# 1 S Parameters and Power gains  Training in 1 day Roberto Antonicelli ST Belgium, Network Division.

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1 S Parameters and Power gains  Training in 1 day Roberto Antonicelli ST Belgium, Network Division

2 S Parameters Network theory Standard amplifier network Bi-port description

3 S Parameters Two-terminal element Conjugate power matching Incident and reflected power Incident power

4 S Parameters Voltages and currents Incident and reflected waves Reflection coefficients Incident and reflected power

5 S Parameters Definition The scatter matrix Scatter matrix The incident wave a depends only on the reference impedance and the source E s, while the reflected wave b depends also on the load, being zero when this is matched. The reflection coefficient  depends on both the circuit impedance Z i and the source impedance Z S. The S parameters depend on both the circuit impedances and a reference impedance Z 0

6 S Parameters Analysis Measurements Scatter matrix |a 1 |² and |a 2 |² are the incident powers at the ports 1 and 2, while |b 1 |² and |b 2 |² are the reflected powers at the two ports s 11 = input reflection coefficient with matched output s 12 = inverse transmission coefficient with matched output s 21 = forward transmission coefficient with matched output s 22 = output reflection coefficient with matched input Ex.: s 11 is the reflection coefficient at port 1, when a 2 = 0, i.e. when the port 2 is terminated over the reference impedance The S parameters depend on both the device and a reference impedance Z 0

7 S Parameters Transmission line’s scatter matrix De-embedding Let l i be the line length, Z i be the line impedance,  the phase constant,  i the electrical length

8 S Parameters Smith chart Freq. [GHz] s11s21s12s22 MagAngleMag AngleMagAngleMagAngle 0.010000.92681-0.458725.73254179.2570.0004476.05570.73428-0.19327 0.100000.92264-2.903325.65939173.5980.0031390.94010.73144-1.71104 0.300000.89887-9.255355.48504160.4800.0089788.34760.72161-5.00982 0.700000.79630-17.80644.63514137.7910.0188086.44460.68115-9.46235 0.900000.74581-19.71454.21554128.9200.0235189.15330.65988-10.6804 1.100000.70000-21.53263.83450121.5950.0274191.53890.64526-11.4538 1.500000.63412-22.74013.20471109.7820.0384196.77700.62723-11.6775 1.900000.58184-23.57622.87512101.1940.05373101.8670.69004-9.52874 2.300000.55180-22.69562.3980489.91900.06715101.0120.62656-21.3144 2.500000.54108-22.79042.2487287.17110.07530102.3660.61271-20.7434 2.700000.53229-22.84772.1296984.38750.08419103.2070.60842-20.9636 2.750000.53082-22.81752.1036083.69950.08659103.5840.60799-21.1857 3.000000.51963-23.38291.9839280.24870.09806103.2500.60700-21.7178

9 S Parameters Input/output reflection coefficients  1 ( 2 ) is the input (output) reflection coefficient that is visible at port 1 (2) when port 2 (1) is terminated on a generically unmatched impedance Z L (Z S ). It is always referred to the reference impedance Z 0.  L ( S ) is the reflection coefficient at the load (source) referred to the reference impedance Z 0.

10 S Parameters Power definitions The available power is the maximum power transferable from the source to the load (conjugate power matching). It depends only on the generator. Source available power

11 S Parameters Power definitions If the source impedance Z S is equal to the reference impedance Z 0, the squared magnitude of the incident wave |a| 2 gives the source available power. Since the available power only depends on the generator, it is often regarded as a source reflected wave, b S :

12 S Parameters Power definitions Intuitively, the net power that is transferred to the load is equal to the available power (referred to a reference impedance Z 0 ) minus the reflected power (referred to the same reference impedance). Power transferred to the load

13 S Parameters Power gain definitions In the reference impedance domain: Transducer gain By definition, the transducer power gain (TG) is the ratio btw the power transferred to the load and the source available power:

14 S Parameters Power gain definitions Maximum available gain The highest transducer gain achievable is called Maximum available gain (MAG). It depends only on the device parameters.

15 S Parameters Power gain definitions Available gain By design strategy, the source is in a controlled mismatch. The available gain depends on the device and the (unmatched) source impedance.

16 S Parameters Power gain definitions Operative gain By design strategy, the load is in a controlled mismatch. The operative gain depends on the device and the (unmatched) load impedance.

17 S Parameters Power gain definitions Unilateral transducer gain Usually, the unilateral approximation is used (s 12 = 0): Here, the mismatch effect on both the source and load sections is in. By simultaneously conjugate matching, we have the maximum unilateral transducer gain:

18 S Parameters Power gain definitions Constant available gain circles By varying the source reflection coefficients on the input Smith Chart, the resulting available power gain changes  Constant gain loci (circles). Centers and radiuses are function of the selected available gain G A. Over all those circles, the available gain is constantly equal to G A (provided the output is in conjugate matching).

19 S Parameters Power gain definitions Constant operative gain circles By varying the load reflection coefficients on the output Smith Chart, the resulting operative power gain changes  Constant gain loci (circles). Centers and radiuses are function of the selected oeprative gain G W. Over all those circles, the operative gain is constantly equal to G W (provided the input is in conjugate matching).

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