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

C Appendix: Calculation of Loss Due to Phase Error

Start by thinking about adding two vectors of equal magnitude. The first vector is at angle of zero degrees and the second is at angle of ϕ. The vector is therefore:

<1+cos ϕ, sin ϕ>

The magnitude can be calculated to be:

Equation 43. 1 + c o s   ϕ ,   s i n   ϕ   =   1 + c o s   ϕ 2 + s i n   ϕ 2 = 2 × 1 + c o s   ϕ

If not intuitive that the angle of the resultant vector is, then this becomes more apparent by recognizing the angle as calculated by this vector relates to the half angle formula for tangent.

Equation 44. t a n     ϕ 2   =   s i n   ϕ 1 + c o s   ϕ

Furthermore, this needs to be intuitive that if one combines an even number of vectors, the angle does not change and the magnitude can be:

Equation 45. N   2 × 2 × 1 + c o s   ϕ =   N × 1 + c o s   ϕ

If there is an odd number of vectors, then add all but the last one and this formula is known. Clearly if the last vector was at angle f/2, then this can give the most power, so without loss of generality, this can be at zero degrees for the worst case. So in the case of an odd number of vectors, the magnitude can be:

Equation 46. 1 + N - 1   2 + N - 1   2 × c o s   ϕ , N - 1   2 ×   s i n   ϕ   =   1 + N - 1   2 + N - 1   2 × c o s   ϕ 2 + N - 1   2 ×   s i n   ϕ 2 = N 2 + 1 2 + N 2 - 1 2 × c o s   ϕ

Of interest is the loss in power compared to the designed for combination of N signals and this can be calculated as:

Equation 47. L o s s   =                             10 × l o g 1 + c o s   ϕ 2                     N   e v e n 10 × l o g 1 n 2 + 1 - 1 n 2 1 + cos ⁡ ϕ 2                     N   o d d