4/18/2011
section 5_8 BJT Internal Capacitances
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5.8 BJT Internal Capacitances
Reading Assignment: 485-490
BJT’s exhibit capacitance between each of its terminals (i.e., base,
emitter, collector). These capacitances ultimately limit amplifier
bandwidth.
Q: Yikes! Who put these capacitors in the BJT? Why did they put
them there? Why don’t we just remove them?
A: These capacitances are parasitic capacitances. Since the
terminals are made of conducting materials (e.g., metal), there will
always be some capacitance associated with any two terminals.
BJT designers and manufacturers work hard to minimize these
capacitances—and indeed they are very small—but we cannot
eliminate them entirely.
If the signal frequency gets high enough, these capacitances can
affect amplifier performance.
HO: BJT INTERNAL CAPACITANCES
Now that we are aware of these internal capacitances, we must
modify our small-signal circuit models.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
section 5_8 BJT Internal Capacitances
2/2
HO: THE HIGH-FREQUENCY HYBRID PI MODEL
The significance of the internal capacitances are typically
specified in terms of a frequency-dependent BJT parameter called
hfe (ω ) . This parameter is often referred to as the “short-circuit
current gain” of the BJT.
From this function hfe (ω ) we can extract a BJT parameter called
the unity-gain bandwidth—a result very analogous to op-amps!
HO: THE SHORT-CIRCUIT CURRENT GAIN
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
BJT Internal Capacitances lecture
1/3
BJT Internal Capacitances
There are very small capacitances in a BJT between the
collector and the base, and the base and the emitter.
Since the capacitor values are very small, their
impedance at low and moderate frequencies is large.
I.E.:
1
ZC =
is large if ω C 1
Cμ
Cπ
j ωC
In other words, at low and moderate frequencies, these capacitor impedances
are approximately open circuits, and thus they can be ignored.
However, at high frequencies, the capacitor impedance can drop to moderate
values (e.g., K Ω s).
In this case, we can no longer ignore these capacitances, but instead must
incorporate them into our small-signal model!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
BJT Internal Capacitances lecture
2/3
The capacitance between base and collector
Q: What are C μ and Cπ ?
A: See below!
Cμ
C μ is a parasitic capacitance between the collector and the
base.
Cμ
This capacitance is due to the pn junction (between
collector and base).
Typical values of C μ are a few picofarads or less.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
BJT Internal Capacitances lecture
3/3
The capacitance between base and emitter
Cπ
Cπ is a parasitic (i.e., small) capacitance between the base and the emitter.
This capacitance actually consists of two parts:
Cπ = C je + Cde
where:
Cπ
Cde = diffusion capacitance ⎫
C je
⎪
⎬ pn junction capacitance
= junction capacitance ⎪⎭
Typically, Cπ is a few picofarads.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/18/2011
The High Frequency Hybrid Pi Model lecture
1/1
The High-Frequency
Hybrid-π Model
B
Combine the internal
capacitances and lead
resistance in a modified
Hybrid-π model.
ic (ω )
ib (ω )
Cμ
rπ
+
vπ
-
Cπ
gmvπ
C
+
vce
-
ro
E
* Therefore use this model to construct small-signal circuit when vi is operating
at high frequency.
* Note since ZC = 1 jωC , all currents and voltages will be dependent on
operating frequency ω.
* Note the voltage across rπ is vπ , but vπ ≠ vbe !!!
* Note at low-frequencies, the model reverts to the original Hybrid-π model.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
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