Let’s talk about the role of MOSFET’s small signal characteristics in analog IC design
MOSFETs are critical to modern analog IC design. However, this article focuses on the large-signal behavior of MOSFETs. Analog ICs typically use MOSFETs for small signal amplification and filtering. In order to fully understand and analyze MOS circuits, we need to define the small signal behavior of MOSFETs.
What is small signal analysis?
What exactly do we mean when we say "small signal"? To define this, let us refer to Figure 1, which shows the output transfer characteristics of the inverter.
(Figure 1. Transmission characteristics of inverter)
VIN and VOUT are both DC voltages.
The value of VIN indicates that we are operating at the bias point (marked in red).
In small-signal analysis, we apply a very small AC signal (ΔVIN) on top of the DC bias voltage. The resulting output AC voltage is amplified based on the slope of the transfer characteristic at the bias point (–AV):
Note that AV is only negative due to the direction of the slope. We will return to AV later in the article. For now, the important conclusion is that the bias point (large signal behavior) affects the amount of gain the output signal receives (small signal behavior).
small signal parameters
Before we model the behavior of the circuit, we need to define our parameters. The main small signal parameters of MOSFET are:
Output resistance (ro)
Intrinsic gain (AV)
Body effect transconductance (gmb)
Unity gain frequency (fT)
Except for fT, which we will not discuss before creating the high frequency MOSFET model, we will define and derive each of the above terms in the following chapters. We'll start with the IV characteristic, transconductance.
As we already know, MOSFET converts input voltage into output current. The ratio of small-signal output current to small-signal input voltage is called transconductance (gm). We can also think of transconductance as the derivative of the output current with respect to the gate-source voltage.
The transconductance in the linear region can be defined as:
For the saturated region, it is:
ID is the drain current
VGS is the gate to source voltage
VDS is the drain to source voltage
Vth is the threshold voltage
μ is the transistor mobility
Cox is the gate oxide capacitance
W is the width of the transistor
L is the length of the transistor.
These two equations lead to several interesting points:
When in the linear region, the transistor's current gain depends on the output voltage. It is completely independent of the input signal. This is not ideal in practice because the gain changes dramatically over the operating range.
At saturation, the transconductance depends only on the input voltage.
For a given input bias voltage, short and wide devices maximize current gain.
The next parameter of interest is the output resistance (ro). This is defined as the change in the drain-to-source voltage of a transistor relative to the drain current. We can find the output resistance by plotting the drain current versus VDS. The slope of the resulting line is equal to the reciprocal of ro.
Let's take a look at the graph in Figure 2. We first saw this number in our previous article on MOSFET structure and operation, where it helped us compare the drain current of NMOS and PMOS transistors.
(Figure 2: Relationship between drain current and VDS of NMOS and PMOS transistors W/L=10μm/2μm)
MOSFET has a small output resistance in the linear region and a large output resistance in the saturation region. In the picture above, both NMOS and PMOS transistors enter saturation at ~1.5V.
Because - as we saw with transconductance - the saturation region provides better small signal performance, we are only concerned with the output resistance of the transistor when it is saturated. We can calculate this as:
where λ is the channel length modulation.
The relationship between ro and λ makes sense when considering that the slope of the IV curve at saturation is caused by channel length modulation. Equation 4 also tells us:
• ro decreases with drain current (ID).
•Due to the above reasons, ro decreases with the overdrive voltage (VD, sat).
•ro increases with transistor length (L).
Now that we know the transistor's output resistance and current gain, we can calculate its maximum voltage gain. This is also known as the intrinsic gain (AV) of the transistor. To better understand the concept of intrinsic gain, let us examine the common source amplifier configuration in Figure 3.
(Figure 3 NMOS transistor configured as common source amplifier)
Since an ideal current source has infinite resistance, the small-signal output transfer function of this circuit can be calculated as:
From Equations 3 and 4, it can be seen that gm and ro are inversely proportional to the drain current. Using this knowledge, we can find the optimal value of drain current that produces the greatest possible gain for a single transistor—in other words, its intrinsic gain. For modern processes, the intrinsic gain is typically between 5 and 10.
body effect transconductance
The last small-signal parameter we need to derive is body effect transconductance (gmb), which describes how body effects affect drain current. We can calculate this as:
where eta is the backgate transconductance parameter, whose value is usually between 0 and 3.
Low and high frequency models
Now that we have defined the parameters, we can build a circuit model to represent the small-signal operation of the transistor. Figure 4 depicts the small-signal behavior of MOSFETs at low frequencies.
(Figure 4 MOSFET small signal model)
At higher frequencies, we need to include the parasitic capacitance of the MOSFET (Figure 5).
(Figure 5 MOSFET structure with parasitic capacitance)
The above representatives are:
•Cgs, gate to source capacitance.
•Cgd, gate to drain capacitance.
•Cgb, gate to body capacitance.
•Csb, source-to-body capacitance.
•Cdb, drain to body capacitance.
•The small-signal transistor model in Figure 6 includes all of these non-idealities except body capacitance.
(Figure 6 MOSFET small signal model with capacitor)
As can be seen from Figure 6, the intrinsic gain of the MOSFET in Figure 3 has a unipolar low-pass transfer function. We can now calculate the bandwidth of the transistor, which in this case will be the frequency at which the voltage gain equals 1 (0dB). This is called unity gain frequency (fT).
To find fT, we short the output to ground and calculate the transconductance in Figure 6. Doing so results in the following equation:
From Equations 4 and 7, we can see that in order to increase the gain, we need to increase the length of the transistor. However, we also see that this results in lower bandwidth. The opposite is also true: reducing the length of the transistor results in higher bandwidth.
Review Editor: Liu Qing
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