Impedance bridging question (resistor network)

[ultravox]

Hi,

How can I calculate the needed resistor values for a impedance bridge like this:

Regards, Ultravox

 

[on1aag]

Hi Ultravox,

You've got three unknown resistors, R1, R2 and R3.
You gonna need three equations to solve it.

First equation: Load the network with a 75 ohm resistor (parallel to R3)
and calculate it's equivalent resistance from the left hand
side and this should be equal to 40 ohms.

Second equation:Load the network with a 40 ohm resistor (parallel to R1)
and calculate it's equivalent resistance from the right hand
side and this should be equal to 75 ohms.

Third equation:The attenuation, (75//R3)/{[(75//R3)+R2]R1/(75//R3)+R2+R1} = A
In other words, the equivalent resistance at the side of the load
(75*R3)/(75+R3) divided by the total equivalent resistance from
the side of the source . . . . . . .and that equals the attenuation.


on1aag.

 

[xxargs]

For π-bridge as

------+--R3--+----
      |      |
Z1   R1      R2  Z2   
      |      |
------+------+----

minimum-loss pad matching using hyperbolic equations:

Z1 > Z2

coshΦ = √(Z1 / Z2) ; calculate minimum loss for matching process

(Φ = attenuate in Neper)

R3 = √(Z1 * Z2) * sinhΦ

1/R1 = (1 / (Z1 * tanhΦ)) - (1/ R3)
1/R2 = (1 / (Z2 * tanhΦ)) - (1/ R3)


for your case give:

Z1 = 75 Ω
Z2 = 40 Ω

coshΦ = √(75/40) = 1.369

use acosh(1.369) give Φ = 0.8350 Neper

(8.686 dB = 1 Neper)

minimum loss for making resistive match is a 0.8350 Neper or 7.25 dB

R3 = √(75 * 40) * sinh(835.0e-3) = 51.23 Ohm

1/R1 = (1/(75 * tanh(835.0e-3)) - (1/51.23) => R1 = ∞ Ohm

1/R2 = (1/(40 * tanh(835.0e-3)) - (1/51.23) => R2 = 58.55 Ohm

ie only two resistace needed.

for your schematic with oppsite order of high/low Z and differ names on resistances as above, give values:

R1 = 58.55 Ohm
R2 = 51,23 Ohm
R3 = not mounted

---

Arithmetic Equations for minimum-loss pad.

R1 is a serie resistance between ports
R2 is a parallell resistance to ground on low impedance side.

Z1 > Z2

R1 = Z1√[1-(Z2/Z1)]
R2 = Z2/√[1-(Z2/Z1)]

 

[ultravox]

@on1aag, I don't get you attenuation calculation...

I fiddled two equations out: ((R3//75) + R2)//R1 = 40
and ((R1//40 +R2)//R3 = 75.

@xxargs that's very clear to me, thank you!

The reason for this is that I want to connect a 75Ω antenna to a TEA5757 (with 40Ω imput impedance). I don't want to use difficult coils to match the impedance, but now I think 7.25 db is quite a lot. I can also accept the mismatch, but I don't know if this is gonna affect the reception. The fm-receiver is near a digital lines...

ultravox

 

[palesha]

better make balun transformer of 2:1 ratio. It will then give u 37.5 ohms to 75 ohms. I think that's ok. U will hv very low loss.

 

[xxargs]

@on1aag, I don't get you attenuation calculation...

I fiddled two equations out: ((R3//75) + R2)//R1 = 40
and ((R1//40 +R2)//R3 = 75.

@xxargs that's very clear to me, thank you!

The reason for this is that I want to connect a 75Ω antenna to a TEA5757 (with 40Ω imput impedance). I don't want to use difficult coils to match the impedance, but now I think 7.25 db is quite a lot. I can also accept the mismatch, but I don't know if this is gonna affect the reception. The fm-receiver is near a digital lines...

ultravox

Γ= 75 - 40 / 75 + 40 = 0.3043

S11 = 10 log10(0.3043^2) = -10.33 dB

S21 = 10 Log10(1 - (0.3043^2)) = -0.422 dB

ie 10 dB return loss and 0.42 dB inserting loss if you connect 75 Ohm antenna directly to 40 Ohm input.

use this solution if you not find other unwanted behavior.

---

If not wideband solution you can try using quarte wave transformer


√[75 * 40) = 54.77 Ohm

make with microstrip/stripline or I think standard 50 Ohm coax cable is enough to make soft impedance step for this type of using.

(checking in Vipec) - yes, give > 20 dB return loss between 90 and 108 MHz if center 100 MHz using 50 Ohm coax as quarterwave transformer between 75 and 40 Ohm port..

Use only if return loss is important (for connecting passive filters, suppress reflection in antenna network etc.) depend of 0.75m small coax cable or microstrip/stripline (for 100 MHz) easly can make 0.5 dB extra inserting loss and you not winning sensitivity compare to direct connect without matching.

Other step is lumped matching with inductance and capacitance.