Activity: MOS Design Parameters

• Instructions: This activity is structured as a tutorial with an activity at the end. Should you have any questions, clarifications, or issues, please contact your instructor as soon as possible.
• At the end of this activity, the student should be able to:

1. Obtain the small signal design parameters of NMOS and PMOS transistors.

Activity 1: Small Signal Models

Figure 1: The MOS low-frequency small signal model.

Most of the time, we are interested in the small signal behavior of our amplifiers, i.e. what happens to the amplifier voltages and currents as we introduce small disturbances (information carrying signals) at the input. To gain intuition and to facilitate circuit analysis using nonlinear devices, we linearize our circuits to obtain the two-port model shown in Fig. 1. For MOSFETs at low frequencies, the two-port model is composed of the transconductance, and the output resistance.

Transconductance

Transconductance is defined as:

$g_{m}={\frac {\partial I_{D}}{\partial V_{GS}}}$

This is the slope of the transfer characteristic curves. Thus, by numerically differentiating the $I_{D}$ vs. $V_{GS}$ curve shown in Fig. 2 for an NMOS transistor with a width $W=10\mathrm {\mu m}$, a length $L=0.15\mathrm {\mu m}$, and with $nf=1$, we can obtain the transconductance plot shown in Fig. 3.

Figure 2: The NMOS transfer characteristics.

Figure 3: NMOS transconductance.

Output Resistance

Similarly, we can take the derivative of the output characteristics, shown in Fig. 4 for $V_{GS}=0.90\mathrm {V}$, to get the output resistance, which is defined as:

$r_{o}=\left({\frac {\partial I_{D}}{\partial V_{DS}}}\right)^{-1}$

The output resistance as a function of $V_{DS}$ is shown in Fig. 5 for $V_{GS}=0.90\mathrm {V}$.

Figure 4: NMOS output characteristic for $V_{GS}=0.90\mathrm {V}$.

Figure 5: NMOS output resistance for for $V_{GS}=0.90\mathrm {V}$.

Answer the following questions:

1. Given the transconductance plot in Fig. 3, why is there a peak in the transconductance curve? What is causing this?
2. Generate Fig. 3, but on a logarithmic y-axis. Is this what you expect for $V_{GS}<V_{TH}$? Explain.
3. Plot $g_{m}$ for $V_{DS}=0.45\mathrm {V}$ and $V_{DS}=1.35\mathrm {V}$. Does the transconductance change with $V_{DS}$? Is this the behavior you expect? Explain.
4. Plot the output resistance for $V_{GS}=0.45\mathrm {V}$ and $V_{GS}=0.1.35\mathrm {V}$. Do you see any changes in the the effects of process variations? What do you think is causing this behavior?

Activity 2: Power Efficiency

Figure 6: NMOS ${\frac {g_{m}}{I_{D}}}$.

Figure 7: NMOS $V_{ov}={\frac {2\cdot I_{D}}{g_{m}}}$.

Activity 3: Transition Frequency

Figure 8: The MOS transistor small signal model.

At high frequencies, the MOS gate-to-source capacitance, $C_{GS}$ and the gate-to-drain capacitance, $C_{GD}$ limits the MOSFET current gain at high frequencies, and thus, needs to be included in the two-port model, as shown in Fig. 8. The short-circuit current gain can be expressed as:

$A_{i}={\frac {i_{\mathrm {out} }}{i_{\mathrm {in} }}}={\frac {g_{m}}{j\omega \left(C_{GS}+C_{GD}\right)}}$

We then define the transition frequency, $f_{T}$ as the frequency when magnitude of the short-circuit current gain is equal to 1, or:

$\left|A_{i}\right|=\left|{\frac {i_{\mathrm {out} }}{i_{\mathrm {in} }}}\right|={\frac {g_{m}}{\omega _{T}\left(C_{GS}+C_{GD}\right)}}={\frac {g_{m}}{2\pi f_{T}\left(C_{GS}+C_{GD}\right)}}=1$

Thus, we get:

$f_{T}={\frac {g_{m}}{2\pi \left(C_{GS}+C_{GD}\right)}}$

Activity 4: PMOS Small Signal Parameters

Repeat Activities 1, 2, and 3 for a 1.8V LVT PMOS transistor with the same dimensions as the NMOS transistor: a width $W=10\mathrm {\mu m}$, a length $L=0.15\mathrm {\mu m}$, and with $nf=1$.