Subsections

• 3.1.1 Admittance, Impedance, Hybrid Matrices and Parameters
• 3.1.2 Scattering Parameters
• 3.1.3 Extraction of the Cut-Off and Maximum Oscillation Frequency

3.1 Introduction

The major objective of small-signal simulations is to extract various figures of merit of the devices or networks. Two-port parameter sets are useful design aids provided by manufactures for high-frequency transistors. In addition, they are used to extract the cut-off frequency or the maximum oscillation frequency, which are further required for characterization of the devices. The following basic definitions are generally used [164]:

Impedance	$\ensuremath{\underline{Z}} = R + \ensuremath{\mathrm{j}}X$,	$Z = \left\vert\ensuremath{\underline{Z}}\right\vert = \sqrt{R^2 + X^2}$ ,
Resistance	$R = \Re(\ensuremath{\underline{Z}})$,	$$R = \frac{G}{Y^2}$$
Reactance	$X = \Im(\ensuremath{\underline{Z}})$,	$$X = -\frac{B}{Y^2}$$
Admittance	$$\ensuremath{\underline{Y}} = \frac{1}{\ensuremath{\underline{Z}}} = G + \ensuremath{\mathrm{j}}B$$,	$Y =\left\vert\ensuremath{\underline{Y}}\right\vert = \sqrt{G^2 + B^2}$,
Conductance	$G = \Re(\ensuremath{\underline{Y}})$,	$$G = \frac{R}{Z^2}$$
Susceptance	$B = \Im(\ensuremath{\underline{Y}})$,	$$B = -\frac{X}{Z^2}$$

with the imaginary unit $\ensuremath{\mathrm{j}}= \sqrt{-1}$.

Furthermore, a distinction between intrinsic and extrinsic parameters has to be made. Measured admittance and scattering parameters are normally different from the simulation results. If the errors are systematically introduced by the measurement environment, it is useful to represent the device as embedded in a parasitic equivalent circuit. Hence, the intrinsic parameters represent the de-embedded device. Based on a standard parasitic equivalent circuit, the simulator can take all parasitics into account and can calculate also extrinsic two-port parameters.

In order to clarify the notation of the various parameters hereafter, the following definition shall be used: $\ensuremath{\underline{Y}}$ refers to a specific complex-valued admittance value, however for the sake of readability Y as in Y-parameters refers to a general admittance quantity. A lower case letter for an intrinsic parameter is sometimes used to distinguish from extrinsic parameters written with an upper case letter. For the sake of clarification, this distinction is not made in this work and the context of the respective parameter is always clearly indicated.

3.1.1 Admittance, Impedance, Hybrid Matrices and Parameters

An $N$-port device/network can be represented by several matrices or parameter sets. At low frequencies, these are usually Y-, Z-, H-, or A-matrices or parameters, because they can be easily measured with open or short circuits.

Figure 3.1: Voltages and currents of a two-port device/network.

For a two-port device/network as depicted in Figure 3.1, they are defined as follows:

$$\displaystyle \left( \begin{array}{c} \ensuremath{\underline{i_1}... ...uremath{\underline{v_1}} \\ \ensuremath{\underline{v_2}} \end{array} \right) \$$ (3.1)

$$\displaystyle \left( \begin{array}{c} \ensuremath{\underline{v_1}... ...uremath{\underline{i_1}} \\ \ensuremath{\underline{i_2}} \end{array} \right) \$$ (3.2)

$$\displaystyle \left( \begin{array}{c} \ensuremath{\underline{v_1}... ...uremath{\underline{i_1}} \\ \ensuremath{\underline{v_2}} \end{array} \right) \$$ (3.3)

$$\displaystyle \left( \begin{array}{c} \ensuremath{\underline{v_1}... ...remath{\underline{v_1}} \\ \ensuremath{\underline{-i_2}} \end{array} \right) \$$ (3.4)

Hybrid (H-) parameters are often used for the description of active devices such as transistors. Like Y-parameters, they are difficult to measure at high frequencies. The absolute value of the ${\ensuremath{\underline{H_{21}}}}$ parameter is used to characterize $f_\textrm {T}$, where the current gain has dropped to unity. The so-called chain or A-parameters, sometimes also referred to as ABCD-parameters, are useful for cascaded circuit topologies, since these parameters allow matrix multiplications of the single elements:

$$\left( \begin{array}{cc} \ensuremath{\underline{A}} & \ensuremath... ...emath{\underline{D_{1}}} \ensuremath{\underline{D_{2}}}) \\ \end{array} \right)$$ (3.5)

Measurements of Z-parameters require (analogously to Y-parameters) open circuit connections which may act as short circuits at RF frequencies due to stray capacitances.

3.1.2 Scattering Parameters

Advanced devices are operated under their originally intended environment conditions for higher frequencies above $100\,$MHz. A steady-state bias is applied to all terminals superimposed by an additional RF excitation. The sinusoidal currents and voltages of all terminals with magnitude and phase should be measured. This would normally involve Z-, Y-, H- and A-parameters of the linear two-port theory, which are able to completely describe the electrical properties of the device. Unfortunately, three main problems can be identified [45]:

1. At high frequencies it is problematic to directly measure voltages and currents. Voltmeters and current probes cannot be simply connected due to the probe impedances and problems related to a correct positioning. It is necessary to apply either AC-wise open or short circuits as part of the Z-, Y-, and H-parameter measurement. Especially at high frequencies short and open circuits are difficult to obtain because of lead inductances and capacitances. Such measurements typically require tuning stubs which are adjusted at each measurement frequency. This is not only inconvenient and cumbersome, but a tuning stub shunting the input or output may cause active devices to oscillate or self-destruct and thus prevent measurements or making them invalid [98].
2. The voltages and currents depend on the length of the cable used to connect the device under test to the measurement setup. Hence, the measured values depend on the position along the cable.
3. At high frequencies, true open and short termination of the device is hard to achieve.

To avoid these drawbacks, S-parameters can be used to characterize a two-port network, which are related to the scattering and reflection of traveling waves (power or equivalent voltage waves). Instead of open and short termination, the ports are terminated by a cable of the characteristic impedance $Z_0$. The device is so embedded into a transmission line of a certain characteristic impedance $Z_0$, usually 50$\,\Omega$. This scattering and reflection is comparable to optical lenses which transmit and reflect a certain amount of light. The traveling waves can so be interpreted in terms of normalized voltage and current amplitudes. S-parameters are the complex-valued reflection coefficients at each port and complex-valued transmission coefficients of the equivalent voltage wave between each pair of ports. Hence, an $N$-port device or circuit with $N^2$ S-parameters has $N$ reflection coefficients and $N^2 - N$ transmission coefficients. An additional advantage is the fact, that traveling waves do not vary in magnitude at points along a lossless transmission line. In contrast to other parameter measurements, S-parameters can be measured at some distance from the measurement transducers [98].

Figure 3.2: Traveling waves at a two-port device/network.

S-parameters provide detailed information on the linear behavior of the two-port. As shown in Figure 3.2, they are basically defined as follows:

$$\displaystyle \left( \begin{array}{c} \ensuremath{\underline{b_1}... ...remath{\underline{a_1}} \\ \ensuremath{\underline{a_2}} \end{array} \right) \ ,$$ (3.6)

with $\underline{a_i}$ and $\underline{b_i}$ as traveling waves. As illustrated in Figure 3.3, the S-parameters are defined as:

$$\displaystyle \left( \begin{array}{c} \ensuremath{\underline{b_1}... ...remath{\underline{a_1}} \\ \ensuremath{\underline{a_2}} \end{array} \right) \ ,$$ (3.7)

with

$\vert\underline{a_i}\vert^2$	$\ldots\$ power wave traveling towards the two-port,
$\vert\underline{b_i}\vert^2$	$\ldots\$ power wave reflected from the two-port,
$$\ensuremath{\vert\underline{S_{11}}\vert^2} = {\frac{\ensuremath... ...erline{b_1}\vert^2}}{\ensuremath{\vert\underline{a_1}\vert^2}}}_{\vert a_2 = 0}$$	$\ldots\$ power reflected from port 1,
$$\ensuremath{\vert\underline{S_{12}}\vert^2} = {\frac{\ensuremath... ...erline{b_1}\vert^2}}{\ensuremath{\vert\underline{a_2}\vert^2}}}_{\vert a_1 = 0}$$	$\ldots\$ power transmitted from port 1 to port 2,
$$\ensuremath{\vert\underline{S_{21}}\vert^2} = {\frac{\ensuremath... ...erline{b_2}\vert^2}}{\ensuremath{\vert\underline{a_1}\vert^2}}}_{\vert a_2 = 0}$$	$\ldots\$ power transmitted from port 2 to port 1,
$$\ensuremath{\vert\underline{S_{22}}\vert^2} = {\frac{\ensuremath... ...erline{b_2}\vert^2}}{\ensuremath{\vert\underline{a_2}\vert^2}}}_{\vert a_1 = 0}$$	$\ldots\$ power reflected from port 2.

Figure 3.3: Definition of S-parameters.

The parameters $\underline{S_{11}}$ and $\underline{S_{21}}$ are obtained by terminating the port 2 by a perfect $Z_0$ load ( $\ensuremath{\vert\underline{a_2}\vert^2} = 0$) and measuring the incident, reflected, and transmitted signals. Parameter $\underline{S_{11}}$ is equivalent to the complex-valued input reflection coefficient (impedance) of the device and $\underline{S_{21}}$ is the complex-valued forward transmission coefficient. In turn, while terminating port 1 by a perfect load ( $\ensuremath{\vert\underline{a_1}\vert^2} = 0$), the parameters $\underline{S_{22}}$ and $\underline{S_{12}}$ are measured, which are the complex-valued output reflection coefficient (output impedance) or reverse transmission coefficient, respectively. The accuracy of the measurements strongly depends on the quality of the terminations. If the perfect load cannot be established, the S-parameter definition requirements are not met.

The magnitudes of the reflection parameters $\underline{S_{11}}$ and $\underline{S_{22}}$, which are always smaller than 1, can be interpreted as follows: In the case of -1, all voltages are inverted and reflected (0$\,\Omega$), zero means perfect impedance matching and no reflections (50$\,\Omega$), and at +1 all voltages are reflected ( $\infty\,\Omega$).

In the case of an active amplification, the magnitudes of the transfer parameter $\underline{S_{21}}$ and reverse parameter $\underline{S_{12}}$ can be larger than $1$. They can also start at a negative value in the case of a phase inversion. If the magnitude is zero, there is no signal transmission, between 0 and $+1$ a damping takes place, at $+1$ there is a unity gain transmission and above $+1$ an input signal amplification.

$\underline{S_{ii}}$ on the real-valued axis characterize Ohmic resistors. $\underline{S_{ii}}$ above the real-valued axis characterize inductive impedances. $\underline{S_{ii}}$ below the real-valued axis characterize capacitive impedances. $\underline{S_{ii}}$ curves in the Smith chart are followed clock-wise to increasing frequencies.

$\underline{S_{ij}}$ curves in the polar chart are followed clock-wise to increasing frequencies.

3.1.3 Extraction of the Cut-Off and Maximum Oscillation Frequency

The cut-off frequency $f_\textrm {T}$ and maximum oscillation frequency $f_\mathrm{max}$ are the most important figures of merit for the frequency characteristics of microwave transistors. They are often used to emphasize the superiority of newly developed semiconductors or technologies. For example, as a rule of thumb, the operating frequency of a transistor, sometimes referred as $f_\mathrm{op}$ should be ten times smaller than $f_\textrm {T}$ [189]. Thus, extraction of these parameters is a commonly performed simulation task usually done by small-signal simulations.

The cut-off frequency $f_\textrm {T}$ is the frequency at which the gain or amplification is unity, thus the absolute value of the short circuit current gain $\underline{H_{21}}$ equals unity:

$$\ensuremath{\left\vert\underline{H_{21}}\right\vert}_{f={\ensuremath{f_\textrm{T}}}}=1 \ .$$ (3.8)

$\underline{H_{21}}$ is defined as the ratio of the small-signal output current to the input current of a transistor with short-circuited output. For a bipolar junction transistor, $\underline{H_{21}}$ basically characterizes the ratio between the small-signal collector current $i_\mathrm{C}$ and the small-signal base current $i_\mathrm{B}$. For a MOS transistor, a similar ratio regarding the small-signal drain and gate currents can be specified:

$$\beta_ = \frac{\left\vert \ensuremath{i_\mathrm{C}}\right\vert}{\left\vert \ensuremath{i_\mathrm{B}}\right\vert} \ ,$$ (3.9)

$$\beta_ = \frac{\left\vert \ensuremath{i_\mathrm{D}}\right\vert}{\left\vert \ensuremath{i_\mathrm{G}}\right\vert} \ .$$ (3.10)

The cut-off frequency is normally extracted for various operating points. Thus, the peak $f_\textrm {T}$ value is the highest frequency for this range of operating points. See the left side of Figure 3.4 for a typical curve. To extract such a complete curve $I_\mathrm{C}$ or $I_\mathrm{D}$ (or $V_\mathrm{B}$ or $V_\mathrm{G}$ respectively) is stepped. Hence, two stepping variables (see Appendix B) are necessary to obtain $f_\textrm {T}$: one for the steady-state operating point and one for the frequency. Whereas the operating point is a matter of ordinary stepping functions, there are several approaches for frequency stepping:

• The frequency can be simply increased until ${\beta}$ is smaller than one. Then, an interpolation algorithm is used to obtain the frequency for which $\beta = 1$.
• This so-called unity current gain frequency can be calculated by extrapolation of $\left\vert\underline{H_{21}}\right\vert$, because for a conventional transistor $\left\vert\underline{H_{21}}\right\vert$ drops with a slope of $-20\,$dB/dec or $-6\,$dB/oct at higher frequencies. This roll-off slope of a one-pole low-pass is depicted on the right side of Figure 3.4. Thus, the frequency is increased until the $-20\,$dB/dec region of the curve is reached. An extrapolation yields the cut-off frequency. Measurement equipments often use this approach to obtain $f_\textrm {T}$, since they are normally not able to measure such high frequencies as required for today's cut-off frequencies. However, this method depends on the frequency chosen, that means which frequency assures the validity of the single-pole approximation [172].
• For simulation purposes it is very inconvenient to run a simulator and a post-processing script in the end. For that reason a conditional stepping approach was developed to use a mathematical iteration algorithm to approximate $f_\textrm {T}$ for a given accuracy (see Section 3.3.3).

Figure 3.4: Complete $f_\textrm {T}$ curve of a bipolar junction transistor (left). Slope of the absolute value of the short circuit current gain and the cut-off frequency at the unity gain point at $\ensuremath{I_\mathrm{C}}= 0.865\,$mA (right).

The second important RF figure of merit is the maximum oscillation frequency $f_\mathrm{max}$, which is related to the frequency at which the device power gain equals unity. The value of $f_\mathrm{max}$ can be determined in two ways. The first one is based on the unilateral power gain $U$ as defined by Mason

$$U(f) = \displaystyle \frac{\left\vert\displaystyle\frac{\ensurema... ...ac{\ensuremath{\underline{S_{21}}}}{\ensuremath{\underline{S_{12}}}}\right)}\ ,$$ (3.11)

where $k$ is Kurokawa's stability factor [94] defined as

$$k(f)=\displaystyle\frac{1-\left\vert\ensuremath{\underline{S_{11}... ...ine{S_{12}}}\right\vert\left\vert\ensuremath{\underline{S_{21}}}\right\vert}\ .$$ (3.12)

Therefore, $f_\mathrm{max}$ is the maximum frequency at which the transistor still provides a power gain [189]. An ideal oscillator would still be expected to operate at this frequency, hence the name maximum oscillation frequency. Like the short-circuit current gain $\underline{H_{21}}$, $U$ drops with a slope of $-20\,$dB/dec.

The second way to determine $f_\mathrm{max}$, which is not entirely correct [189], is based on the maximum available gain (MAG) and the maximum stable gain (MSG). Whereas MAG shows no definite slope, MSG drops with $-10\,$dB/dec.

$f_\mathrm{max}$ does not have to be necessarily larger than $f_\textrm {T}$. Generally, transistors have useful power gains up to $f_\mathrm{max}$, that above they cannot be used as power amplifiers any more. However, the importance of $f_\textrm {T}$ and $f_\mathrm{max}$ depends on the specific application. Thus, there is no general answer whether $f_\mathrm{max}$ should be priorized over $f_\textrm {T}$. Both figures should be as high as possible, and manufactures often strive for ${\ensuremath{f_\textrm{T}}} \approx {\ensuremath{f_\mathrm{max}}}$ in order to enter many different markets for their transistors [189].