# Miller Effect - Emitter Degeneration

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 = VIp/2. With Vp = Vpp/2 and Ip = Ipp/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 V0/2 (where p indicates peak). Therefore, V+pp= V-p V0 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 Vc 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 Vb through Re to ground gives Vb = IbRb +Vf + IeRe

With Ic ? Ie then, Vb ? Ib Rb +Vf We will choose Vb with some Ic bias in mind ( Ic = ? Ib ).
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) Re ? 0: Vb ? Ib Rb f c e +Vf + I cRR
The first term is usually small wrt the third term. This leaves us with
V b?Vf + Ic Rc
This is a good design since we can set Vb for a desired Ic 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 Rb = 0. Using KVL in the base and emitter circuit gives
vi = ibrb + ieR
With ib rb small and ie ? ic 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 Gv by changing RG and Re. 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 ic = ?ib 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 ?Re in parallel with the effective input capacitance from (1)

This has the effect of reducing the input impedance magnitude from the huge value of ?Re.
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
zc= rc|| Zc= rC||( j?CC)-1
This collector impedance is the parallel combination of the finite output resistance rC 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 Z0 in (9.46) is often very large for CE amplifiers with emitter degeneration, which makes for a good current source.