MOSFET High Frequency Model and Amplifier Frequency Response

MOSFET High Frequency Model and Amplifier Frequency Response

Objectives

To review the small signal MOSFET and BJT models at low frequencies

To study the high frequency MOSFET and BJT models

To estimate the MOSFET and the BJT unity-gain frequency

Introduction

In the last two lectures we examined the time and frequency response of the STC circuits.

As we said the importance of studying the STC circuits is that the analysis of a complex amplifier circuit can be usually reduced to the analysis of one or more simple STC circuits. The frequency response of the amplifier circuits will be explored starting next lecture. Before starting this study it will be constructive to reviewin this lecture the MOSFET and BJT small-signal models, and examining the high-frequency models. Also, the transistor cut-off frequency which is considered a figure of merit at high frequency operation will be estimated for both transistors.

The MOSFET small signal model

Fig-1 in (Lec_03_Ver_01.vsd)

Figure 1 MOSFET small signal model

The small signal model of the MOSFET is shown in figure 1.
Since the gate is insulated from the channel by the gate-oxide the input resistance of the transistor is infinite.
The small-signal parameters are controlled by the Q-point (operating point).

The NMOS small-signal parameters may be summarized in Table 1
For PMOS the small-signal model will be the same as NMOS after replacing the electron mobility with the hole mobility and the NMOS threshold by the PMOS threshold

Table 1 MOSFET small-signal parameters

Symbol / Parameter / Value
gm / Transconductance /
VTN is the NMOS threshold voltage
Kn = μn CoxW/L
μn is the electron mobility
Cox is the gate oxide capacitance/area= εox/tox
εox is the oxide permittivity
tox is the oxide thickness
W is the transistor width
L is the transistor length
ro / Output resistance /
λ is the channel length modulation parameter
gmb / Back-gate transconductance /
0<η<1 is called the back-gate transconductance parameter.

The MOSFET high frequency model

The MOSFET small-signal model discussed earlier is valid at low frequencies. This model will fail at high frequencies.
At high frequencies the transistor internal capacitances should be considered.
The MOSFET's internal capacitances limit the high-frequency performance of the MOSFET that means:
Limit the switching speed of the circuits in digital applications
Limit the frequency at which useful amplification can be obtained in the amplifiers.

MOSFET capacitances depend on operation region and are non-linear functions of voltages at device terminals.
The MOSFET’s internal capacitances are associated to either reverse biased junctions or changes of charges due to voltage difference.
A physical capacitance may also exist formed by a dielectric between two conductors. Refer to Figure 2.

Fig-2 in (Lec_03_Ver_01.vsd)

Figure 2 NMOS Transistor Capacitances in the Saturation Region

The complete high frequency MOSFET modelin the saturation region is shown in Figure 3-a.

Fig-3a in (Lec_03_Ver_01.vsd)

Figure 3-a MOSFET high-frequency model

The different high frequency internal capacitances of MOSFETare summarized in Table 2.

Table 2 MOSFET internal capacitances

Symbol

Name

Origin

Value

Cgs

Gate-Source capacitance

Physical (gate oxide)

Cgd

Gate-Drain capacitance

Physical (drain diffusion overlap under gate oxide)

/ Lov is the overlap length between the drain and the gate

Csb

Source-Substrate Capacitance

Depletion capacitance between the source and the substrate

/ Csb0 is the source-substrate capacitance when VSB = 0
Vo is the junction built-in potential (0.6V to 0.8V)

Cdb

Drain-Substrate Capacitance

Depletion capacitance between the drain and the substrate

/ Csb0 is the drain-substrate capacitance when VDB = 0

Symbol

Name

Origin

Value

Csb

Source-Substrate Capacitance

Depletion capacitance between the source and the substrate

/ Csb0 is the source-substrate capacitance when VSB = 0
Vo is the junction built-in potential (0.6V to 0.8V)

Cdb

Drain-Substrate Capacitance

Depletion capacitance between the drain and the substrate

/ Csb0 is the drain-substrate capacitance when VDB = 0

The model in Figure 3-amay be simplified when we have the source and the substrate connected (no body effect) to the model shown in Figure 3-b

Fig-3b in (Lec_03_Ver_01.vsd)

Figure 3-b Simplified MOSFET model when the source and the substrate are connected

Further simplification may be achieved by neglecting Cdb as shown in Figure 3-c.

Fig-3c in (Lec_03_Ver_01.vsd)

Figure 3-c Simplified MOSFET high-frequency model

The MOSFET Unity-Gain Frequency (fT)

As an application to the MOSFET high frequency model, let us calculate the MOSFET unity-gain frequency fT.
fT is considered a figure of merit for the high-frequency operation of the MOSFET as an amplifier.
fT is defined as the frequency at which the short-circuit current-gain of the common-source amplifier becomes unity.
Figure 4 shows the circuit used to calculate fT, in which we used the simplified model in Figure 3c.

Please note that since we are now dealing with quantities (currents and voltages) which are frequency dependent. We will use the s-domain analysis discussed in Lecture 2. Also, we will use capital letters with lowercase subscripts to identify these quantities.

Fig-4 in (Lec_03_Ver_01.vsd)

Figure 4 Determining the short-circuit current gain

Applying the nodal analysis at the output node we can write

Recalling that Cgd (overlap drain capacitance) is small, at the frequencies of interest we can neglect the second term

Applying the nodal analysis at the input node we may write

Combining the input and output current equations we can write

For physical frequencies s=jω, it can be seen that the magnitude of the current gain becomes unity at the frequency

As we can see from the last equation. Higher fT means higher gm and lower internal MOSFET capacitances which means better amplifier operation.
Typically, fT is ranging from about 100MHz for older technologies (e.g., a 5-μm CMOS technology) to many GHz for current high-speed technologies (e.g., a 0.13-μm CMOS technology).

The BJT small-signal model

The small signal model of the BJT amplifier is shown in figure 5. Figures 5-a,b are for the π-model, where Figures 5-c,d are for the T-model.
These models are valid for both NPN and PNP transistors.
For the same operating point, the BJT has higher transconductance and higher output resistance that the MOSFET.

Fig-5 in (Lec_03_Ver_01.vsd)

Figure 5 small signal-models of the BJT

The small-signal parameters are controlled by the Q-point (operating point).
The BJT small-signal parameters may be summarized in Table 3

Table 3 BJT small signal parameters

Symbol / Parameter / Value
gm / Transconductance /
VT is the thermal voltage = kT/q, which equals 25mV at room temperature.
k is Boltzman's constant
T is the absolute temperature in Kelvins
q is the electron charge
rπ / Base input resistance /
β is the common-emitter current gain
re / Emitter input resistance /
α is the common-base current gain
ro / Output resistance /
VA is the early voltage.

The BJT high-frequency model

Fig-6 in (Lec_03_Ver_01.vsd)

Figure 6 The high-frequency hybrid-π model of the BJT

•The high frequency hybrid-πmodel for the BJT is shown in Figure 6.

•This model is useful for signal frequencies up to a several tens of megahertz, after which a more detailed model becomes necessary.

•Typically, the base-emitter junction capacitance Cπis in the range of few pF to few tens of pF, while the collector-base junction capacitance Cμis in the range of fraction of pF to few pF

•The base resistor rx is added partly to account for the comparatively long internal connection from the base external connection and the actual internal base connection.

•A representative resistance value for this lumped resistor is in the range of 50Ω to perhaps 200Ω.

•This resistor ordinarily can be neglected for hand estimates.

•Note that rx becomes the dominant input resistance for frequencies so high that Cπ effectively short-circuits rπ.

•A second base-width modulation effect, characterized by a resistor connected between the base and collector is omitted; its influence is dominated by the collector junction reverse-bias capacitance Cμ.

•The emitter junction (diffusion) capacitance Cπ represents the charge store to support the current flow across the base.

The BJT Cutoff frequency

As defined earlier, it is the frequency at which the current gain ofthe transistor becomes one. (i.e. no more active element). It is calculated by finding the short circuit collector current in terms of the base current.
Using the high frequency model of BJT we can draw the circuit to estimate the cut-off frequency of the BJT as shown in Figure 7.

Fig-7 in (Lec_03_Ver_01.vsd)

Figure 7 Circuit used to estimate the BJT cutoff frequency

Applying nodal analysis at the input and output nodes as we did earlier. We can estimate the cut-off frequency as follows:

We can observe from the last analysis that the common-emitter current gain (hfe) frequency response is similar to a simple pole withωp as the pole frequency. This may be drawn as shown in Figure 8

Fig-8 in (Lec_03_Ver_01.vsd)

Figure 8 Bode plot of |hfe|

As we can see from the last equation. Higher ωT means higher gm and lower internal BJT capacitances which means better amplifier operation.
Typically, fT is ranging from about 100MHz to Tens of GHz.