Transmission Line Principle: Observing the Reflection Coefficient and Standing Wave
11 min read
Numerous kinds of waves in nature behave basically alike. Like a voice that echoes off of a cliff, electrical waves replicate once they encounter a change within the impedance of the medium they’re touring in. Wave reflection can result in an attention-grabbing phenomenon known as the standing wave. Standing waves are important to the way in which most musical devices produce sound. For instance, string devices wouldn’t perform with out the predictability and amplification results of the standing waves.
Nonetheless, in RF design, standing waves are undesirable after we goal to switch energy from one block to the following one within the sign chain. Actually, standing waves can have an effect on the efficiency of various RF and microwave techniques, from anechoic chambers to on a regular basis home equipment reminiscent of microwave ovens.
Whereas the ideas of wave propagation and reflection usually are not terribly difficult, they could be just a little complicated at first. One of the simplest ways to visualise how the waves propagate and replicate off of a discontinuity is to plot the wave equations for various configurations.
On this article, we’ll first derive the required equations and use them to clarify the standing wave phenomenon by means of a number of instance waveforms.
Transmission Line Voltage and Present Wave Equations
First, let’s derive our equations. I do know it’s boring, however they actually assist us perceive how waves propagate and work together with one another on a transmission line. Within the earlier article on this collection, we examined the sinusoidal steady-state response of a transmission line and derived the voltage and present equations. Making use of vs(t) = Vscos(ωt) to a line, the voltage and present waves are:
[v(x,t)= A cos(omega t-beta x) + B cos(omega t+beta x)]
[i(x,t)=fracAZ_0 cos(omega t-beta x)- fracBZ_0 cos(omega t+beta x)]
The place:
- A and B are constants that may be discovered from boundary situations on the enter and output ports of the road
- Z0 is the attribute impedance
- β is the part fixed
These equations correspond to the configuration proven in Determine 1(a), the place the constructive x-axis course is chosen to be from the supply to the load. If we symbolize these waves with their phasors, the forward-traveling (or incident) wave and the backward-traveling (or mirrored) voltage waves shall be, respectively, Ae-jβx and Bejβx, as proven in Determine 1(a).
Determine 1. Diagrams displaying the constructive axis course are from the supply to the load (a) after which from the load to the supply (b).
Relating to transmission line issues, it’s normally extra handy to decide on the constructive axis course from the load to the supply, as proven in Determine 1(b). To seek out the brand new equations, we have to exchange x within the authentic equations with l-d. As expressed within the new variable, d, the forward-traveling wave turns into:
[Ae^-j beta x = Ae^-j beta (l-d)=Ae^-j beta le^j beta d = A_1 e^j beta d]
The place A1 = Ae-jβl is a brand new fixed. From right here, you may confirm that, within the new coordinate system, the mirrored wave is B1e-jβd, the place B1 = Bejβl. Due to this fact, the whole voltage and present phasors are proven in Equations 1 and a couple of.
[V(d)=A_1e^j beta d+B_1e^-j beta d]
Equation 1.
[I(d)=fracA_1Z_0e^j beta d-fracB_1Z_0e^-j beta d]
Equation 2.
These equations make it simpler to look at the load impact on the wave reflection as a result of, on this case, the load is at d = 0, simplifying the equations. Letting d = 0, the next equations are obtained on the load finish, as seen in Equations 3 and 4.
[V(d=0)=A_1+B_1]
Equation 3.
[I(d=0)=fracA_1Z_0-fracB_1Z_0]
Equation 4.
For instance, let’s take into account the case the place the road is terminated in an open circuit. With the output open-circuited (ZL = ∞), the output present is clearly zero. From Equation 4, we now have A1 = B1, and thus, the whole voltage is V(d = 0) = 2A1.
Due to this fact, for an open circuit line, the mirrored voltage is the same as the incident voltage on the output, and the whole voltage at this level is double the incident voltage. Equally, we will use Equations 3 and 4 to search out the ratio of the mirrored wave to the incident wave for an arbitrary load impedance ZL. This ratio is a vital parameter often known as the reflection coefficient, which we’ll get into shortly.
Enter Impedance and Reflection Coefficient Components
Utilizing Equations 1 and a couple of, we will discover the ratio of voltage to present (i.e., the enter impedance of the transmission line) at completely different factors alongside the road. This results in Equation 5.
[Z_in(d) = fracV(d)I(d)=Z_0 fracA_1e^j beta d+B_1e^-j beta dA_1e^j beta d-B_1 e^-j beta d]
Equation 5.
Noting that the road impedance on the load finish of the road (d = 0) is the same as the load impedance ZL, we receive:
[Z_L = Z_0 fracA_1+B_1A_1-B_1]
Utilizing just a little algebra, the above equation offers us the ratio of the mirrored voltage wave to the incident voltage wave (B1/A1), which is outlined because the reflection coefficient Γ in Equation 6.
[Gamma = fracZ_L – Z_0Z_L + Z_0]
Equation 6.
The above dialogue reveals that for a terminated line, there’s a particular relationship between the incident and mirrored waves. Be aware that, basically, a mirrored image coefficient is a fancy quantity, and each magnitude and part data of Γ are essential. For energy switch, we try to have a matched load (ZL = Z0), resulting in Γ = 0. Below this situation, a wave utilized to the enter is totally absorbed by the load, and no reflection happens. It’s instructive to think about two different particular circumstances right here: an open circuit line and a brief circuit line that we’ll get into shortly.
Whereas the ideas of wave propagation and reflection usually are not principally difficult, they could be complicated at first. One of the simplest ways to visualise how the waves propagate and replicate off of a discontinuity is to plot the equations we now have developed above. Additionally, it’s value mentioning that there are numerous on-line simulators which may enable you to develop a greater understanding of wave propagation ideas.
Quick Circuit Strains
Subsequent, let’s go over quick circuit strains. With a brief circuit, the whole output voltage needs to be zero always. Moreover, from Equation 6, we now have Γ = -1. The incident voltage wave is given by:
[v_i (d,t)=Real text Part text of Big (A_1 e^j beta d e^j omega t Big ) = A_1 cos(omega t+beta d)]
The highest curve in Determine 2 supplies the plot of this equation at three completely different deadlines t1, t2, and t3, the place t1 < t2 < t3.
Determine 2. Instance curves for the ahead voltage (prime), backward voltage (center), and whole voltage (backside) of a brief circuit.
The above curves breakdown the place:
- The size of the transmission line is 0.2 meters
- The load is at d = 0
- β is 50 radian/meter
- The sign frequency is 2 GHz
Be aware how the incident wave step by step strikes towards the load (at d = 0) as time passes. The center curve within the above determine reveals the mirrored voltage that strikes away from the load. The mirrored voltage equation is:
[v_r (d,t)=Real text Part text of Big (Gamma A_1 e^-j beta d e^j omega t Big ) = A_1 cos(omega t – beta d)]
The place Γ is ready to -1 to take the quick circuit into consideration. The entire voltage is the sum of the incident and mirrored voltages which can be given within the decrease curve. The ahead voltage fluctuates between its minimal and most values in any respect factors alongside the road, together with the load finish of the road. Nonetheless, the mirrored voltage takes the other worth of the incident voltage in order that the whole voltage is all the time zero on the load finish.
The entire voltage wave has an attention-grabbing characteristic: it stands nonetheless, and in contrast to its constituent waves, the whole voltage wave just isn’t touring in both course. For instance, the utmost and nil voltage factors don’t shift with respect to time. To raised illustrate this, Determine 3 plots the whole voltage for 36 completely different deadlines.
Determine 3. A plot displaying the whole voltage for 36 completely different deadlines.
As will be seen, the zero-crossings (nodes) and the positions of most amplitude (antinodes) are some fastened positions alongside the road. For the reason that wave just isn’t touring in both course, it’s known as a standing wave.
Open Transmission Circuit Line
For an open circuit line (ZL = ∞), Equation 6 yields Γ = 1. On this case, the magnitude and part of the mirrored voltage are equal to the incident voltage. The highest and center curves in Determine 4, respectively, present the incident and mirrored voltage waves on an open circuit line at three completely different deadlines.
Determine 4. Instance plots displaying the ahead voltage (prime), backward voltage (center), and whole voltage (backside) of an open circuit.
Be aware that each the incident and mirrored waves have the identical worth at d = 0. Due to this fact, the whole voltage (the underside curve) is double the incident voltage on the load finish. Since Γ = 1, the mirrored present Ir additionally has the identical magnitude and part because the incident present Ii. Nonetheless, the whole present is Ii – Ir = 0 on the load finish, which isn’t any large shock because the load is an open circuit.
Moreover, we will once more observe that the whole voltage is a standing wave. That is finest illustrated in Determine 5, which plots the whole voltage wave for 36 completely different deadlines.
Determine 5. Instance plot displaying the whole voltage wave for 36 completely different deadlines for an open circuit.
Calculating an Arbitrary Load of a Terminated Line
Subsequent, let’s use our equations to look at a terminated line with Γ = 0.5. The incident and mirrored voltage waves at an arbitrary time are plotted in Determine 6.
Determine 6. Plot displaying the incident and mirrored voltage waves.
These two waves are touring in reverse instructions. It’s best to be capable to think about that at a sure cut-off date and at some particular place alongside the road, the peaks of the 2 waves will coincide, producing the utmost worth of the whole voltage wave. That is illustrated in Determine 7.
Determine 7. Instance plot displaying the utmost worth of the whole voltage wave when the peaks of the incident and mirrored waves coincide.
Additionally, at another cut-off date, a specific place alongside the road will “see” the height of the bigger wave and the minimal of the smaller one, as proven in Determine 8.
Determine 8. Instance plot displaying the whole voltage wave the place the incident and mirrored waves have opposing peaks and valleys.
At these factors, the amplitude of the whole voltage wave is at its minimal. In our instance, the ahead and mirrored waves respectively have an amplitude of 1 and 0.5. Due to this fact, the whole voltage wave has a minimal amplitude of 1 – 0.5 = 0.5. To raised observe the voltage amplitude at completely different factors alongside the road, Determine 9 plots the whole voltage wave at 36 completely different situations.
Determine 9. Instance plot displaying the whole voltage wave at 36 completely different situations.
This determine offers you an thought of the fluctuation amplitude at completely different factors on the road. Be aware that whereas factors reminiscent of d = 0.1881 m fluctuate between ±1.5 V, there are different factors. For instance, d = 0.1568 m, which has a a lot smaller amplitude and fluctuates between ±0.5 V.
One query you may ask is, is the whole wave touring or standing nonetheless? Determine 10 reveals a smaller variety of the whole voltage plots at some consecutive deadlines (t1 < t2 < …< t6) to reply this query.
Determine 10. Instance displaying fewer whole voltage plots at consecutive deadlines.
The determine reveals that, as time passes, the wave travels towards the load. Be aware that whereas the amplitude of the incident and mirrored waves are fixed, the amplitude of the mixed voltage rises and falls over time.
Incident, Reflection, and Standing Waves Abstract
Let’s summarize our observations:
- With a matched load, the incident wave travels towards the load, and there’s no reflection. On this case, the wave has fixed amplitude alongside the road.
- With quick circuit and open circuit strains, the incident wave is completely mirrored (Γ = -1 or 1). On this case, the mixed voltage just isn’t touring in both course and is known as a standing wave.
- With standing waves, we now have nodes and antinodes at fastened positions alongside the road. Nodes don’t fluctuate in any respect, whereas antinodes fluctuate on the most amplitude.
- With hundreds aside from the above three circumstances, we now have a touring wave that rises and falls over time (though it’s truly a touring wave, we would nonetheless glibly often seek advice from this wave as a standing wave). On this case, we don’t have any nodes, however some factors have a smaller amplitude than others. This example is between the perfect case of no reflection (Γ = 0) and the worst case of whole reflection (Γ = ±1).
Due to this fact, with all that in thoughts, it is necessary for us to know at which level throughout this spectrum our transmission line is working. The parameter VSWR (voltage standing wave ratio), which is outlined because the ratio of the wave’s most amplitude to its minimal amplitude, permits us to characterize how shut we’re to having a standing wave. When there’s whole reflection, VSWR is infinite; for a matched load, VSWR is 1.
As for different circumstances, VSWR is someplace between these two excessive values. The VSWR supplies us with another manner of characterizing the quantity of reflection. This shall be mentioned in better element within the subsequent article.
Featured picture used courtesy of Adobe Stock
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