EECS 242 Scattering Parameters Prof Niknejad University of California Berkeley University of California Berkeley EECS 242 p 1 43 Scattering Matrix V1 1 V1 3 V2 2 V3 V3 V2 Voltages and currents are difficult to measure directly at microwave freq Z matrix requires opens and it s hard to create an ideal open parasitic capacitance and radiation Likewise a Y matrix requires shorts again ideal shorts are impossible at high frequency due to the finite inductance Many active devices could oscillate under the open or short termination It s important to realize that although we associate S parameters with high frequency and wave propagation the concept is valid for any frequency S parameters are easier to measure at high frequency The measurement is direct and only involves measurement of relative quantities such as the SWR or the location of the first minima relative to the load University of California Berkeley EECS 242 p 2 43 Power Flow in an One Port The concept of scattering parameters is very closely related to the concept of power flow For this reason we begin with the simple observation that the power flow into a one port circuit can be written in the following form Pin Pavs Pr where Pavs is the available power from the source Unless otherwise stated let us assume sinusoidal steady state If the source has a real resistance of Z0 this is simply given by Pavs Vs2 8Z0 Of course if the one port is conjugately matched to the source then it will draw the maximal available power from the source Otherwise the power Pin is always less than Pavs which is reflected in our equation In general Pr represents the wasted or untapped power that one port circuit is reflecting back to the source due to a mismatch For passive circuits it s clear that each term in the equation is positive and Pin 0 University of California Berkeley EECS 242 p 3 43 Power Absorbed by One Port The complex power absorbed by the one port is given by Pin 1 V1 I1 V1 I1 2 which allows us to write Pr Pavs Pin the factor of 4 instead of 8 is used since we are now dealing with complex power The average power can be obtained by taking one half of the real component of the complex power If the one port has an input impedance of Zin then the power Pin is expanded to Pin 1 Vs2 V1 I1 V1 I1 4Z0 2 1 2 Zin Vs Vs Zin V Vs s Zin Z0 Zin Z0 Zin Z0 Zin Z0 which is easily simplified to where we have assumed Z0 is real Pin Vs 2 2Z0 Z Z0 Zin Zin 0 Zin Z0 2 University of California Berkeley EECS 242 p 4 43 Definition of Reflection Coefficient With the exception of a factor of 2 the premultiplier is simply the source available power which means that our overall expression for the reflected power is given by Vs2 Pr 4Z0 Z Z0 Zin Zin 0 1 2 Zin Z0 2 which can be simplified Zin Z0 2 Pavs 2 Pr Pavs Zin Z0 where we have defined or the reflection coefficient as Zin Z0 Zin Z0 From the definition it is clear that 1 which is just a re statement of the conservation of energy implied by our assumption of a passive load University of California Berkeley EECS 242 p 5 43 Scattering Parameter This constant also called the scattering parameter of a one port plays a very important role On one hand we see that it is has a one to one relationship with Zin Given we can solve for Zin by inverting the above equation Zin Z0 1 1 which means that all of the information in Zin is also in Moreover since 1 we see that the space of the semi infinite space of all impedance values with real positive components the right half plane maps into the unit circle This is a great compression of information which allows us to visualize the entire space of realizable impedance values by simply observing the unit circle We shall find wide application for this concept when finding the appropriate load source impedance for an amplifier to meet a given noise or gain specification More importantly expresses very direct and obviously the power flow in the circuit If 0 then the one port is absorbing all the possible power available from the source If 1 then the one port is not absorbing any power but rather reflecting the power back to the source Clearly an open circuit short circuit or a reactive load cannot absorb net power For an open and short load this is obvious from the definition of For a reactive load this q is pretty clear if we substitute 2 Zin jX jX Z0 X 2 Z0 q X 1 jX Z0 X 2 Z 2 0 University of California Berkeley EECS 242 p 6 43 Relation between Z and The transformation between impedance and is a well known mathematical transform see Bilinear Transform It is a conformal mapping meaning that it preserves angles which maps vertical and horizontal lines in the impedance plane into circles We have already seen that the jX axis is mapped onto the unit circle Since 2 represents power flow we may imagine that should represent the flow of voltage current or some linear combination thereof Consider taking the square root of the basic equation we have derived p p Pr Pavs where we have retained the positive root We may write the above equation as b1 a1 where a and b have the units of square root of power and represent signal flow in the network How are a and b related to currents and voltage University of California Berkeley EECS 242 p 7 43 Definition of a and b Let a1 V1 Z0 I1 2 Z0 b1 V1 Z0 I1 2 Z0 and It is now easy to show that for the one port circuit these relations indeed represent the available and reflected power V1 I1 V1 I1 Z0 I1 2 V1 2 a1 4Z0 4 4 2 Now substitute V1 Zin Vs Zin Z0 and I1 Vs Zin Z0 we have Z Z Z Zin 2 Z0 Vs 2 Vs 2 Zin Vs 2 0 in 0 a1 4Z0 Zin Z0 2 4 Zin Z0 2 4Z0 Zin Z0 2 2 or Vs 2 2 a1 4Z0 Z Z Z Zin 2 Z02 Zin 0 in 0 Zin Z0 2 University of California Berkeley Vs 2 4Z0 Zin Z0 2 Zin Z0 2 Pavs EECS 242 p 8 43 Relationship betwen a b and Power Flow In a like manner the square of b is given by many similar terms b1 2 2 Vs 4Z0 Z Z Z Zin 2 Z02 Zin 0 in 0 Zin Z0 2 a1 2 2 Zin Z0 2 Pavs 2 Pavs Zin Z0 as expected We can now see that the expression b a is analogous to the expression V Z I or I Y V and so it can be generalized to an N port circuit In …