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and it is now possible to determine the quiescent drain-to-source voltage from the load-line equation for the drain circuit: VDD = VDSQ + RD IDQ (1031)
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In a MOSFET, it is possible to approximate the small-signal behavior of the device as a linear relationship by using the transconductance parameter gm , where gm is the slope of the iD -vGS curve at the Q point, as shown in Figure 1025 Formally, we can write gm = iD vGS (1032)
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(b) Common-source amplifier
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Figure 1024 Typical common-drain and common-source MOSFET ampli ers
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Since an analytical expression for the drain current is known (equation 1026), we can actually determine gm as given by gm = [k(vGS VT )2 ] vGS (1033)
at the operating point (IDQ , VCSQ ) Thus, evaluating the expression for the transconductance parameter, we obtain the expression
gm = 2k (vGS VT )|IDQ ,VGSQ
IDQ gm
= 2 kIDQ
IDSS IDQ =2 VT
(1034)
VGSQ
Figure 1025 MOSFET transconductance parameter
where gm is a function of the quiescent drain current, as expected, since the tangent to the iD -vGS curve of Figure 1025 has a slope that is dependent on the value of IDQ The next two examples illustrate the calculation and signi cance of the transconductor parameter in MOSFETs
Part II
Electronics
EXAMPLE 106 MOSFET Transconductance Calculation
Problem
Compute the value of the transconductance parameter for a MOSFET
Solution
Known Quantities: MOSFET quiescent voltage and current values; MOSFET
parameters
Find: gm
VT = 10 V; k = 0125 mA/V2
Schematics, Diagrams, Circuits, and Given Data: VDSQ = 35 V; VGSQ = 24 V; Assumptions: Use the MOSFET model of equation 1029 Analysis: First, we compute the quiescent drain current:
IDQ = k VGSQ VT
= 0125 (24 1)2 = 0245 mA
Next, we evaluate the transconductance parameter gm = 2 kIDQ = 2 0125 0245 10 3 = 035 mA V
Comments: The transconductance parameter tells us that for every volt increase in
gate-source voltage, the drain current will increase by 035 mA The MOSFET clearly acts as a voltage-controlled current source
EXAMPLE 107 Analysis of MOSFET Ampli er
Problem
Determine the gate and drain-source voltage and the drain current for the MOSFET ampli er of Figure 1026
R1 RD vD D vG G vS S VDD R2 RS
Solution
Known Quantities: Drain, source, and gate resistors; drain supply voltage; MOSFET
parameters
Find: vGS ; vDS ; iD
RS = 6 k ; VDD = 10 V VT = 1 V; k = 05 mA/V
Schematics, Diagrams, Circuits, and Given Data: R1 = R2 = 1 M ; RD = 6 k ;
Assumptions: The MOSFET is operating in the saturation region All currents are expressed in mA and all resistors in k Analysis: The gate voltage is computed by applying the voltage divider rule between Figure 1026
resistors R1 and R2 (remember that no current ows into the transistor): vG = R2 1 VDD = VDD = 5 V R1 + R 2 2
10
Transistor Ampli ers and Switches
Assuming saturation region operation, we write: vGS = vG vS = vG RS iD = 5 6iD The drain current can be computed from equation 1029: iD = k (vGS VT )2 = 05 (5 6iD 1)2 leading to
2 36iD 50iD + 16 = 0
with solutions: iD = 089 mA and iD = 05 mA To determine which of the two solutions should be chosen, we compute the gate-source voltage for each For iD = 089 mA, vGS = 5 6iD = 034 V For iD = 05 mA, vGS = 5 6iD = 2 V Since vGS must be greater than VT for the MOSFET to be in the saturation region, we select the solution: iD = 05 mA vGS = 2 V
The corresponding drain voltage is therefore found to be: vD = vDD RD iD = 10 6iD = 7 V And therefore vDS = vD vS = 7 3 = 4 V
Comments: Now that we have computed the desired voltages and current, we can verify
that the condition for operation in the saturation region is indeed satis ed: vDS > vGS VT leads to 4 > 2 1; since the inequality is satis ed, the MOSFET is indeed operating in the saturation region
The transconductance parameter allows us to de ne a very simple model for the MOSFET during small-signal operation: we replace the input circuit (gate) by an open circuit, since no current can ow into the insulated gate, and model the drain-to-source circuit by a controlled current source, gm VGS This small-signal model is depicted in Figure 1027
+ VGS S
gm VGS = ID G S MOSFET circuit symbol
MOSFET small-signal model
Figure 1027 MOSFET small-signal model
It is important to appreciate the fact that the transconductance, gm , is dependent on the quiescent value of the drain current, and therefore any MOSFET ampli er design is going to be very strongly dependent on the operating point
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