SNAA434 March   2025 LMX2820

 

  1.   1
  2.   Abstract
  3.   Trademarks
  4. 1Introduction
  5. 2Creating Multiple Copies of the Input Signal
    1. 2.1 Skew and Slew Rate Considerations
    2. 2.2 Buffers vs. Resistive Splitters
    3. 2.3 Phase Noise Considerations With Buffers
  6. 3Considerations with Combining Outputs
    1. 3.1 Isolation Between Sources
    2. 3.2 Single-Ended vs. Differential Outputs
    3. 3.3 Losses Due to Combining
  7. 4Resistive Method for Combining Multiple Signals
    1. 4.1 General Case Where Source Output Impedance can be Different Than Load Impedance
    2. 4.2 Special Case Where Source and Load Impedance are the Same
    3. 4.3 Increasing R1 to Improve Isolation
  8. 5Impedance Matching With Reactive Circuit
  9. 6Loss Due to Phase Error
  10. 7Phase Noise Improvement by Combining Multiple Signals
    1. 7.1 Theoretical Improvement for Multiple Signals Designed for in Phase
    2. 7.2 Combining Multiple Signals With a Phase Error
  11. 8Summary
  12. 9References
  13.   A Appendix: Calculations for Resistive Matching Network
  14.   B Appendix: Calculations for Reactive Matching Network
  15.   C Appendix: Calculation of Loss Due to Phase Error

A Appendix: Calculations for Resistive Matching Network

The two key constraints are:

  1. Impedance as seen looking out from the load is equal to the load impedance
  2. Impedance as seen looking out from any source is equal to that source output impedance

To simplify this, first introduce these two variables, x and y.

Equation 17. x =   R 1 +   R S o u r c e
Equation 18. y =   R 2 +   R L o a d

Now realizing that the combination of N things in parallel has an impedance that is equal to the original impedance divided by N and using the above two definitions yields the following key equations:

Equation 19. R L o a d =   R 2 +   x N
Equation 20. R S o u r c e =   R 1 +   x N - 1   | | y   = R 1 +   x × y x + n - 1 × y

These equations can be rearranged as follows:

Equation 21. x N   + y = 2 × R L o a d
Equation 22. x × y x + n - 1 × y + x   =   2 × R S o u r c e

Equation Equation 22 can be simplified to

Equation 23. 2 × N × R L o a d × x   =   2 × R S o u r c e × 2 × N × R L o a d - y

Equations Equation 21 and Equation 23 can be combined to get

Equation 24. x   =   2 × R S o u r c e × R L o a d × N × N - 1 N 2 × R L o a d - R S o u r c e

Equations (11), (12), (18), and (19) can be combined to get the values for R1 and R2

Equation 25. R 1   =   R S o u r c e × 2 × R L o a d × N × N - 1 N 2 × R L o a d - R S o u r c e - 1
Equation 26. R 2   = R L o a d × 1 - 2 × R S o u r c e × N - 1 N 2 × R L o a d - R S o u r c e

By setting the following condition, this can make sure that R2 >=0, which also makes sure R1 >=0

Equation 27. R L o a d R S o u r c e   =   2 × N - 1 N 2

In many cases, the sources and load impedance are the same. In such cases, equations Equation 25 and Equation 26 simplify to:

Equation 28. R 1   =   R 2 =   R M a t c h × N - 1 N + 1