(1) The open and short circuit required for the Z and Y parameters cannot usually be

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ECE 580 Network Theory Scattering Matrix 76 The Scattering Matrix Motivation for introducing the SM: () The open and short circuit required for the Z and Y parameters cannot usually be implemented in

1 ECE 580 Network Theory Scattering Matrix 76 The Scattering Matrix Motivation for introducing the SM: () The open and short circuit required for the Z and Y parameters cannot usually be implemented in actual high-frequency measurements (parasitic C and L); () There may be biasing and/or stability problems for active devices. Hence, it is preferable to measure the two-port under its actual operating conditions; (3) At microwave frequencies, V and I are hard to measure; only power and phase are detectable. (4) Always exist. D.M Pozar, John Wiley; Microwave Engineering A photograph of the Hewlett-Packard HP850B Network Analyzer. This test instrument is used to measure the scattering parameters (magnitude and phase) of a one ore two-port microwave network from 0.05 GHz to 6.5 GHz. Built-in microprocessors provide error correction, a high degree of accuracy, and a wide choice of display formats. This analyzer can also perform a fast Fourier transform of the frequency domain data to provide a time domain response of the network under test.

2 ECE 580 Network Theory Scattering Matrix 77 Definitions We shall use variables, which are directly related to power. The inputs (incident signals) are related to the known maximum available power of the generator at the port; outputs (reflected signals) are related to the actual output power at the port. Assume resistive terminations R k at all ports, with or without a generator E k. The actual port variables are V k and I k, considered peak value phasors of steady-state sine-wave signals. We define the following hypothetical quantities: Incident voltage V ik at port k: the output voltage E k / of the generator at port k under matching (max-power) conditions. Incident current I ik at port k: the output current Ek/(R k ) of the generator at port k under matching conditions. Incident wave signal a k at port k: a k = V ik I ik = V ik R k = I ik R k Hence, a k = E k = P 8R max k k

3 ECE 580 Network Theory Scattering Matrix 78 Reflected voltage V rk at port k: the difference between the actual and incident voltages; V rk = V k -V ik. If there is no generator, V rk equals the actual voltage at port k. Reflected current I rk at port k: The difference between the incident and actual currents (notice the sign difference!); I RK = I ik -I k. If there is no generator at port k, I rk =-I k. V rk =R k I rk. Reflected wave (signal) b k at port k: b k = V rk I rk = V rk / R k = I rk R k Hence b k / = V rk I rk /. If there is no generator, this is the actual power exiting at port k. As will be shown, the actual power entering at pork k where there is a generator is [ a k b k ] /. Scattering Matrix S connects the column vectors a and b formed from all incident and reflected signals: b = S a From this definition it follows that the lm element of S is given by S lm = b l = V r l / R l a m E m / R m ( ) = V / R l l E /( R m ), l m Where E k =0 is set for all k m. S lm has a clear physical meaning: S lm = P rl P max m = actual power leaving port l maximum power from port m E k =0 for k m.

4 ECE 580 Network Theory Scattering Matrix 79 For the diagonal element s ll, with the only generator E l. V l = E l Z l Z l + R l Which gives b l a l = s ll = Z l R l Z l + R l Reflection Coefficient at port l: Gives the ratio of the reflected and incident powers. The power entering port l is therefore given by: P l = P il P rl = P il [ S ll ] Noted that the scattering relations given above describe not only the N-port (like e.g., the open-circuit impedance equations) but also its terminations and sources. Scattering variables

5 ECE 580 Network Theory Scattering Matrix 80 By KVL, E= V+RI ρe = V-RI = V RI V + RI E So ρ= V RI V + RI = Z R Z + R the Reflectance (Reflection Coefficient) By superposition: V i = (V + RI) V r = (V RI) I i = R (V + RI) I r = (V RI) R V i = RI i V r = RI r We define: Incident signal = a = V i I i Reflected signal = b = V r I r = R V i = R I i = R V r = R I r

6 ECE 580 Network Theory Scattering Matrix 8 a = R V + R b = R V R I I V = R (a + b) I = R (a b) Given that V = ZI R (a + b) = ZR (a b) a + b = z n (a b) (Z n +)b = (Z n )a b = (Z n +) (Z n )a where Z n R ZR b = Sa where S = (Z n +) (Z n ) = ρ = (Z R) /(Z + R) If the -port is passive, Z & Z n are positive functions. i.e. Re{ Z} 0 for σ 0 & the same for Z n Then S satisfies S 0 for σ 0 S is then called a bounded function Z n is positive iff S is bounded.

7 ECE 580 Network Theory Scattering Matrix 8 Power relations The active power absorbed by the -port is W a = Re{ IVeσt } = Re (a b)r = eσt Re {(a b)(a + b) } = eσt Re {(aa bb) + (ab ba) } R (a + b)e σt Real Imag. W a = { eσt a b } Since b =Sa & b = S a W a = eσt ( S ) a In the harmonic state σ = 0 & W a = P P = ( S ) a P is a maximum when S =0 i.e. Z = R So P max = a is maximum available power from source. Hence P = (- S ) P max

8 ECE 580 Network Theory Scattering Matrix 83 N-port case V = n-vector of port voltages I = n-vector of port current R = diag (R, R,,R n ) & R - R = diag ( R,R,...,R n ) & R Exactly as in the -port case a = R V + R I V = R (a + b) b = R V R I I = R (a b) V i = R a V r = R b I i = R a I r = R b The scattering matrix is defined by b = S a

9 ECE 580 Network Theory Scattering Matrix 84 Relation between S and Z V = ZI R (a + b) = ZR (a b) a + b = R ZR (a b) = Z n (a b) Z n = R ZR (Z n +)b = (Z n )a b = (Z n +) (Z n )a S = (Z n +) (Z n ) = (Z n )(Z n +) S = (Z n )(Z n +) = (Z n + )(Z n +) = (Z n +) = Y an Where Y an = (Z n +) = (R ZR + R RR ) = R (Z + R)R = R (Z + R) R (Normalized & augmented Y matrix) As Y a, and hence Y an, always exists for any well-defined n-port, it means that S also always exists.

10 ECE 580 Network Theory Scattering Matrix 85 Examples Ideal transformer n Y a = n n n R + n R n n n n Y an = R n n R R n R + n R n n R R n R n S = Y an = R n R n n R R n R + n R n n R R n R n R If R = n R n or n R = n R Then S = 0 0

11 ECE 580 Network Theory Scattering Matrix 86 Alternative method

12 ECE 580 Network Theory Scattering Matrix 87 Going back to S = (Z n +) (Z n ) = R ZR + R RR = R (Z + R)R R ZR R RR R (Z R)R = R (Z + R) R R (Z R)R S = R (Z + R) (Z R)R Normalization R ZR = Z n Divides row i by R i Divides col j by R j (i, j) th element of Z n is Z ij R i R j

13 ECE 580 Network Theory Scattering Matrix 88 Gyrator Y a = R R R R R R = R + R R R R R Y an = R R R R R + R R R R R R R R R S = Y an = R R R R R + R R R R R R R R If R R = R Then S = 0 0 Note: We do not require R =R =R

14 ECE 580 Network Theory Scattering Matrix 89 Direct calculation: Z = R R V = E Z ER =, V Z + R R + R R i E, a = E R b = V r / R = (V V i ) / R S = b a = R R R R + R R V = V r = RI = R E R + Z b = V r R = E R R R + Z = E R R R R + R S = b a = R R R R R + R To find S & S, move E is series with R Then, R R, R -R, hence S = S, S = -S

15 ECE 580 Network Theory Scattering Matrix 90

16 ECE 580 Network Theory Scattering Matrix 9 Find the scattering matrix S(s) of the circuit shown using direct analysis. (Note: the circuit is the same as on p.74 of the notes.) From the circuit, Z = 3s3 + 9s + s + 3 3s + 9s + S = Z R Z + R = 3s3 + 6s 7s + 3s 3 +s +s + 4 By circuit analysis, V = 3E g 3s 3 +s +s + 4 = V r S = R R V r E g = 3 3s 3 +s +s + 4

17 ECE 580 Network Theory Scattering Matrix 9 Method of using Y an

18 ECE 580 Network Theory Scattering Matrix 93

19 ECE 580 Network Theory Scattering Matrix 94 Method of using Y an

20 ECE 580 Network Theory Scattering Matrix 95

21 ECE 580 Network Theory Scattering Matrix 96

Scattering Parameters

Berkeley Scattering Parameters Prof. Ali M. Niknejad U.C. Berkeley Copyright c 2016 by Ali M. Niknejad September 7, 2017 1 / 57 Scattering Parameters 2 / 57 Scattering Matrix Voltages and currents are