Problems Four

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Scattering parameters

Question 7

Show that a condition for stability of a 2 port is that there is no reflection gain for connections looking in to either port; and that such a circuit is unconditionally stable for any passive load impedance and passive generator source impedance.

• Here, the important point is that there must be no reflection gain at either port for any conditions of passive load attached to the other port. If this condition is not met, the return power is more than the incident power; we have in effect made a "reflection amplifier" which can cause oscillations in a reactive load formed by a length of transmission line attached to an open or short circuit.
• The condition for a length of line to oscillate is that the round trip gain, accounting for the losses of the line and the reflection coefficients at each end, be greater than one.
• For a passive load the magnitude of the reflection coefficient is always less than one. Therefore, for there to be round trip gain we require the other reflection to have magnitude greater than one. If it doesn't, no oscillations will occur.

A certain 2-port has s parameters s11 = 0.1 angle -30 degrees, s12 = 0.05 angle -60 degrees, s21 = 30 angle -60 degrees, and s22 = 0.15 angles -20 degrees. All these s parameters are taken at a spot frequency.

•        let us write these s parameters formally as
  
  
          {0.1 -30}             {0.05 -60}
          {30  -60}             {0.15 -20}
  
         the defining equations are
  
         b1 = s11a1 + s12a2
         b2 = s21a1 + s22a2
  
  

•           let us call the reflection coefficient at the load
            on port 2 gamma.
  
            Then a2 = gamma b2 because the output b2 of the 2-port
            is reflected from the load to form a2.
  
            Using the equation b2 = s21a1 + s22a2
            then we find a2(1/gamma - s22) = s21a1
  
            Using the equation b1 = s11a1 + s12a2
            then we find b1 = a1[s11 + s12s21/(1/gamma - s22)]
  
            We want b1 to have magnitude less than a1
            for there to be no reflection gain at port 1.
  
            Thus the complex number s11 + s12s21/(1/gamma -s22)
            has to lie inside the unit circle.
  
            One can substitute the values above; I estimate
            that if the reflection coefficient is less than
            about |gamma| = 0.55 the circuit will be stable.
  
            That puts the output transmission line VSWR as
            being less than about 3.5
  
  

Question 8.

Write down the S matrices of the following microwave components.... You may assume they have ideal characteristics.

A 30 dB amplifier

An isolator with 20dB front to back transmission ratio

A perfect 3 port circulator

A perfect 4 port dual directional coupler, with coupling strength -30 dB

A 2 metre length of 50 ohm coaxial cable at 400MHz, assuming a velocity factor of 2/3

• answers.....

  
  
                        The 30dB amplifier
  
                     0                     0
                     {31.6 -theta}         0
  
  
  
  
                        The 20 dB isolator
                     
                     0                {0.1 - phi}
                     {1 -theta)            0
  
  
  
  
                        The circulator
  
                     0            0          {1 -phi}
                     {1 -phi}     0          0
                     0            {1 -phi}   0
   
                   Here the circulation is 123123123....
  
  
  
                        The dual directional coupler
  
      0            {0.9995 -theta}        {0.0316 -rho}         0
     {0.9995 -theta}    0                     0           {0.0316 -phi} 
     {0.0316 -rho}      0                     0           {0.9995 -psi}
      0            {0.0316 -phi}          {0.9995 -psi}         0  
    
                  Here we see the S matrix is reciprocal.
                  Adding the moduli squared in the columns gives 1
                  Also, theta-rho = phi-psi - 180 degrees
                  and,  theta-phi = rho-psi - 180 degrees
  
  
  
  
                 
                       The 2 metre length of cable
  
           Well, the velocity is 20 cm/nsec so a wavelength
           at 0.4GHz is 50cm and the line is 4 wavelengths long.
  
                       0             1
                       1             0
  
  

A generator is connected via the isolator described above to one of the ports of the three port circulator. The circulator has a mismatch on its two other ports giving a reflection coefficient of 0.2 angle 0 on each load connected to the ports. The phase shift between successive ports of the circulator is -45 degrees.

Calculate the magnitude of the reflection coefficient first at the driven port of the circulator, and second, at the generator terminals.

• The reflection coefficient at port 2 has size 0.2 so the wave incident on port 3 is 0.2 of the size of the incident wave. It is reflected by the load on port 3 with another factor 0.2 so the size of the reflection coefficient at port 1 is 0.04. Now the phase shift is -90 degrees if you need to calculate the effective input impedance.
• The isolator reduces the reflection coefficient by another 20 dB or factor of 10. Thus at the generator terminals the reflection coefficient is 0.004 in size.