4/18/2011
The Short Circuit Current Gain lecture
1/8
The Short-Circuit
Current Gain hfe
Consider the common emitter “low-frequency” small-signal model with its output
short-circuited.
B
+
-
ib (ω )
vi (ω )
rπ
ic (ω )
+
+
vbe
gmvbe
-
vce
-
C
ro
E
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
2/8
Boring! Tell me something
I don’t already know
In this case we find:
ic (ω ) = gm vbe (ω )
= gm rπ ib (ω )
But we know that:
gm rπ =
Therefore:
IC VT IC
=
=β
VT IB IB
ic (ω )
=β
ib (ω )
Just as we expected!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
3/8
When the input signal is changing rapidly
Now, contrast this with the results using the high frequency model:
- vce +
ib (ω )
B
+
-
vi (ω )
ic (ω )
Cμ
rπ
C
gmvπ
+
vbe
+
vce
Cπ
-
-
ro
E
Evaluating this circuit, it is evident that the small-signal base current is:
⎛1
⎞
+ jω Cπ + C μ ⎟
⎝ rπ
⎠
ib (ω ) = vπ (ω ) ⎜
(
)
While the small-signal collector current is:
ic (ω ) = vπ (ω ) ( gm − jωC μ )
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
4/8
Here’s something you did not know
Therefore, the ratio of small-signal collector current to small-signal base
current is:
r g − jω rπ C μ
ic (ω )
= π m
ib (ω ) 1 + jω rπ C μ + Cπ
(
)
Typically, we find that gm ωC μ , so that we find:
rπ gm
ic (ω )
≈
ib (ω ) 1 + jω rπ (C μ + Cπ )
and again we know:
rπ gm =
IC VT IC
=
=β
VT IB IB
Therefore:
ic (ω )
β
≈
ib (ω ) 1 + jω rπ (C μ + Cπ )
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
5/8
Your BJT is frequency dependent!
We define this ratio as the small-signal BJT (short-circuit) current gain,
hfe (ω ) :
hfe (ω ) ic (ω )
β
≈
ib (ω ) 1 + jω rπ (C μ + Cπ )
Note this function is a low-pass function, were we can define a 3dB break
frequency as:
ωβ =
Therefore:
hfe (ω ) Jim Stiles
1
rπ (Cπ + C μ )
ic (ω )
β
≈
ib (ω ) 1 + j ω ωβ
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
6/8
This is how we define BJT bandwidth
2
Plotting the magnitude of hfe (ω ) (i.e., hfe (ω ) ) we find that:
2
h (ω ) (dB)
fe
logβ 2
20 dB/decade
0 dB
logω
ωβ
ωT
We see that for frequencies less than the 3 dB break frequency, the value of
hfe (ω ) is approximately equal to beta:
hfe (ω ) ≈ β
Jim Stiles
ω < ωβ
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
7/8
This should SO remind you of op-amps
Note then for frequencies greater than this break frequency:
hfe (ω ) =
β
1+ j ω
≈j
β ωβ
ωβ
ω
ω > ωβ
Note then that hfe (ω ) = 1 when ω = β ωβ .
We can thus define this frequency as ωT , the unity-gain frequency:
ωT β ωβ
so that:
Jim Stiles
hfe (ω = ωT ) = 1.0
The Univ. of Kansas
Dept. of EECS
4/18/2011
The Short Circuit Current Gain lecture
8/8
Déjà vu; all over again
We can therefore state that:
hfe (ω ) ≈
β ωβ
ω
=
ωT
ω
ω > ωβ
and also that:
hfe (ω ) ≈ β
Jim Stiles
ω < ωβ
The Univ. of Kansas
Dept. of EECS