Instrumentation & Measurement Magazine 24-2 - 65

Table 3 - Transformation matrices T used in the correlation matrix conversion
Original Representation
Admittance CY

Resulting
Representation

	

1 0
TYY  

0 1​
​

y
TYZ   11
 y21
​

Impedance CZ

z
TZY   11
 z21
​

1 0
TZZ  

0 1​
​

0  z11 
TZA  

 1  z21 ​
​

Chain CA

 0 a12 
TAY  

 1 a22 ​
​

 0  a11 
TAZ  

 1  a21 ​
​

1 0
TAA  

 0 1 ​
​

Rn 

C11 A
,	(28)
2 kTo

and
C11 AC 22 A  Im C12 A   j  Im  C12 A 
2

	

Yopt 

C11 A

.	(29)

Here Re( ) and Im( ) stand for the real and imaginary parts
of elements in CA, respectively, and j is the imaginary unit.
In many occasions, I need to convert the correlation matrix
from one form to the other. For example, if I want to convert
the correlation chain matrix CA (original representation) to its
admittance form CY (resulting representation), I can use the
conversion equation [67]:
	

†

CY  TYAC ATYA	(30)

where transformation matrix TYA is given by:
	

y
TYA   11
  y21

1
 .	(31)
0 

Here y11 and y21 are the elements in the admittance matrix Y.
For other conversions, you can find their corresponding transformation matrices in Table 3.

z12 

z22 ​

y12 

y22 ​

Chain CA

Admittance CY

2 
1 
NFmin 
1
Re  C12 A   C11 AC 22 A  Im C12 A   ,	(27)
kTo 


	

Impedance CZ

y
TYA   11
  y21
​

1

0 ​

continuous downscaling of the transistor dimensions, the impact of the surrounding parasitics introduced by the probe
pads and metal interconnections on the transistor's noise characteristics has gained much importance. Therefore, I need
to apply the de-embedding algorithm to remove the impact
of the surrounding parasitics and obtain the intrinsic noise
performance of the transistor. Depending on the DUT layout, I also need to design dummy OPEN, SHORT, or THRU
structures, as shown in Fig. 5 to obtain the information of the
parasitics for the de-embedding procedure.
In general, if the characterization frequency is lower than
26.5 GHz or the distributed effects of the metal interconnections are not pronounced, I use OPEN and SHORT dummy
structures. Here I assume that the probe pads connect with
the metal interconnections and transistor in parallel, and the
metal interconnections connect with the transistor in series.
The de-embedding procedure assuming a parallel-series configuration is as follows:
Step 1: Measure the scattering parameters SDUT, SOPEN, and
SSHORT of the DUT, OPEN, and SHORT dummy structures.
Convert each of them to their admittance matrices YDUT, YOPEN,
and YSHORT, respectively.
Step 2: Measure the noise parameters NFmin,DUT, Rn,DUT, and
Yopt,DUT of the DUT, and calculate the correlation chain matrix
CA,DUT using (25).
Step 3: Convert the correlation chain matrix CA,DUT to its
correlation admittance matrix CY,DUT using YDUT and the transformation matrix TYA in Table 3.
Step 4: Substitute YOPEN into (24), calculate the correlation
admittance matrix CY,OPEN of the OPEN dummy structure.

Noise Parameter
De-embedding
To measure the thermal
noise of a nanoscale transistor, I lay out the DUT in
a Ground-Signal-Ground
(GSG) configuration, as
shown in Fig. 5, which includes a transistor, probe
pads, and metal interconnections to access
the transistor. With the
April 2021	

Fig. 5. The DUT and dummy structures of OPEN, SHORT, and THRU for high-frequency noise characterization laid out in a
Ground-Signal-Ground (GSG) configuration.

IEEE Instrumentation & Measurement Magazine	65



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