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U
niversity of
S
outhern
C
alifornia
USC Viterbi School of Engineering
Ming Hsieh Department of Electrical Engineering
EE 536a:
Solutions, Homework #01
Spring, 2009
Due: 01/26/2009
Choma
Solution
−
Problem #01:
Problem 1.2, pages 9091, Textbook
(a).
The circuit for calculating the Thévenin output voltage,
V
ot
, which drives capacitance
C
l
, is offered
in Figure (P1.1).
This circuit is simply the original structure with capacitance
C
l
removed.
KVL
applied around the source circuit loop in this schematic diagram gives
+
−
V
s
R
s
r
b
I
(+
1
)
I
β
V
ot
Figure (P1.1)
()
ss
b
π
e
VR
r
r
β
1R I,
⎡
=+
+
+
+
⎣
⎤
⎦
(P11)
while at the output port,
ot
l
V
β
RI.
=−
(P12)
If we solve the first of these two relationships for the input port current,
I
, and then substitute this
solution into (P12), we learn that
ls
ot
sb
π
e
β
RV
V
Rrr
,
β
1R
++++
(P13)
which for large values of the current gain metric,
β
, reduces to
l
ot
s
π
ee
β
R
V
β
R
⎛⎞
≈−
⎜⎟
⎝⎠
V
.
(P14)
We can confirm this result intuitively by observing that for
>>1
, the current,
I
, flowing through
the load resistance,
R
l
, is roughly the same as the current,
(
+1)I
, conducted by resistance
R
e
.
If
parameter
is indeed large, we can additionally argue that most of the applied signal voltage,
V
s
, is
dropped across resistance
R
e
, which conducts a current that exceeds the current conducted by re
sistances
r
b
and
r
π
by a factor of
(
+1)
.
Thus,
V
s
≈
(
+1)R
e
I
R
e
I
, while
V
ot
=
−
R
l
I
.
It follows
that
V
ot
/V
s
−
R
l
/R
e
, as we more formally established in the analysis leading to (P14).
(b).
At low signal frequencies, the impedance,
1/j
ω
C
l
, attributed to capacitance
C
l
has a very large
magnitude and consequently, we can rationalize that it behaves as an approximate open circuit.
But remember that we evaluated the Thévenin output voltage,
V
ot
, by physically removing capaci
tance
C
l
; that is, we effectively replaced
C
l
by an open circuit.
Thus, the low frequency gain,
V
o
/V
s
,
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View Full DocumentEE 536a
University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #01
3
Spring Semester, 2009
of the entire amplifier closely approximates what we might cleverly term to be the
Thévenin volt
age gain
, say
A
ot
, of the circuit; that is,
V
ot
/V
s
.
In short,
oo
t
ot
ss
low frequencies
VV
A.
≈
±
(P15)
We should record for posterity that at zero signal frequency, where capacitance
C
l
is literally an
open circuit, the approximation in the last result becomes an identity; that is
t
ot
zero frequency
≡=
(P16)
(c).
We can determine the indicated output resistance by inspection, without resorting to the ohmmeter
method of evaluating a port resistance.
In particular,
R
out
is computed with the signal voltage,
V
s
,
set to zero.
But from (P11),
V
s
= 0
forces current
I
to zero, whence
β
I = 0
.
With
I = 0
, the entire
circuit to the left of resistance
R
l
in Figure (P1.1) is effectively disconnected, whence via casual in
spection,
out
l
R
R.
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 Electrical Engineering, Integrated Circuit

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