Quest for the Ideal Transistor?

-March 11, 2015

It’s the first day of spring.  For a change, the San Francisco fog does not dare to come to campus.  The quad is thick with students in cutoffs and tee shirts, sitting or lying on the grass.  It’s about noon, and it looks as if I am the only soul lecturing this time of day.  Physically my students are in class, pretending to pay attention; mentally they are outside, soaking up sun.  Today’s lecture is meant to be an initiation to the transistor as an amplifier, a very unsuitable topic for a classroom saturated with the hormones of 20-year-olds (plus, I must confess, the hormones of a no-longer-so-young).  I start out by pointing out that since the transistor is a one-quadrant device, we need first to bias it at a suitable operating point in the active region.  “If you say so”, seems to be the classroom’s mute reaction.

Next, we must keep the variations about the operating point suitably small to ensure that the small-signal approximation holds.  Again, “If you say so” is the uniform response.  Finally, we must include capacitors for ac-coupling and bypassing.  This seems to add insult to injury because it elicits no response whatsoever, not even “If you say so”.  I try making the above points via a coarse analysis of the circuit of Figure 1, but realizing how pointless my effort is, I announce that this is it for today.  At which point students spring to their feet in unison to check for SMSs and head for the exit.  No sooner than I reach for the eraser and turn around to say goodbye that they are all gone, some through the door, others must have left through the windows…

Figure 1
A circuit that shouldn’t be covered on the first day of spring.

I am left alone to ponder how much easier my own teaching and my students’ learning would be if the transistor were a four-quadrant device instead of a one-quadrant one.  Presently such a device does not exist, but we can try mimicking one using a whole bunch of transistors (for good measure, let’s pretend we’re doing this in a garage – you never know…).

A Pseudo-Ideal BJT

BJTs are normally-off devices, requiring base-emitter voltage drops of 0.6~0.7 V to draw convincing currents.  We’d like the npn BJT of Figure 2a to start drawing significant current as soon as we raise vB above zero.  We meet this requirement by compensating for the B-E drop of Q1 with the E-B drop of the emitter follower Q3, as depicted in Figure 2b.  Ignoring base currents, we have

Figure 2
Step-by-step quest for the ideal transistor.

where Is1 and Is3 are the saturation currents, assumed identical, and VT is the thermal voltage.

To ensure symmetric operation about vB = 0, we need to complement the Q1-Q3 pair with the Q2-Q4 pair of Figure 2c.  Assuming again matched saturation currents and ignoring base currents, we now have the symmetric counterpart of Equation (1),

Finally, to complete our quest for the pseudo-ideal BJT, we need to take the difference.

Figure 3
Last step, leading to the four-quadrant transistor; also shown is one of its circuit symbols.

We accomplish this task by means of two current mirrors with the outputs tied together, as depicted in Figure 3 for the case of Wilson-type current mirrors.  The Q5-Q6-Q7 mirror replicates iC1 and sources it into the output node, whereas the Q8-Q9-Q10 mirror replicates iC2 and sinks it out of the output node.  For vB = 0, the exponentials of Equation (3) cancel each other out to give iC = 0.  For vB > 0, iC1 prevails to give iC > 0, and for vB < 0, iC2 prevails to give iC < 0.  Clearly, the circuit allows for full four-quadrant operation.  Moreover, it exhibits high input resistance thanks to the Darlington function provided by the Q3 and Q4 emitter followers, and high output resistance because of the Wilson mirrors.

Well, this pseudo-ideal BJT has been around for quite some time.  Variously called a transductor, a macro transistor, a diamond transistor, and a Current Conveyor II+ [1], it is also available in IC form as the OPA861 [2].  Figure 4 shows how much simpler the amplifier of Figure 1 would be if implemented with a transductor.  Note that the amplifier of Figure 1 provides signal inversion, whereas that of Figure 3 is of the non-inverting type.

Figure 4
Implementing a common-emitter amplifier with a transductor.

The Current-Feedback Amplifier (CFA)

As we know, the range of applications of an amplifier can be expanded dramatically through the use of negative feedback, and a transductor-based amplifier is no exception.  Since the transductor exhibits high output resistance, we need to use an output buffer to prevent loading by the feedback network.  This leads us to the circuit of Figure 5, where the output buffer, consisting of Q11 through Q14, is similar to the input buffer Q1 through Q4.  This circuit too has been around for quite some time [3].  Called a current-feedback amplifier (CFA), it replaces the conventional op amp in certain high-speed applications.  Unlike the ordinary BJT, which we configure for negative-feedback operation by connecting the feedback network between the collector and the base, the non-inverting nature of the transductor requires that the feedback network be connected between its (buffered) collector and the emitter, that is, between the vO and vN nodes of Figure 5

To investigate feedback operation, refer to the simplified equivalent of Figure 6a, showing explicitly the net impedance zc presented by the C node towards ground (for reasons that will become clear shortly, the C node is also called the gain node).  To a first approximation, zc can be modeled with a resistance Rc in parallel with a capacitance Cc, so expanding gives

Typically, Rc in the range of 105~106 Ω and Cc is in the pF range.  Now, any current imbalance In created by the external network will get replicated by the Wilson mirrors at the C node to give

Vo = zcIn                                                                                                                        (5)

Figure 5
Using an output buffer to turn a transductor into a current-feedback amplifier (CFA).

Turning next to the typical feedback interconnection of Figure 6b, we sum currents into the Vn node to get

Letting Vn = Vp = Vi, solving for In, and inserting into Equation (5) gives the closed-loop voltage gain

Figure 6
(a) Simplified equivalent of the CFA.  (b) CFA symbol and interconnection for negative-feedback operation as a non-inverting amplifier.

In a well-designed circuit, R2 is on the order of 103 Ω, so with Rc in the range of 105~106 Ω, we can ignore the term R2/zc at dc and state that at low frequencies A tends to the familiar op amp expression

The advantages of CFAs compared to ordinary op amps (also called voltage-feedback amplifiers or VFAs) are fast dynamics.  Equation (6) indicates that the loop gain of this circuit is

T = zc/R2                                                                                                                       (8)

so the closed-loop bandwidth is given by the frequency at which |zc| = R2, also called the crossover frequency fx (see Ref. [4]).  So long as R2 << Rc, this frequency is fx = 1/(2πR2Cc).  With R2 on the order of 103 Ω and Cc on the order of 10–12 F, fx will be on the order of 108 Hz.  Note also that fx is set by R2 regardless of R1.  Compared to VFAs, where fx is inversely proportional to the noise gain 1 + R2/R1, the fx of a CFA appears to be independent of the noise gain (for a more detailed analysis of higher-order effects, see Ref. [3]).  Another dynamic advantage of CFAs is their relative immunity from slew-rate limitations because Cc is driven directly by the input buffer, which can supply virtually any current to rapidly charge/discharge Cc

Back to the Voltage-Feedback Amplifier (VFA)

Let us admit, we are so used to VFAs, that the presence of a buffer directly across the input terminals tends to make us feel uneasy (which is exactly how I felt the first time I run into a CFA).  Yet, the fast dynamics of CFAs are quite tempting…  Wouldn’t be possible to modify a CFA so as to obtain a VFA that retains at least some of the dynamic advantages of the original CFA?  This issue too has been addressed quite sometime ago by adding a third voltage buffer (see Q15-Q16-Q17-Q18 of Figure 7) to turn node vN into a high resistance input, along with a resistance R between the outputs of the first buffer and this new buffer to generate the control current previously denoted as iN

Figure 7
CFA-derived VFA.

To analyze the circuit, consider the current through R, assumed to flow from left to right, which is (VpVn)/R.  The current mirrors convey this current to the gain node C, where it produces the voltage zc(VpVn)/R.  This voltage is then buffered to the output node to give Vo, so the open-loop voltage gain is

where Eq. (4) has been used.  Again, a well-designed circuit has R << Rc, in turn implying a large dc value for a.  Owing to its inherently fast current-mode operation, this op amp type is especially suited to high-speed applications.  A popular example is the LT1363 - 70MHz, 1000V/µs op amp.

Closure

Our quest for the ideal transistor has brought us to rediscover a series of circuits that have already been around for quite some time.  Does this imply that when you sit down to try inventing something new, you better keep in mind the notorious dictum: "Everything that can be invented has [already] been invented"?  Or that…?  Well, I won’t try answering now, because I’m going to lock my office door, pull out from a secret drawer my own tee shirt and cutoffs, change my clothes, and go out to soak up some sun myself.

References

1. Analog Circuit Design: Discrete and Integrated, Sergio Franco, San Francisco State University
2. Demystifying the Operational Transconductance Amplifier, Xavier Ramus, Texas Instruments
3. Design with Operational Amplifiers and Analog Integrated Circuits, 4th edition, Sergio Franco, San Francisco State University
4. Inverting vs. Noninverting, EDN