Available Power Distortion Emitter Degeneration Miller Effect

Available Power Distortion Emitter Degeneration Miller Effect
While the efficiency of an amplifier, as discussed in the previous
lecture, is an important quality, so is the gain of the amplifier.
Transducer gain, which we simply call Gain, G, is defined as

as we’ve seen previously. With transistor amplifiers, we want to characterize the gain of an ac input signal as in the following circuit:

Consequently for this amplifier, the numerator in (1.22) is the ac
output power P = VI_{p}/2. With V_{p} = V_{pp}/2 and I_{p} = I_{pp}/2, then

Now, what about the “input” power for (1.22)? For this
amplifier, we’re only interested in the ac signal. The maximum
ac power possible from the source Vo with a matched load as in

is the available power P+ given by

In other words, how well the amplifier and load are matched to
the source dictates how much power is “available,” i.e., input to
the amplifier. Recall that the displayed voltage on an AWG with a matched load is V_{+p} V_{0}/2 (where p indicates peak). Therefore, V_{+pp}= V-p V_{0} which yields

where pp V+ is the displayed peak-to-peak voltage on the AWG.
In summary, the ac gain of an amplifier in (1.22) contains the
ratio of two power terms. The ac output power to a resistive load
in (9.14) forms the numerator. The denominator can be defined a
number of ways. Here we have chosen a conservative measure:
the available power from the source, given in (9.16). Distortion
You will most likely discover in Prob. 21 (Driver Amplifier)
that when the input voltage amplitude becomes too large, the
output voltage waveform will be distorted. An example is shown
in

Recall that the Driver Amplifier is (almost) a CE amplifier with
a transformer coupled resistive load:

The slight nonlinear behavior of V_{c} in Fig. 9.6a is due to the base-emitter diode. As illustrated in

The distortion in Fig. 9.7b is due to the nonlinear behavior of the
base-emitter junction at large signals (not because of the base
resistance as stated in the text). Other distortions you may encounter are illustrated in

In (a) the distortion is caused by improper input biasing, while in
the (b) the distortion is from an input amplitude that is too large.
(You should understand what is happening with the transistor to
cause these distortions.) Emitter Degeneration
The CE amplifiers we’ve considered have all had the emitter
tied directly to ground. Notice that the Driver Amplifier has the
additional resistance R12+R13 connected to the emitter (and
eventually to ground through Key Jack J3 when transmitting).
Adding an emitter resistance is called emitter degeneration. This
addition has two very important and desirable effects:
1. Simpler and more reliable bias (dc),
2. Simpler and more reliable gain (ac).
Let’s consider each of these points individually:
1. Bias (dc) – assuming an active transistor, then using KVL
from V_{b} through Re to ground gives
V_{b} = I_{b}R_{b} +V_{f} + I_{e}R_{e}

With I_{c} ? I_{e} then, V_{b} ? I_{b} R_{b} +V_{f}
We will choose V_{b} with some I_{c} bias in mind ( I_{c} = ? I_{b} ).
There are two cases to consider here:
(a) Re = 0:
Here we see that the bias current Ic will depend on the
transistor ?. This is not a good design since ? can vary
considerably among transistors.
(b) R_{e} ? 0: V_{b} ? I_{b} R_{b} f c e +V_{f} + I _{c}R_{R}
The first term is usually small wrt the third term. This
leaves us with
V _{b}?V_{f} + I_{c} R_{c}
This is a good design since we can set V_{b} for a desired I_{c} without explicitly considering the transistor ?.
2. Gain, G – To determine ac gain we use a small signal
model of the BJT in the circuit shown above

Note that we’ve chosen R_{b} = 0.
Using KVL in the base and emitter circuit gives
v_{i} = i_{b}r_{b} + i_{e}R_{}
With i_{b} r_{b} small and i_{e} ? i_{c} then
v ? i R
In the collector arm,
v = ?i R
Dividing (9.30) by (9.29) gives the small-signal ac gain Gv
of this common-emitter amplifier to be

Notice that this gain depends only on the external resistors
connected to this circuit and not on ?. Hence, we can easily
control G_{v} by changing R_{G} and R_{e}. Nice design! Input and Output Impedance. Miller Effect.
The last topics we will consider in this lecture are the
determination of the ac input and output impedances of this CE
amplifier. It is important to know these values to properly match
sources and loads to the amplifier.
1. AC Input Impedance of the CE Amplifier with Emitter
Degeneration.
Referring to Fig. 9.9a again, the ac input impedance is defined
As

Using (9.29) and i_{c} = ?i_{b} gives

Notice that i Z is the product of two large numbers.
Consequently, the ac input impedance could potentially be very
large, which is desirable in certain circumstances.
However, you will see in Prob. 22 that this high input
impedance is often not observed because of the so-called Miller
capacitance effect.
To understand this effect, we construct the small signal model of
a CE amplifier and include the base-to-collector capacitance:

This b-to-c capacitance arises due to charge separation at the
CBJ. Other junction capacitances are also present in the
transistor, but are not manifest at the “lower” frequencies of
interest here.
While Cm, the Miller capacitance, is usually quite small (a few
pF), its effect on the circuit is magnified because of its direct
connection from the output to input terminals of this amplifier
with high gain.
Let’s now re-derive the input impedance while accounting for
this Miller capacitance. Referring to the figure above, the
capacitor current is

From (9.31) we know that

Substituting this into (9.35) we find that

We see from this expression that the effects of the capacitance
Cm are magnified by the gain of the amplifier! This is the socalled Miller effect. Therefore, considering this Miller effect the input impedance of the CE amplifier will be ?R_{e} in parallel with the effective input capacitance from (1)

This has the effect of reducing the input impedance magnitude
from the huge value of ?R_{e}.
2. AC Output Impedance of the CE Amplifier with Emitter
Degeneration.
As shown in the text, the output impedance of a CE amplifier
with emitter degeneration is given by the approximate
expression

Rs is the source resistance and zc is the collector impedance
z_{c}= r_{c}|| Z_{c}= r_{C}||( j?C_{C})^{-1}
This collector impedance is the parallel combination of the finite
output resistance r_{C} of the BJT (from the Early effect illustrated in Fig. 9.10) and the finite output capacitance of the BJT, labeled Cc in the text. The output impedance Z_{0} in (9.46) is often very large for CE amplifiers with emitter degeneration, which makes for a good current source.