A Determination of the eight-port noise parameters

Figure 2 shows the constituted eight-port network and the associated noise sources. The relationships among measured noise power spectra densities, b appearing at the output of the eight-port network (ports 1–4 via the SP4T switch), on the one hand and external and internal noise sources coming from switch terminations and the eight-port network, a and c respectively on the other hand, can be expressed as in (1).

(1) $${\bf b} = \left( {\matrix{ \matrix{{\bf b}_e \hfill \cr {\bf b}_s \hfill} \cr {{\bf b}_i} \cr}} \right) = \left( {\matrix{ {{\bf S}_{ee}} & {{\bf S}_{es}} & {{\bf S}_{ei}} \cr {{\bf S}_{se}} & {{\bf S}_{ss}} & {{\bf S}_{si}} \cr {{\bf S}_{ie}} & {{\bf S}_{is}} & {{\bf S}_{ii}} \cr}} \right)\left( {\matrix{ \matrix{{\bf a}_e \hfill \cr {\bf a}_s \hfill} \cr {{\bf a}_i} \cr}} \right) + \left( {\matrix{ \matrix{{\bf c}_e \hfill \cr {\bf c}_s \hfill} \cr {{\bf c}_i} \cr}} \right).$$

S _ij in (1) represent the partitioned S-parameters of the eight-port network. The measured noise power spectra density, b is deduced from the measured noise figure and gain for the respective output ports. Subscripts e, i, and s represent the measuring, DUT noise input, and noise source ports, respectively.

The authors have shown in [Reference Ahmed and Yeom23], that the output noise spectral density b _e , is related to all other noise waves in the setup, shown in Fig. 2, by

(2) $${\bf b}_e = {\bf S}_t{\bf a}_e + {\bf P}{\bf a}_s + {\bf Q}{\bf c}_i + {\bf \Lambda} {\bf c}_D + {\bf c}_e.$$

The embedded matrices can be expressed in terms of the partition matrices as

(4 × 2) $${\bf \Lambda} = {\bf S}_{ei}\left( {{\bf I} - {\bf S}_{ii}{\bf S}_D} \right)^{ - 1}, $$

(4 × 4) $${\bf S}_t = {\bf S}_{ee} + {\bf \Lambda} {\bf S}_D{\bf S}_{ie}, $$

(4 × 2) $${\bf Q} = {\bf \Lambda} {\bf S}_D, $$

(2 × 2) $${\bf P} = {\bf S}_{es} + {\bf \Lambda} {\bf S}_D{\bf S}_{is}. $$

S _t a _e represents the thermal noise contribution from terminations inside the SP4T switch to the output noise waves b _e and Pa _s represents the thermal noise contribution to the output noise waves b _e from the smart noise source and 50 Ω termination at ports 5 and 6, respectively. c _e represents the noise wave intrinsic to the eight-port network and which appears directly at the measuring ports, while c _i on the other hand is part of the intrinsic noise of the eight-port network appearing at ports 7 and 8. The term Qc _i represents the c _i that is reflected back into the six-port network and which appears at its outputs. Qc _i is thus due to mismatch between ports 7 and 8 of the eight-port network on the one hand and the input terminals of the DUT on the other hand, respectively. The term Λc _D in equation (2) represents the desired noise wave contribution to the output noise wave exclusively by the DUT, which is the term that needs to be known in order to determine the noise wave parameters of a given DUT. Once the undesired noise contributions to b _e , i.e. c _e , c _i , and a _e are determined, the noise wave correlation matrix of the DUT, c _D can then be obtained from Λc _D .

B Determination of c _e , c _i , and correlations c _ie

From equation (2), c _e can be determined by measuring noise wave b _e across a 50 Ω load when a _e  = a _s  = a _i  = 0. Thus, ports 7 and 8 are terminated in 50 Ω loads and the noise source is applied to port 5 with port 6 terminated in 50 Ω load. Note that the 50 Ω loads also generate thermal noise. As previously stated, the output noise wave b _e is obtained from ports 1 to 4. The noise source is alternately applied to port 6 and the noise power measurement can be taken at ports 1–4.

Since the noise contributions due to the 50 Ω terminations as well as the smart noise source are thermal, the normalized noise contributions can be determined from their respective gains, G _ij . Thus, the normalized noise power of the eight-port network emerging from ports 1 to 4 is given by

(3) $${\bf C}_{ee} = \left( {\matrix{ {n_{1,av} - ( {G_{15} + G_{16} + G_{17} + G_{18}} )} \cr {n_{2,av} - ( {G_{25} + G_{26} + G_{27} + G_{28}} )} \cr {n_{3,av} - ( {G_{35} + G_{36} + G_{37} + G_{38}} )} \cr {n_{4,av} - ( {G_{45} + G_{46} + G_{47} + G_{48}} )} \cr}} \right).$$

n _i,av is the average noise powers measured at the output ports with the termination conditions described above in place. The gains G _ij are determined from the S-parameters of the eight-port network.

C Determination of c_i and correlations with c _e

The term c _i and its correlation can be determined using three different terminations, standard open termination, O, standard short termination S, and offset-open D, which is the case when the port in question (port 7 or 8) is left open without any termination.

In Fig. 2, c _o1 and c _o2 are defined as the noise waves of the LNAs (referred to their respective inputs) incident on the eight-port network at the DUT input plane. Then, c _e can be expressed as a linear combination of c _o1 and c _o2 as

(4) $${\bf c}_e = c_{o1}{\bf r}_1 + c_{o2}{\bf r}_2 + {\bf N}_S.$$

Noise waves c _o1 and c _o2 are mainly due to the output noises of the LNAs in the DUT ports, which are transformed to the DUT input plane. The column vectors r ₁ and r ₂ are the column partition of S _ei . The remaining noise contributions to c _e , are represented by column vector N _s .

Consider the cases that one of the DUT ports, say port 7 is terminated in short, open, and offset open while the other port 8 is terminated in 50 Ω. The respective DUT S-parameters for these terminations are

$${\bf S} = \left( {\matrix{ { - 1} & 0 \cr 0 & 0 \cr}} \right),\quad {\bf O} = \left( {\matrix{ 1 & 0 \cr 0 & 0 \cr}} \right),\quad \quad {\bf D} = \left( {\matrix{ {e^{j\theta}} & 0 \cr 0 & 0 \cr}} \right).$$

In addition, the correlation matrix C _DD for the case that port 8 is terminated by 50 Ω is

$${\bf C}_{DD,7} = \left( {\matrix{ 0 & 0 \cr 0 & 1 \cr}} \right).$$

Similarly when port 7 is terminated in 50 Ω

$${\bf C}_{DD,8} = \left( {\matrix{ 1 & 0 \cr 0 & 0 \cr}} \right).$$

The corresponding measured noise powers for ports 1–4 when the DUT ports are terminated in short, open, and offset open are denoted β _S7 and β _S8, β _O7 and β _O8, β _D7 and β _D8, respectively. The subscripts 7 and 8 denote the port shorted or opened or offset opened.

Noting that the Q-matrix of equation (1) will change according to the nature of S _D , which in this case are the terminations described above, q _ij in equations (5) represents element of the new Q-matrix formed with those terminations. Thus, the excess noise power at port 1 for terminations open, short, and offset open can be expressed as

(5a) $$\eqalign {& \left ( {\matrix{ {{\left \vert {q_{11,O7}} \right \vert} ^2} & {q_{11,O7}r_{11}^*} & {r_{11}q_{11,O7}^*} \cr {{\left \vert {q_{11,S7}} \right \vert} ^2} & {q_{11,S7}r_{11}^*} & {r_{11}q_{11,S7}^*} \cr {{\left \vert {q_{11,D7}} \right \vert} ^2} & {q_{11,D7}r_{11}^*} & {r_{11}q_{11,D7}^*} \cr}} \right)\left( {\matrix{ {{\left \vert {c_{i1}} \right \vert} ^2} \cr {c_{i1}c_{o1}^*} \cr {c_{i1}^* c_{o1}} \cr}} \right) \cr & \quad = \left( {\matrix{ {\beta _{O7,1}} \cr {\beta _{S7,1}} \cr {\beta _{D7,1}} \cr}} \right).}$$

Similarly, when port 8 is terminated in the three terminations mentioned above and port 7 is terminated in matched load, the following equations are obtained:

(5b) $$\eqalign {& \left( {\matrix{ {{\left \vert {q_{12,O8}} \right \vert} ^2} & {q_{12,O8}r_{12}^*} & {r_{12}q_{12,O8}^*} \cr {{\left \vert {q_{12,S8}} \right \vert} ^2} & {q_{12,S8}r_{12}^*} & {r_{12}q_{12,S8}^*} \cr {{\left \vert {q_{12,D8}} \right \vert} ^2} & {q_{12,D8}r_{12}^*} & {r_{12}q_{12,D8}^*} \cr}} \right)\left( {\matrix{ {{\left \vert {c_{i2}} \right \vert} ^2} \cr {c_{i2}c_{o2}^*} \cr {c_{i2}^* c_{o2}} \cr}} \right) \cr & \quad = \left( {\matrix{ {\beta _{O8,1}} \cr {\beta _{S8,1}} \cr {\beta _{D8,1}} \cr}} \right).}$$

Solving equations (5a) and (5b) yields |c _i1|², c _i1 c _o1*, c _i1*c _o1 and |c _i2|², c _i2 c _o2*, c _i2*c _o2. Similar results can be obtained for the three remaining measuring ports, ports 2–4. Thus by averaging the values, c _i and its correlation with c _e can be determined.

Once these noise waves and respective correlations are determined, then all the internal noise sources of the eight-port network as well as the external noise source coming from switch terminations are known and a calibrated noise power, which is exclusively due to a DUT being measured, β _e can be obtained by accounting for the effect of these noise contributions; and the result can be expressed as

(6) $${\bf \beta} _e = {\bf \Lambda} {\bf c}_D.$$

This relationship between the calibrated measured noise powers and the DUT noise waves is of the form:

(7) $${\bf y} = {\bf Ax}{\rm,} $$

(8a) $${\bf A} = \left( {\matrix{ {{\left \vert {\lambda _{11}} \right \vert} ^2} & {{\left \vert {\lambda _{12}} \right \vert} ^2} & {2{\mathop{\rm Re}\nolimits} \left( {\lambda _{11}\lambda _{12}^*} \right)} & { - 2{\mathop{\rm Im}\nolimits} \left( {\lambda _{11}\lambda _{12}^*} \right)} \cr {{\left \vert {\lambda _{21}} \right \vert} ^2} & {{\left \vert {\lambda _{22}} \right \vert} ^2} & {2{\mathop{\rm Re}\nolimits} \left( {\lambda _{21}\lambda _{22}^*} \right)} & { - 2{\mathop{\rm Im}\nolimits} \left( {\lambda _{21}\lambda _{22}^*} \right)} \cr {{\left \vert {\lambda _{31}} \right \vert} ^2} & {{\left \vert {\lambda _{32}} \right \vert} ^2} & {2{\mathop{\rm Re}\nolimits} \left( {\lambda _{31}\lambda _{32}^*} \right)} & { - 2{\mathop{\rm Im}\nolimits} \left( {\lambda _{31}\lambda _{32}^*} \right)} \cr {{\left \vert {\lambda _{41}} \right \vert} ^2} & {{\left \vert {\lambda _{42}} \right \vert} ^2} & {2{\mathop{\rm Re}\nolimits} \left( {\lambda _{41}\lambda _{42}^*} \right)} & { - 2{\mathop{\rm Im}\nolimits} \left( {\lambda _{41}\lambda _{42}^*} \right)} \cr}} \right),$$

(8b) $${\bf x} = \left( {{\left \vert {c_{d,1}} \right \vert} ^2,{\left \vert {c_{d,2}} \right \vert} ^2,{\mathop{\rm Re}\nolimits} \left( {c_{d,1}c_{d,2}^*} \right),{\mathop{\rm Im}\nolimits} \left( {c_{d,1}c_{d,2}^*} \right)} \right)^t,$$

(8c) $${\bf y} = \left( {{\left \vert {\beta _1} \right \vert} ^2,{\left \vert {\beta _2} \right \vert} ^2,{\left \vert {\beta _3} \right \vert} ^2,{\left \vert {\beta _4} \right \vert} ^2} \right)^t.$$

The column matrix x can then be reordered into a 2 × 2 noise wave correlation matrix of the DUT, C _DD given by

(9) $${\bf C}_{{\bf DD}} = \left( {\matrix{ {{\left \vert {c_{d,1}} \right \vert} ^2} & {c_{d,1}c_{_{d,2}} ^*} \cr {c_{_{d,1}} ^* c_{_{d,2}}} & {{\left \vert {c_{d,2}} \right \vert} ^2} \cr}} \right) = \left( {\matrix{ {c_{11}} & {c_{12}} \cr {c_{21}} & {c_{22}} \cr}} \right) $$

The noise current correlation matrix C _Y and ABCD noise correlation matrix C _A are obtained by applying the transformations in [Reference Hillbrand and Russer25] to the obtained noise wave correlation matrix in (9). The noise parameters are then computed from the ABCD noise correlation matrix obtained using the formulae in [Reference Hillbrand and Russer25].

A S-parameters measurement of the eight-port network

It must be noted that, the eight-port network consists of the six-port network, the two flexible cables connecting the couplers to the six-port network via the LNA amplifiers, the bias-T and the onwafer probe. Since there is no direct way of connecting the tips of the probe (here representing ports 7 and 8) to the network analyzer, a challenge arises in the measurement of the S-parameters of the eight-port network.

To do this, the wafer probes are disconnected making the eight-port network an all-coaxial connector-type network. The S-parameters of such a network is easily measured using a four-port network analyzer. Since the output ports of the eight-port network (ports 1–4) are connected to an SP4T switch, the eight-port network essentially reduces to a five-port network. Thus, by varying the switch state, four (4) datasets of S-parameters can be obtained which can be combined to form the eight-port S-parameters.

To obtain the S-parameters of the wafer-probes, a two-port network analyzer is used. Port 1 of the network analyzer is calibrated for one-port coaxial connector type DUT measurement, while the other port is calibrated with a wafer probe calibration kit for one-port wafer probe DUT measurement. The wafer probe is then connected to the coaxial connector of the network analyzer and a two-port S-parameters measurement is carried out to obtain the S-parameters of the wafer probes. The S-parameters of the wafer probes thus obtained is combined with the eight-port S-parameters earlier measured to form the complete eight-port S-parameters inclusive of the wafer probes.

B Noise parameters measurement

The measurement setup is as shown in Fig. 1. For the purposes of calibration, onwafer-type 50 Ω terminations, open, short, and offset-open were used as the DUT. The respective noise powers measured at ports 1–4 enabled the calibration of the eight-port network as set out in Section III, which involves the determination of the noise waves emanating from the switch terminations as well as those emerging from the eight-port network. The following are the steps involved in the calibration. The reference to port numbers is as designated in Fig. 1(a).

Step 1: The NFA is calibrated with a 14–16 dB ENR noise source only. Mismatch at the input of the NFA can be reduced using an attenuator. Step 2: The setup for the calibration is wired as shown in Fig. 1, but without the DUT.

The following set of steps is for determining c _e , the noise wave emitted by the eight-port network, directly into the measurement ports.

Step 3: Two 50 Ω terminations are used to terminate ports 7 and 8. Step 4: The noise source is then connected to port 5, while port 6 is terminated in matched load. Step 5: Switching the output from ports 1 to 4, the noise figure and gain for this state is measured for each of the four measuring ports. Step 6: The noise source is then switched to port 6, while port 5 is terminated in matched load. Step 7 is then repeated.

This set of steps is for determining c _i , the noise wave emitted by the eight-port network into ports 7 and 8, and its correlation with c _e .

Step 8: One of the 50 Ω terminations is replaced with an open termination. Step 9: Steps 4–6 are then repeated for this state of termination. Step 10: The open and 50 Ω terminations at ports 7 and 8 are switched and steps 4–6 are again repeated. Steps 8–10 are then repeated for the remaining sets of termination; short, and offset open.

The noise powers measured in the steps above for the different conditions of terminations, are used to determine all the noise sources emanating from within and without the eight-port network.

Having thus calibrated the eight-port network, the DUTs shown in Fig. 4, a 10-dB attenuator and an MGA-82563 MMIC amplifier, were then employed in the setup to determine their respective noise waves correlation matrices and thus their noise parameters. With the DUTs connected, the noise spectral densities at the output of the eight-port network are imported into the equations set above in Section III to determine their respective noise parameters.

A S-parameters measurement of the eight-port network

The results of the S-parameters measurement for the eight-port extension of the six-port network, which includes the wafer probes, are as shown in Fig. 3. In total, eight (8) such plots could be obtained, considering that there are four (4) output ports and the noise source can alternately be applied to ports 5 and 6. For want of space however, only three (3) samples of such results are presented here. It must be noted that, in noise figure measurements with an NFA, the gains, G of the respective two-ports being measured can also be obtained along with the respective noise figures, F. The gains can then be compared with the S-parameters results to assess the accuracy of the constituted eight-port S-parameters. Figure 3(a) shows the gain of the two-port formed between ports 5 and 1; i.e. when the smart noise source is applied to port 5 and the noise power is measured at port 1. Figure 3(b) on the other hand shows the gain of the two-port formed between ports 5 and 2. The last, Fig. 3(c) shows the gain obtained when the noise source is applied to port 6 and the noise power is measured from port 3.

Fig. 3. Comparison of measured gains from S-parameters measurement and from noise figure measurements: (a) Ports 5–2; (b) Ports 5–3; and (c) Ports 6–3.

A very good agreement can be observed between the two results (network analyzer and NFA measurements), deviating by < ±0.1 dB, which confirms that the S-parameters of the eight-port network including the wafer probes were accurately obtained.

B Noise parameters measurement

Figures 5 and 6 show the results of the noise parameters measurement of two DUTs, respectively [an MGA-82563 MMIC amplifier (Fig. 4(a)) and a 10-dB attenuator (Fig. 4(b))]. In the case of the 10-dB attenuator which is a passive device, Fig. 5, the reference noise parameters were obtained from simulation, using the measured S-parameters of the 10 dB attenuator; in accordance with the Bosma theory [Reference Bosma26]. The agreement between the reference and obtained results is remarkably very good. The deviation from the reference trace is observed to be within ±0.1 dB for the minimum noise figure, NF _min , while that for the noise resistance, R _n is within 1%. It is also remarkable that, the optimum reflection coefficient, the magnitude of which is between 0.12 and 0.14 could be estimated by the proposed technique.

Fig. 4. DUT: (a) MGA-82563 MMIC amplifier and (b) 10 dB attenuator.

Fig. 5. Noise parameters of 10 dB attenuator: (a) Minimum noise figure, NF _min ; (b) Noise resistance, R _n ; (c) Magnitude of optimum reflection coefficient, S _opt ; and (d) Phase of optimum reflection coefficient, S _opt .

Fig. 6. Noise parameters of MGA-82563 MMIC amplifier. (a) Minimum noise figure, NF _min ; (b) Noise resistance, R _n ; (c) Magnitude of optimum reflection coefficient, S _opt ; and (d) Phase of optimum reflection coefficient, S _opt .

The reference noise parameters in the case of the MGA-82563 MMIC amplifier were obtained from the datasheet supplied by the manufacturer. It must be noted that, only three (3) frequency points for the given frequency range is available from the datasheet, whereas our measurement were taken over 51 frequency points. Furthermore, quoting from the datasheet of the MGA-82563 by Avago Technologies [27], “the values for parameters are based on comprehensive product characterization data, in which automated measurements are made on a minimum of 500 parts taken from 3 non-consecutive process. The data derived from product characterization tend to be normally distributed, e.g. fits the standard ‘bell curve’. For parameters where measurements or mathematical averaging may not be practical, such as the Noise and S-parameter tables or performance curves, the data represents a nominal part taken from the ‘centre’ of the characterization distribution”. Thus, the results obtained from the measurement of only one such MGA-82563 MMIC amplifier using the proposed technique shows deviation in noise parameters from the nominal values in the datasheet. In the light of these observations, the departure of the measured results from the datasheet, is reasonable and demonstrates the capability of the technique presented in estimating the noise parameters of an onwafer-type DUT. It must be added that, factors that affect the accuracy of the extracted noise parameters include input and output mismatches of the DUT as well as the gain of the DUT. A portion of the noise waves emanating from the DUT can be reflected at the input and output planes of the noise parameters test set, back into the DUT, a situation that has not been factored in this work.