(a) Schematic of the cryogenic S-parameter measurement circuit. The signals a 1 and a 2 travel down to the RF switches A and B, located at the mixing chamber (T ⩽ 20 mK) where they are then routed to a set a of calibration standards and DUTs. The reflected and transmitted signals, b 1 and b 2 are routed through a HEMT amplifier at 4 K and a subsequent post-amplifier mounted at room temperature. A calibration algorithm is then used to de-embed the entire measurement setup and extract the DUT scattering parameters. (b) Detailed view of the microwave switch plate assembly.
Error model of the measurement system. The b 1, 2 and a 1, 2 waves (going in and out of the VNA) correspond to the labelling in our schematic circuit in Figure 1(a) . The error matrix E encompasses the effects of intervening cryogenic circuitry which is de-embedded in the calibration.
6-port switch cable symmetry. We show the (a) measured return and transmission loss for 6 nominally identical short (6 in.) commercial M/M (male/male) subminiature microwave cables (SMA) attached to one switch and (b) the maximum variation between the curves in (a). The variations in (b) are defined as , where i varies across the different cables and is plotted on a linear scale. The red and blue curves correspond to the separate switches.
Simulation of DUT input return loss error caused by a variation in the thru calibration standard path as a function of the phase difference between the line and the thru standards. In the simulation the thru had zero length. The DUT is a 20 dB attenuator with 40 dB return loss. The input and output error matrices had unit transmission and 20 dB return loss. The input return loss of the error matrix in front of the thru standard was modified by ε11 to simulate a systematic error in the thru input path.
Measured maximum phase error between the cables connecting the two 6-port switches to the device and calibration standards. The maximum phase error is plotted separately for each switch in red and blue.
Return loss measured at the 6-port common port, when switched to each of the 6 positions of a single RF switch.
Time dependence of transmission of the measurement chain and standards. Measured variation in input return loss for the reflect calibration standard (a) and transmission loss for the thru calibration standard (b) over multiple days. The values shown are expressed as a ratio of the corresponding day's S-parameters to those measured on day 0.
Time dependence of the phase drift in the measurement chain and standards. Measured phase stability of the thru standard over multiple days. Transmission coefficient (a) and reflection coefficient (b).
(a) Directional coupler DUT used to test calibration system. (b) De-embedded return loss and (c) transmission loss. Measurements taken at 20 mK (red) and 300 K (green) using our TRL system mounted on the dilution refrigerator (schematic shown in Figure 1 ) are compared with standard room temperature automated SOLT calibrated measurements taken at the bench top with short test cables at room temperature (black).
Calibrated cryogenic measurements of a commercial double-circulator. (a) Calibrated forward transmission (red) and return loss (blue) of a double-circulator configured as an isolator, measured at T = 21 mK in our dilution refrigerator setup (Figure 1 ). We can compare the performance at T = 3.5 K and 21 mK for the (b) reverse isolation and (c) forward transmission. This device was specified for performance over the 4–8 GHz band, but only measured by the manufacturer at 77 K.
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