COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS
In the previous thread on 3 dB Hybrid Couplers we have seen that one of the advantages of these devices is ISOLATON. It is isolation that avoids Local Oscillator interference among receivers fed from the same antenna. But another extremely important advantage is the fact that the use of 3 dB hybrid couplers allows power splitting with MINIMAL LOSS. By using the simple Ohm's law the numerical example below compares the power loss by using a 4 way 3 db Hybrid (loss that we have measured & demonstrated in the previous post to be 1 dB) to the power loss experienced by paralleling the input of the receivers with T's.
With a 3 dB/4 way Hybrid, starting with a power of 80 mW (2 V on 50 Ohm, as the value for the example) each receiver sees 63.49 mW, e.g. minimal loss. Paralleling the 4 receivers, each receiver sees only 3.2 mW and the insertion loss becomes 13 dB!
The reader, looking at calculations below, will ask the following question: where is the power gone with the parallel connection of 4 receivers?
Most of the power is dissipated across the 50 Ohm source impedance, because this load is no more 50 Ohm (as it is with the 3 dB Hybrid) but 12.5 Ohm, for reason of the parallel connection. So current out of antenna tends to increase and there is more voltage drop across its 50 Ohm source impedance. A second reason of the loss is the fact that each of the parallel receivers sees the voltage relative to 12.5 Ohm, BUT ITS IMPEDANCE IS 50 OHM, so the power in dBm is sees is lower in the ratio of the two impedances squared.
Please appreciate that this is a very optimistic calculation, valid only if the antenna is connected directly to the 4 paralleled receivers without coaxial cable, so the negative effect of REFLECTIONS is not taken into consideration. In a future post we shall define VSWR, explain its meaning and calculate what the difference in insertion loss would be.
Antenna system impedance Zo, Ohm 50 Ohm
Number of loads, n 4
Single load impedance, Zk, Ohm 50 Ohm
Antenna Voltage, Va, 2 V
Antenna nominal power Pa, Pa=(Va^2)/50 = 0.08 W
Antenna nominal power Pa1, 80 mW
3dB Hybrid loss, dB / ratio r1 1.26
Power in loads, Pl, Pl=Pa/r = 63.49 mW
Composite load impedance Zkc, Zkc=Zk/n = 12.5 Ohm
Antenna current Ia, Ia=Va*1000/(Zo+Zkc)= 32mA
Voltage across composite load Vc, Vc=Zkc*Ia/1000 = 0.4 V
Power delivered to composite loads Pll, Pll=(Vc^2)/Zkc = 12.8 mW
Effective power loss, ratio Lr Lr=Pa1/Pll = 6.25
Effective power loss Lrd 7.96 dB
Power seen by each load Pa2 Pa2=(Vc^2)*1000/Zk = 3.2 mW
Difference in power, ratio dk dk=Pl/Pa2 = 19.84126984
Difference in power dB, Dd 12.97 dB
THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
This Forum makes corrections on the first post impossible: I forgot this time. I shall do corrections, if necessary, in the next post.
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
Well done, glovisol, I think it could not be explained simpler!
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
Well done, thanks Sdrom33, but with transcription mistakes, due to transfer from an EXCEL file to the Forum. I cannot correct the first draft, as universally known, so I have decided to integrally rewrite the text and transfer the EXCEL calc table as a Jpeg.
ERRATA CORRIGE 27/04/2019
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  4 RECEIVERS
In the previous thread on 3 dB Hybrid Couplers we have seen that one of the advantages of these devices is ISOLATION. It is isolation that avoids Local Oscillator interference among receivers fed from the same antenna. But another extremely important advantage is the fact that the use of 3 dB hybrid couplers allows power splitting with MINIMAL LOSS. By using the simple Ohm's law the numerical example below compares the power loss by using a 4 way 3 db Hybrid (loss that we have measured & demonstrated in the previous post to be 1 dB) to the power loss experienced by paralleling the input of the receivers with T's.
With a 3 dB/4 way Hybrid, starting with a power of 80 mW (2 V on 50 Ohm, as the value for the example) each receiver sees 15.87 mW, for a total of 63.49 mW e.g. minimal loss of 1 dB. Paralleling the 4 receivers, each receiver sees only 3.2 mW and the insertion loss becomes 7 dB for each receiver.
The reader, looking at calculations below, will ask the following question: where has the power gone with the parallel connection of 4 receivers?
Most of the power is dissipated across the 50 Ohm source impedance, because the load is 50 Ohm no more (as it is with the 3 dB Hybrid) but 12.5 Ohm, for reason of the parallel connection. So current out of antenna increases and there is more voltage drop across its 50 Ohm source impedance. A second reason of the loss is the fact that each of the parallel receivers sees the voltage relative to 12.5 Ohm, BUT ITS IMPEDANCE IS 50 OHM, so the power in dBm it sees is lower in the ratio of the two impedances squared.
CORRECTION OF 05/05/2019: WHAT FOLLOWS NOT ENTIRELY TRUE, AS SHOWN IN SUBSEQUENT POSTS
** Please appreciate that this is a very optimistic calculation, valid only if the antenna is connected directly to the 4 paralleled receivers without coaxial cable, so the negative effect of REFLECTIONS is not taken into consideration. In a future post we shall define VSWR, explain its meaning and calculate what the difference in insertion loss would be. **
ERRATA CORRIGE 27/04/2019
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  4 RECEIVERS
In the previous thread on 3 dB Hybrid Couplers we have seen that one of the advantages of these devices is ISOLATION. It is isolation that avoids Local Oscillator interference among receivers fed from the same antenna. But another extremely important advantage is the fact that the use of 3 dB hybrid couplers allows power splitting with MINIMAL LOSS. By using the simple Ohm's law the numerical example below compares the power loss by using a 4 way 3 db Hybrid (loss that we have measured & demonstrated in the previous post to be 1 dB) to the power loss experienced by paralleling the input of the receivers with T's.
With a 3 dB/4 way Hybrid, starting with a power of 80 mW (2 V on 50 Ohm, as the value for the example) each receiver sees 15.87 mW, for a total of 63.49 mW e.g. minimal loss of 1 dB. Paralleling the 4 receivers, each receiver sees only 3.2 mW and the insertion loss becomes 7 dB for each receiver.
The reader, looking at calculations below, will ask the following question: where has the power gone with the parallel connection of 4 receivers?
Most of the power is dissipated across the 50 Ohm source impedance, because the load is 50 Ohm no more (as it is with the 3 dB Hybrid) but 12.5 Ohm, for reason of the parallel connection. So current out of antenna increases and there is more voltage drop across its 50 Ohm source impedance. A second reason of the loss is the fact that each of the parallel receivers sees the voltage relative to 12.5 Ohm, BUT ITS IMPEDANCE IS 50 OHM, so the power in dBm it sees is lower in the ratio of the two impedances squared.
CORRECTION OF 05/05/2019: WHAT FOLLOWS NOT ENTIRELY TRUE, AS SHOWN IN SUBSEQUENT POSTS
** Please appreciate that this is a very optimistic calculation, valid only if the antenna is connected directly to the 4 paralleled receivers without coaxial cable, so the negative effect of REFLECTIONS is not taken into consideration. In a future post we shall define VSWR, explain its meaning and calculate what the difference in insertion loss would be. **
 Attachments

 Comparison Hybrid Vs. Parallel.jpg (95.85 KiB) Viewed 1572 times
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  3 RECEIVERS
A composite 3 dB Hybrid for splitting an odd number of ways, such as 3, can be constructed using two identical two way Hybrids in cascade, as shown in the schematic uploaded below. Output B delivers half the input power at A, less 0.5 dB insertion loss. Outputs D and E deliver 25% of input power at A, less 1 dB insertion loss.
The enclosed EXCEL calculation shows that, using the described 3 dB Hybrid arrangement, with an antenna power of 80 mW, 31.75 mW are delivered to the "main" receiver, e.g output 1 B, while 12.6 mW are delivered to the remaining two receivers, e.g. outputs D and E.
If we just parallel connect the three receivers to the antenna, each receiver will only get a power of 5 mW. All other considerations exposed for the 4 way Hybrid split are also valid in this case of 3 outputs, in the sense that this is a very optimistic comparison, because we have not taken VSWR ((e.g. reflectons) into consideration.
A composite 3 dB Hybrid for splitting an odd number of ways, such as 3, can be constructed using two identical two way Hybrids in cascade, as shown in the schematic uploaded below. Output B delivers half the input power at A, less 0.5 dB insertion loss. Outputs D and E deliver 25% of input power at A, less 1 dB insertion loss.
The enclosed EXCEL calculation shows that, using the described 3 dB Hybrid arrangement, with an antenna power of 80 mW, 31.75 mW are delivered to the "main" receiver, e.g output 1 B, while 12.6 mW are delivered to the remaining two receivers, e.g. outputs D and E.
If we just parallel connect the three receivers to the antenna, each receiver will only get a power of 5 mW. All other considerations exposed for the 4 way Hybrid split are also valid in this case of 3 outputs, in the sense that this is a very optimistic comparison, because we have not taken VSWR ((e.g. reflectons) into consideration.
 Attachments

 3 way 3 dB Hybrid.jpg (53.79 KiB) Viewed 1486 times

 Comparison N=3 Vs. parallel.jpg (85.25 KiB) Viewed 1486 times
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  4 RECEIVERS WITH dBM VALUES
In order to give a better idea of the disadvantage of paralleling the inputs of 4 receivers on 50 Ohm, instead of using 4 way Hybrids, the table below shows the same calculatìons done at an antenna output level of 0.5 uV and with power levels expressed in dBm. Looking at the table we see that with an antenna output of 113 dBm, the Hybrid delivers 120 dBm to each receiver: signals at this levels are on the threshold, but can still be copied, depending on general noise conditions.
By paralleling on 50 Ohm, the power seen by each receiver drops down to 127 dBm and goes below threshold of audibility.
In order to give a better idea of the disadvantage of paralleling the inputs of 4 receivers on 50 Ohm, instead of using 4 way Hybrids, the table below shows the same calculatìons done at an antenna output level of 0.5 uV and with power levels expressed in dBm. Looking at the table we see that with an antenna output of 113 dBm, the Hybrid delivers 120 dBm to each receiver: signals at this levels are on the threshold, but can still be copied, depending on general noise conditions.
By paralleling on 50 Ohm, the power seen by each receiver drops down to 127 dBm and goes below threshold of audibility.
 Attachments

 Comparison Hybrid dBm N=4 Vs. Parallel.jpg (112.98 KiB) Viewed 1290 times
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  4 RECEIVERS IN THE REAL WORLD
Contrary to what it was expected, in the real world the difference is not as significant as we could find by simply applying the ohms law! Just read on....
To find the effective loss difference in the real world, we must consider the loss resultant from loading the 50 Ohm transmission line with 12.5 Ohm for the parallel load of four receivers.
We have an antenna with an output impedance of 50 Ohm, connected to a 50 Ohm coaxial line length which is loaded by a 12.5 Ohm load represented by the four receivers. Calculation is simplified because the RSP’s inputs do not have a reactive component. The problem we have to solve is to find what the input impedance Zin will be, at the antenna side, when the 50 Ohm feedline is loaded by 12.5 Ohm resistive.
Note the following tractation is based on the mathematical development available here:
https://ac6la.com/swrloss.html
The fundamental TRANSMISSION LINE EQUATION [1] calculates Zin:
Zin=Zo*((ZL*Cosh(γ*l) + Zo*Sinh(γ*l))/(ZL*Sinh(γ*l) + Zo*Cosh(γ*l))) [1]
Where, in our case:
ZL = 12.5 Ohm
Zo = 50 Ohm
l = coaxial line length in ft.
α = coaxial line loss attenuation constant in Nepers/ft (1 Neper = 8.69 dB)
γ = Complex loss coefficient = α + jβ
In our case the RSP’s inputs being devoid of reactive components, β = 0 and γ = α
Once we know Zin, Total Loss is calculated by [2] described by Steve Stearns, K6OIK, in a presentation titled "A Transmission Line Power Paradox and Its Resolution" given at Pacificon 2014.
Total Loss = 20*LogCosh(γ*l)  ((Zin/Zo)*Sinh(γ*l)) + 10*Log(Zin/ZL) [2]
Equations [1] and [2] have been entered into Excel and Table 1 below shows that the calculation system is correct: in fact introducing a Zo = 50 Ohm system impedance, a cable with a loss of Cl = 3 db/100 ft, a load impedance of ZL = 50 Ohm, Excel correctly calculates a Zin = 50 Ohm and no extra loss, but only the loss of the correctly terminated cable.
In Table 2, with all data as above, but with load impedance ZL = 12.5 Ohm, Excel calculates a total loss of 7.2 dB, of which 3 dB are due to the normal terminated cable loss and 4.2 dB are excess loss, due to the VSWR resulting from the mismatch.
In addition, in Table 2 we come to the unexpected result that, considering a theorical 100 ft coaxial feedline with 3 dB/100ft loss, the difference in insertion loss between the two cases under examination is only 3 dB (less than that calculated with the Ohm's law) while it was expected that the loss difference would become greater once the effect of the VSWR on the coaxial feedline was calculated.
In the next posts we shall see how the system parameters of an existing cable (RG58/U) influence the differential in signal loss and how system VSWR relates to these results.
Contrary to what it was expected, in the real world the difference is not as significant as we could find by simply applying the ohms law! Just read on....
To find the effective loss difference in the real world, we must consider the loss resultant from loading the 50 Ohm transmission line with 12.5 Ohm for the parallel load of four receivers.
We have an antenna with an output impedance of 50 Ohm, connected to a 50 Ohm coaxial line length which is loaded by a 12.5 Ohm load represented by the four receivers. Calculation is simplified because the RSP’s inputs do not have a reactive component. The problem we have to solve is to find what the input impedance Zin will be, at the antenna side, when the 50 Ohm feedline is loaded by 12.5 Ohm resistive.
Note the following tractation is based on the mathematical development available here:
https://ac6la.com/swrloss.html
The fundamental TRANSMISSION LINE EQUATION [1] calculates Zin:
Zin=Zo*((ZL*Cosh(γ*l) + Zo*Sinh(γ*l))/(ZL*Sinh(γ*l) + Zo*Cosh(γ*l))) [1]
Where, in our case:
ZL = 12.5 Ohm
Zo = 50 Ohm
l = coaxial line length in ft.
α = coaxial line loss attenuation constant in Nepers/ft (1 Neper = 8.69 dB)
γ = Complex loss coefficient = α + jβ
In our case the RSP’s inputs being devoid of reactive components, β = 0 and γ = α
Once we know Zin, Total Loss is calculated by [2] described by Steve Stearns, K6OIK, in a presentation titled "A Transmission Line Power Paradox and Its Resolution" given at Pacificon 2014.
Total Loss = 20*LogCosh(γ*l)  ((Zin/Zo)*Sinh(γ*l)) + 10*Log(Zin/ZL) [2]
Equations [1] and [2] have been entered into Excel and Table 1 below shows that the calculation system is correct: in fact introducing a Zo = 50 Ohm system impedance, a cable with a loss of Cl = 3 db/100 ft, a load impedance of ZL = 50 Ohm, Excel correctly calculates a Zin = 50 Ohm and no extra loss, but only the loss of the correctly terminated cable.
In Table 2, with all data as above, but with load impedance ZL = 12.5 Ohm, Excel calculates a total loss of 7.2 dB, of which 3 dB are due to the normal terminated cable loss and 4.2 dB are excess loss, due to the VSWR resulting from the mismatch.
In addition, in Table 2 we come to the unexpected result that, considering a theorical 100 ft coaxial feedline with 3 dB/100ft loss, the difference in insertion loss between the two cases under examination is only 3 dB (less than that calculated with the Ohm's law) while it was expected that the loss difference would become greater once the effect of the VSWR on the coaxial feedline was calculated.
In the next posts we shall see how the system parameters of an existing cable (RG58/U) influence the differential in signal loss and how system VSWR relates to these results.
 Attachments

 TABLE 1
 Total loss calculator verification.jpg (76.42 KiB) Viewed 1207 times

 TABLE 2
 Real world comparison.jpg (120.85 KiB) Viewed 1207 times
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  4 RECEIVERS IN THE REAL WORLD  RG58/U EXAMPLE
The effect of introducing coaxial cable between antenna and RSP receivers, as it happens in the real world, is that the resultant attenuation tends to reduce the advantage of the 3 dB Hybrid coupler.
Table 3 below, based on the Excel calculator, which unfortunately cannot be uploaded, but is available to anyone upon direct request to the writer, shows that, when attenuation of the antenna signal increases, because of a longer cable run or of because of a higher operating frequency (coaxial cable attenuation increases with frequency) the 3 dB Hybrid splitter advantage over direct paralleling diminishes, while, of course, the L.O. isolation advantage stays the same.
The step from 50 to 12.5 Ohm implies a VSWR of 4 (seen by the antenna, e.g. the other way around, compared to a transmitting installation). For instance, at a frequency of 10 Mhz, a run of 100 ft of RG58/U will attenuate an antenna signal of 100 dBm to 108.4 dBm at the input of each RSP receiver using a 3dB Hybrid and to 112.5 dBm with parallel direct, for an advantage for the Hybrid of 4.1 dB.
Table 3 shows that the advantage is significant at lower frequencies and with short coaxial cable runs(*) so it could very well be that someone, using a high insertion loss splitter (not necessarily a 3 dB Hybrid) with a long coax cable run at high HF might find that indeed direct paralleling is advantageous as far as signal attenuation is concerned.
(*) In fact the initial calculations using Ohm's law are equivalent to the transmission line calculations if one considers zero cable length. With zero cable length and zero loss the 3 dB hybrid advantage over the parallel connection is maximum.
The effect of introducing coaxial cable between antenna and RSP receivers, as it happens in the real world, is that the resultant attenuation tends to reduce the advantage of the 3 dB Hybrid coupler.
Table 3 below, based on the Excel calculator, which unfortunately cannot be uploaded, but is available to anyone upon direct request to the writer, shows that, when attenuation of the antenna signal increases, because of a longer cable run or of because of a higher operating frequency (coaxial cable attenuation increases with frequency) the 3 dB Hybrid splitter advantage over direct paralleling diminishes, while, of course, the L.O. isolation advantage stays the same.
The step from 50 to 12.5 Ohm implies a VSWR of 4 (seen by the antenna, e.g. the other way around, compared to a transmitting installation). For instance, at a frequency of 10 Mhz, a run of 100 ft of RG58/U will attenuate an antenna signal of 100 dBm to 108.4 dBm at the input of each RSP receiver using a 3dB Hybrid and to 112.5 dBm with parallel direct, for an advantage for the Hybrid of 4.1 dB.
Table 3 shows that the advantage is significant at lower frequencies and with short coaxial cable runs(*) so it could very well be that someone, using a high insertion loss splitter (not necessarily a 3 dB Hybrid) with a long coax cable run at high HF might find that indeed direct paralleling is advantageous as far as signal attenuation is concerned.
(*) In fact the initial calculations using Ohm's law are equivalent to the transmission line calculations if one considers zero cable length. With zero cable length and zero loss the 3 dB hybrid advantage over the parallel connection is maximum.
 Attachments

 TABLE 3
 Cable results.jpg (63.72 KiB) Viewed 1155 times
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
COMPARISON BETWEEN PARALLELING LOADS CONNECTED TO ANTENNA OUTPUT OR SPLITTING WITH 3 db HYBRIDS  4 RECEIVERS IN THE REAL WORLD  RG58/U EXAMPLE
With RG58/U coaxial cable, at a frequency of 10 MHz (loss 1.4 dB/100 ft) the Zin vs. cable length is given by the graph below: the longer the cable, the better the match, but at the cost of additional insertion loss per receiver. As obvious, when length and insertion loss become very large, Zin asimptotically reaches the value of 50 Ohm.
With RG58/U coaxial cable, at a frequency of 10 MHz (loss 1.4 dB/100 ft) the Zin vs. cable length is given by the graph below: the longer the cable, the better the match, but at the cost of additional insertion loss per receiver. As obvious, when length and insertion loss become very large, Zin asimptotically reaches the value of 50 Ohm.
 Attachments

 Figure 1. dB values shown are insertion losses per receiver.
 Zin Vs. length.jpg (114.43 KiB) Viewed 1072 times
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Re: THE POWER LOSS ADVANTAGE OF USING 3 dB COUPLERS
Glovisol, from your calculations it seems that ON5HB, apart strange theories on local oscillators, mixers, RSP receivers, vswr and so on, was not completely wrong in practice, after all!
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