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

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

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

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

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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

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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

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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

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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

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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.

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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

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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 :

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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

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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:

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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.

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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.

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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.

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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:

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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).

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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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