6.012  Microelectronic Devices and Circuits – Fall 2009
Inverter Analysis and Design
The inverter stage is a basic building block for digital logic circuits and memory
cells. A generic inverter stage is illustrated below on the left. It consists of two devices,
a pullup device, which is typically either a bipolar junction transistor or an
enhancement mode field effect transistor, and a pulldown device, which might be
another transistor, or a resistor, current source, diode, etc. The stage load which is
shown in the figure represents the input resistance of the following stage, which is
typically a stage (or n stages) just like the original stage.
i
SL
(
v
OUT
)
[ = n
i
IN
(v
OUT
)]
i
IN
(
v
IN
)
+

v
IN
+

v
OUT
Pull
Up
Pull
Down
Stage
Load
+ V
DD
i
PU
(
v
IN,
v
OUT)
i
PD
(
v
IN,
v
OUT
)
+

v
IN
+

v
OUT
Pull
Up
Pull
Down
Stage
Load
+ V
DD
(a) A generic inverter stage
(b) The static currents to calculate v
OUT
(v
IN)
An important piece of information about an inverter stage is its static transfer
characteristic, v
OUT
(v
IN
). To calculate this characteristic we sum the currents into the
output node of the inverter, as is illustrated above on the right.
With all of these
currents written as functions of v
IN
and v
OUT
, this sum yields the desired relationship:
i
PU
(v
IN
, v
OUT
) = i
PD
(v
IN
, v
OUT
) + i
SL
(v
OUT
)
As an example, consider the MOSFET inverter circuit shown at the top of the
next page with an nchannel MOSFET pulldown and a resistor pullup. The MOSFET
is characterized by its Kvalue and by its threshold voltage, V
T
(we will assume for
simplicity that
α
is 1). To analyze this circuit we not first that with a MOSFET pull
down, the static input current is zero and if the stage output is connected to the input of
a similar stage, the static stage load current will also be zero, and the equation above is
simply i
PU
= i
PD
. With a resistor pullup, the pullup current, i
PU
, is (V
DD
 v
OUT
)/R
and the pulldown current, i
PD
, is the MOSFET drain current. This current depends on
the gatetosource voltage, v
GS
, which is the same as v
IN
, and the draintosource
voltage, v
DS
, which is the same as v
OUT
. With v
IN
less than V
T
, the pulldown current
is zero and v
OUT
is V
DD
. As v
IN
increases past V
T
, v
OUT
will initially be larger than
1
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(v
IN
 V
T
), and the device will be in saturation so that i
PD
will be K(v
IN
 V
T
)
2
/2. v
OUT
is found by setting i
PD
equal to i
PU
:
i
PU
= i
PD
+

v
IN
+

v
OUT
+ V
DD
R
V
T
, K
i
IN
= 0
i
SL
= 0
i
PU
= (
V

v
OUT
)/R
i
PD
= 0
if
(
v
IN

V
T
) < 0 <
v
OUT
= K(
v
IN

V
T
)
^2
/2
if
0 < (
v
IN

V
T
) <
v
OUT
= K(
v
IN

V
T

v
OUT
/2)
v
OUT
if
0
<
v
OUT
< (
v
IN

V
T
)
A MOSFET inverter with an nchannel MOSFET pulldown and resistor pullup.
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 Fall '09
 CharlesG.Sodini
 Electric charge, Bipolar junction transistor, VLO, Vin, IPU

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