad” as the SWR can be. If, instead, the load were equal to the characteristic impedance of the feed line, say 50 ohms, no power would be reflected and only a constant incident wave would appear at all points along the cable. The ratio of any two voltages would therefore be 1 which is as “good” as the SWR can be. The SWR for terminations between these two extremes may be calculated by considering the interaction of the reflected wave with the incident wave to determine the minimum and maximum voltages. But, as it turns out, the SWR is simply the ratio of the resistance of the termination and the characteristic impedance of the line. For example, a 75 ohm load will give an SWR of 1.5 when used to terminate a 50 ohm cable since 75/50 = 1.5. A 25 ohm resistor will give an SWR of 2 since 50/25 = 2. Note that the larger resistance is always used in the numerator by convention.
Consider that the concept of a reflected wave also works at “DC”. Suppose that a long pair of superconducting jumper cables are connected to a 12 volt car battery and the far end of the cables are touched together. The battery will be “shorted out” as long as the cables are touching: that is, the battery voltage will fall to zero and the current will be limited only by the internal resistance of the battery. Another way to describe what is happening is to say that 6 volts travels down the cables where is encounters the short and is reflected back inverted in “phase”. This -6 volt reflected wave cancels the +6 volts at all points on the cables. In this example, the characteristic impedance of this system is the battery’s internal resistance: if a resistor of the same value is connected to the ends of the cable then the voltage will drop to 6 volts and maximum power will be delivered to the resistor. When the short is removed the 6 volts reflects off of the open circuit without inversion and it adds constructively bringing the voltage on the cable up to 12 volts. A 12 volt battery could be said to be a 6 volt source driving a poorly matched load. The battery is a 6 volt source when it is loaded by its characteristic impedance which rarely happens – most batteries couldn’t withstand a “good” match for very long! The point is that SWR, return loss, etc. are valid concepts for long cables, short cables, no cables, or even ideal non-dimensional parts. And perhaps more importantly, simple Ohm’s Law computations at DC will give the same results as the more mysterious RF equations involving magically reflecting signals and characteristic impedance.
For example, consider a 2 volt battery with a 50 ohm internal resistance driving a 50 ohm load through 50 ohm coax cable. (Follow along on a piece of scratch paper!) The match is optimum and the maximum power of 20 mW is delivered to the load. (1 volt squared / 50 ohms.) Now consider a 100 ohm load. The current is 2/150 = 13.3ma and the resulting voltage across the 100 ohms is 1.33 volts. The power dissipated in the resistor is 1.33 x 13.3 ma = 17.8mw. Since incident and reflected power concepts are valid at DC it could be said that 20 mW arrives at the 100 ohm resistor which absorbs 17.8 mW and reflects the remaining 2.2 mW. The reflected 2.2 mW has a voltage of 0.33 volts in the 50 ohm cable. This reflected voltage adds to the 1 volt incident wave to give 1.33 volts. For a very low frequency there would also be a point along a sufficiently long cable where the voltages would subtract giving 0.67 volts. (As the frequency approaches DC, the required cable length approaches infinity!) The SWR is therefore: 1.33/.67 = 2. It is indeed easier to calculate the ratio of the resistors as mentioned earlier! Obviously, at DC the wavelength is infinite and only the voltage addition is observed. Note that the reflected wave is not inverted when the resistance is greater than the characteristic impedance of the cable! (Here is a memory aid: remember the DC example where a short reflected a canceling negative voltage. Obviously, a lower resistance reflects an inverted wave.) Also, notice that the voltage across the 100 ohm resistor (1.33 volts) is equal to the voltage that would appear across a 50 ohm resistor (1 volt) added to the reflected voltage (0.33 volts). Although this description may seem like an artificial construction, consider what happens when the battery is first connected. With a fast oscilloscope connected at a midpoint on the cable, the 1 volt could be observed as it passes as a step increase. When the 1 volt arrives at the load, 0.33 volts is reflected and is observed a short time later bumping the 1 volt up to 1.33 volts where the scope is connected. The voltage does not simply go up to 1.33 volts in one step!
In cases involving RF signals, some time will pass during the ’round trip of the reflected energy and the phase of the reflection will also depend upon this length of time. Imagine that a resistor in a black box is at the end of a length of cable. From the outside world this length of cable will give the reflection from the resistor a phase shift since the signal must make a round trip through the length. If a 100 ohm resistor has an SWR of 2, a cable long enough to invert the signal after the round trip will make it look like a 25 ohm resistor, also with an SWR of 2 but with inversion (a cable with a multiple of 1/4 wavelength would do the trick). Since the impedance looking into this black box is a function of the SWR and the cable length, it can be seen that intentionally mismatched lines can be used to transform one impedance into another. Notice that the 1/4 wave cable inverts the impedance and preserves the SWR. This impedance inversion may be used to match two impedances at a particular frequency by connecting them with a 1/4 wave cable with an impedance equal to the geometric mean of the two impedances. (The geometric mean is the square-root of their product.) A 50 ohm, 1/4 wave cable will match a 25 ohm source to a 100 ohm load : sqrt(25 x 100) = 50. Of course, it is not always easy to find the desired impedance cable!
Multiples of 1/2 wavelength will give enough delay that the reflection is not inverted and the impedance will be the same as the load. Such cables may be used to transfer the load impedance to a remote location without changing its value (at one frequency).
Other cable lengths will transform an impedance which differs from the cable’s impedance with a reactive component. If the load is a lower impedance than the cable, a length below 1/4 wave will have an inductive component and above 1/4 (but below 1/2) wave a capacitive component. If the load is a higher impedance than the cable, the reverse is true. Above 1/2 wavelength, the reactance will alternately look capacitive and inductive in 1/4 wave multiples. This reactance will combine with the load’s reactance and offers the possibility of resonating the reactive component of the load. Therefore, a cable with the “right” length and impedance can match a source and load with different resistance and reactance values. Obviously, these calculations can become quite involved and most engineers resort to a Smith chart, a computer program or perhaps the most common method, trial and error with a SWR meter or return loss bridge! In most cases, it is most desirable to match every component of a system to the chosen system impedance so that device matching is not frequency sensitive and critically dependent upon the cable lengths.
SWR is a useful number for evaluating the actual voltages and currents present along transmission lines and SWR can be directly measured in many cases but it is often more convenient to work with other, equivalent measures. For example, the voltage reflection coefficient is the fraction of the incident voltage that is reflected. If 0.2 volts reflects from a load with a 1 volt incident wave then the reflection coefficient is 0.2. This number conveys the same information as the SWR but is often more easily calculated and observed. And the terms ‘power transmitted’, ‘transmission loss’ and ‘power reflected’ need no explanation beyond explaining that they are usually percentages. The return loss is simply the amount of power that is “lost” to the load and does not return as a reflection. Clearly, high return loss is usually desired even though “loss” has negative connotations. Return loss is commonly expressed in decibels. If one-half of the power does not reflect from the load, the return loss is 3 dB.
Return loss is a convenient way to characterize the input and output of signal sources. For example, it is desireable to drive a power splitter with its characteristic impedance for maximum port-to-port isolation and , therefore, it may be desireable to check the output return loss of an oscillator or other source. Theoutput return loss is measured by applying a test signal to the oscillator through a directional coupler or circulator:
Any reflected energy appears at the test port and will be x dB below the input. This dB drop is the return loss (after correcting for the coupler’s loss). The test signal frequency is swept through or adjusted to be near the oscillator’s output frequency. A spectrum analyzer connected to the test port of the coupler will display the output of the oscillator and the reflected test signal. The dB drop in the reflected test signal below the applied level is the output return loss. The baseline is easily determined by disconnecting the oscillator so that nothing is connected to the coupler’s test port. Since there is no load all of the energy will reflect and a 0 dB return loss reference may be established. In situations where an open is unacceptable due to high power levels an intentional mismatch will provide a known return loss. For example, a 75 ohm resistor will exhibit a 14 dB return loss in a 50 ohm system while reflecting only 4% of the test power.
An isolator is a seemingly magical device which allows energy to flow in only one direction so reflected energy from a load at the test port does not return to the signal generator but is passed on to the output port regardless of the impedance at any of the ports! This “magic” defies linear “common sense” for passive networks but isolators are highly non-linear devices employing special ferrite in powerful magnetic fields. Engineers who design circuits and systems operating above 500 MHz enjoy the utility of the ferrite isolator but these marvelous devices become impractical below about 200 MHz. Circuits are available in the technical library which simulates the ferrite isolator for frequencies below 300 MHz. The RF op-amps can handle signals approaching 12 dBm so this isolator is only suitable for bench testing low-power RF devices. The attenuation through a directional coupler or return loss bridge can make measurements difficult when the return loss is high and the test signal is small but the isolator has no “loss” and will work well with very small signals. It is also desirable to use small signals when testing antennas for obvious reasons. The isolator exhibits a good output return loss at its test impedance and its good input return loss provides an excellent termination for a long cable from a generator with a questionable SWR.