Transmission Line Idea: Observing the Reflection Coefficient and Standing Wave

Varied varieties of waves in nature behave essentially alike. Like a voice that echoes off of a cliff, electrical waves mirror once they encounter a change within the impedance of the medium they’re touring in. Wave reflection can result in an attention-grabbing phenomenon referred to as the standing wave. Standing waves are important to the best way most musical devices produce sound. For instance, string devices wouldn’t operate with out the predictability and amplification results of the standing waves.

Nevertheless, in RF design, standing waves are undesirable after we intention 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 similar to microwave ovens.

Whereas the ideas of wave propagation and reflection should not terribly sophisticated, they could be a bit of complicated at first. The easiest way to visualise how the waves propagate and mirror 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 way 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 v_s(t) = V_scos(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 optimistic x-axis route is chosen to be from the supply to the load. If we characterize these waves with their phasors, the forward-traveling (or incident) wave and the backward-traveling (or mirrored) voltage waves will likely be, respectively, Ae^-jβx and Be^jβx, as proven in Determine 1(a).

Determine 1. Diagrams displaying the optimistic axis route are from the supply to the load (a) after which from the load to the supply (b).

Concerning transmission line issues, it’s often extra handy to decide on the optimistic axis route 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 possibly can confirm that, within the new coordinate system, the mirrored wave is B₁e^-jβd, the place B₁ = Be^jβl. Subsequently, the full voltage and present phasors are proven in Equations 1 and a pair 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 think about the case the place the road is terminated in an open circuit. With the output open-circuited (Z_L = ∞), the output present is clearly zero. From Equation 4, we’ve A₁ = B_1, and thus, the full voltage is V(d = 0) = 2A₁.

Subsequently, for an open circuit line, the mirrored voltage is the same as the incident voltage on the output, and the full 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 Z_L. This ratio is a crucial parameter generally known as the reflection coefficient, which we’ll get into shortly. 

Enter Impedance and Reflection Coefficient System

Utilizing Equations 1 and a pair 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 Z_L, we get hold of:

[Z_L = Z_0 fracA_1+B_1A_1-B_1]

Utilizing a bit of algebra, the above equation provides 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 exhibits that for a terminated line, there’s a particular relationship between the incident and mirrored waves. Notice that, usually, a mirrored image coefficient is a posh quantity, and each magnitude and part data of Γ are vital. For energy switch, we try to have a matched load (Z_L = Z₀), resulting in Γ = 0. Beneath 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 should not mainly sophisticated, they could be complicated at first. The easiest way to visualise how the waves propagate and mirror off of a discontinuity is to plot the equations we’ve developed above. Additionally, it’s value mentioning that there are a lot of on-line simulators which may provide help to develop a greater understanding of wave propagation ideas.

Quick Circuit Traces

Subsequent, let’s go over quick circuit traces. With a brief circuit, the full output voltage must be zero always. Moreover, from Equation 6, we’ve 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 t_3, the place t₁ < t₂ < t₃.

Determine 2. Instance curves for the ahead voltage (high), 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

Notice how the incident wave steadily strikes towards the load (at d = 0) as time passes. The center curve within the above determine exhibits 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 about to -1 to take the quick circuit under consideration. The full 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. Nevertheless, the mirrored voltage takes the other worth of the incident voltage in order that the full voltage is all the time zero on the load finish.

The full voltage wave has an attention-grabbing characteristic: it stands nonetheless, and in contrast to its constituent waves, the full voltage wave shouldn’t be touring in both route. For instance, the utmost and nil voltage factors don’t shift with respect to time. To raised illustrate this, Determine 3 plots the full voltage for 36 completely different deadlines.

Determine 3. A plot displaying the full voltage for 36 completely different deadlines.

As could be seen, the zero-crossings (nodes) and the positions of most amplitude (antinodes) are some fastened positions alongside the road. Because the wave shouldn’t be touring in both route, it’s referred to as a standing wave. 

Open Transmission Circuit Line

For an open circuit line (Z_L = ∞), 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 (high), backward voltage (center), and whole voltage (backside) of an open circuit.

Notice that each the incident and mirrored waves have the identical worth at d = 0. Subsequently, the full voltage (the underside curve) is double the incident voltage on the load finish. Since Γ = 1, the mirrored present I_r additionally has the identical magnitude and part because the incident present I_i. Nevertheless, the full present is I_i – I_r = 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 full voltage is a standing wave. That is greatest illustrated in Determine 5, which plots the full voltage wave for 36 completely different deadlines.

Determine 5. Instance plot displaying the full 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 full voltage wave. That is illustrated in Determine 7.

Determine 7. Instance plot displaying the utmost worth of the full voltage wave when the peaks of the incident and mirrored waves coincide.

Additionally, at another cut-off date, a selected 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 full voltage wave the place the incident and mirrored waves have opposing peaks and valleys. 

At these factors, the amplitude of the full voltage wave is at its minimal. In our instance, the ahead and mirrored waves respectively have an amplitude of 1 and 0.5. Subsequently, the full 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 full voltage wave at 36 completely different situations.

Determine 9. Instance plot displaying the full voltage wave at 36 completely different situations.

This determine provides you an thought of the fluctuation amplitude at completely different factors on the road. Notice that whereas factors similar to 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 full wave touring or standing nonetheless? Determine 10 exhibits a smaller variety of the full 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 exhibits that, as time passes, the wave travels towards the load. Notice 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 traces, the incident wave is completely mirrored (Γ = -1 or 1). On this case, the mixed voltage shouldn’t be touring in both route and known as a standing wave.
• With standing waves, we’ve nodes and antinodes at fastened positions alongside the road. Nodes don’t fluctuate in any respect, whereas antinodes fluctuate on the most amplitude.
• With masses apart from the above three circumstances, we’ve a touring wave that rises and falls over time (though it’s truly a touring wave, we would nonetheless glibly sometimes 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 case is between the best case of no reflection (Γ = 0) and the worst case of whole reflection (Γ = ±1).

Subsequently, 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 approach of characterizing the quantity of reflection. This will likely be mentioned in higher element within the subsequent article.

Featured picture used courtesy of Adobe Stock

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