ES 330 Electronics II Supplemental Topic #1 (August 2015)

Unifying Principle for Active Devices: Charge Control Principle

Donald Estreich

An active device is an electron device, such as a transistor, capable of delivering

power amplification by converting dc bias power into time varying signal power. It

delivers a greater energy to its load than if the device were absent. The charge control

framework [1-3] presents a unified understanding of the operation of all electron devices

and simplifies the comparison of the several active devices used in compound

semiconductor analog and digital integrated circuits.

Figure 1. Generic charge control device consisting of three electrodes embedded around a charge transport region.

Consider the generic electron device shown in Fig. 1. It consists of three

electrodes encompassing a charge transport region. The transport region is capable of

supporting charge flow (the electrons shown in the figure) between an emitting electrode

and a collecting electrode. A third electrode, called the control electrode, is used to

-

-

-

-

(electrons)

Charge -Q

Collecting

Electrode Emitting

Electrode

Controlling

Electrode Charge +Qc

Transport Region

+ + + +

2

establish the electron concentration within the transport region. Placing a control charge,

QC, on the control electrode establishes a controlled charge, denoted as -Q, in the

transport region. The operation of active devices depends upon the charge control

principle [1]:

Each charge placed upon the control electrode can at most introduce an equal

and opposite charge in the transport region between the emitting and

collecting electrode.

At most we have the relationship, │-Q│ = │QC│. Any parasitic coupling of the control

charge to charge on the other electrodes, or remote parts of the device, will decrease the

controlled charge in the transport region, that is, │-Q│ < │QC│more generally. For

example, charge coupling between the control electrode and the collecting electrode

forms a feedback or output capacitance, say Co. Time variation of QC leads to the

modulation of the current flow between emitting and collecting electrodes.

The generic structure in Fig. 1 could represent any one of a number of active

devices (e.g., vacuum tubes, unipolar transistors, bipolar transistors, photoconductors,

etc.). Hence, charge control analysis is very broad in scope and it applies to all electronic

transistors.

Starting with the charge control principle, we associate two characteristic time

constants with an active device, thereby, leading to a first-order description of its

behavior. Application of a potential difference between the emitting and collecting

electrodes, say VCC, establishes an electric field in the transport region, although this

applied field is not always needed when diffusion or internal fields from doping profiles

are effective. Electrons in the transport region respond to the electric field and move

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across this region with a transit time r. The transit time1 is the first of the two important

characteristic times used in charge control modelling. With charge -Q in the transit

region, the static (dc) current Io between emitting and collecting electrodes is

Io = -Q/r = Qc/r (1)

A simple interpretation of r is as follows: r is equal to the length l of the transport region

divided by the average velocity of transit (i.e., r = l/v). From this perspective a charge

of -Q (coulombs) is swept out the collecting electrode every r seconds.

Figure 2. Generic charge control device of Fig. 2.2.1 connected to input and output resistors, Rin and RL, respectively, with bias voltage and input signal applied.

Consider Fig. 2 showing the common-emitting electrode connection of the active

device of Fig. 2.2.1 connected to input and output (i.e., load) resistances, say Rin and RL,

1The transit time r is best interpreted as an average transit time per carrier (in our case the electron). We

note that 1/r is common to all devices – it is related to a device’s ultimate capability to process information.

-

-

-

-

Collecting

Electrode Emitting

Electrode

Controlling

Electrode

Transport

Region

Ci

VCC

vin

+ -

- RL vout

+

Rin

+ + + +

4

respectively. The second characteristic time of importance can now be defined – it is the

“lifetime” time constant and we denote it by the symbol . It is a measure of how long a

charge placed on the control electrode will remain on the control terminal. The “lifetime”

time constant is established in one of several ways depending upon the physics of the

active device and its connection environment. The controlling charge may “leak away”

by (1) discharging through the external resistor Rin as typically happens with FET

devices, (2) recombining with intermixed oppositely charged carriers within the device

(e.g., base recombination in a bipolar transistor), or (3) discharging through an internal

shunt leakage path within the device. The dc current flowing to replenish the lost control

charge is

Iin = -Q/ = Qc/ (2)

The static (dc) current gain GI of a device is defined as the current delivered to

the output divided by the current replenishing the control charge during the same time

period. Where in seconds charge -Q is both lost and replenished, charge Qc times the

ratio /r has been supplied to the output resistor RL. In symbols, the static current gain is

GI = Io/Iin = /r (3)

provided -Q = QC holds.

In the dynamic case the process of small-signal amplification consists of an

incremental variation of the control charge Qc directly resulting in an incremental change

in the controlled charge, -Q. The resulting variation in output current flowing in the load

resistor translates into a time varying voltage vo. The charge control formalism holds just

as well for large-signal situations. In the large-signal case the changes in control charge

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are no longer small incremental changes. Charge control analysis under large charge

variations is less accurate due to the simplicity of the model, but still very useful for

approximate switching calculations in digital circuits.

An important dynamic parameter is the input capacitance Ci of the active device.

Capacitance Ci is a measure of the work required to introduce a charge carrier in the

transport region. Capacitance Ci is given by the change in charge Q from a corresponding

change in input voltage vin. It is desirable to maximize Ci in an active device. The

transconductance gm is calculated from

gI

v

I

Q

Q

vm

o

in v

o

ino

(4)

The first partial derivative on the right-hand-side of Eq. (2.2.4) is simply (1/r) and the

second partial derivative is Ci . Hence, the transconductance gm is the ratio

gm

i

r

C

(5)

A physical interpretation of gm is the ratio of the work required to introduce a charge

carrier to the average transit time of a charge carrier in the transport region. The

transconductance is one of the most commonly used device parameters in circuit design

and analysis.

In addition to Ci another capacitance, say Co, is introduced and associated with

the collecting electrode. Capacitance Co accounts for charge on the collecting electrode

coupled to either static charge in the transport region or charge on the control electrode.

A non-zero Co indicates that the coupling between the controlling electrode and the

charge in transit is less than unity (i.e.,-Q < QC).

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For small-signal analysis the capacitance parameters are usually taken a fixed

numbers evaluated about the device’s bias state. When using charge control in the large-

signal case, the capacitance parameters must include the voltage dependencies. For

example, the input capacitance Ci can be strongly dependent upon the control electrode to

emitting electrode and collecting electrode potentials. Hence, during the change in bias

state within a device the magnitude of the capacitance Ci is time varying. This variation

can dramatically affect the switching speed of the active device. Parametric dependencies

upon the instantaneous bias state of the device are at the heart of accurate modelling of

large-signal or switching behavior of active devices.

Figure 3. A two-port small-signal admittance parameter model for an active device. Notice the

forward admittance yf is the transconductance of the device.

We introduce the small-signal admittance charge control model shown in Fig. 3.

This model uses the emitting electrode as the common terminal in a two-port connection.

yo = Co

r

yi = Ci

yf gm = Ci

r

yr = - Co

yf vi -yr vo yo yi vo +

vi +

Controlling Electrode

Collecting Electrode

Emitting Electrode

io ii

7

The transconductance gm is the magnitude of the real part of the forward admittance yf

and is represented as a voltage-controlled current source positioned from collecting-to-

emitting electrode. The input admittance, denoted by yi, is equivalent to (Ci/), where

is the control charge “lifetime” time constant. Parameter yi can be expressed in the form

(gi + sCi) where s = j. An output admittance, similarly denoted by yo, is given by

(Co/r) where r is the transit time and, in general yo = (go + sCo) in general. Finally, the

output-to-input feedback admittance yr is included using a voltage-controlled current

source at the input. Often yr is small enough to approximate as zero (the model is then

said to be unilateral).

Consider the frequency dependence of the dynamic (ac) current gain Gi. The

low-frequency current gain is interpreted as follows: An incremental charge qc is

introduced on the control electrode with “lifetime” . This produces a corresponding

incremental charge -q in the transport region. Charge -q is swept across the transport

region every “transit time” r seconds. In time charge -q crosses the transit region r

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times, which is identically equal to the low-frequency current gain.

Figure 4. Circuit used to calculate the small-signal current gain Gi for our active device.

The “lifetime” associated with the control electrode arises from charge “leaking

off” the controlling electrode. This is modeled as an RC time constant at the input of the

equivalent circuit shown in Fig. 4(a) with equal to RinCi. The break frequency B

associated with the control electrode is (it’s 3dB below the low frequency value)

B

in iR

1 1

C

(6)

When the charge on the control electrode varies at a rate less than B, Gi is equal to /r

because charge “leaks off” the controlling electrode faster than 1/. Alternatively, when

gm vi Ci yo gi vi +

Short- circuit

io ii

/r

log Gi

0 dB log

-20 dB/decade

T

B

(a)

(b)

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is greater than B, Gi decreases with increasing because the applied signal charge

varies more rapidly than 1/. Hence, Gi is inversely proportional to

G i

r

T 1

(7)

where T is the common-emitter unity current gain frequency. At T (= 2fT) the ac

current gain equals unity as illustrated in Fig. 4(b).

Now consider the current gain-bandwidth product Gi f. Purely capacitive input

impedance can’t define a bandwidth. However, a finite, real impedance always appears at

the input terminal in any practical application. Let Ri be the effective input resistance if

the device (i.e., Ri will be equal to (1/gi) in parallel with the external input resistance Rin).

Since the input current is equal to qc/ and the output current is equal to q/r, the current

gain-bandwidth product is

G fq

qi

r

c

/

/

2 (8)

For B, at =1/, and assuming |qc| = |-q|,

G fi

r

T

1

2 2

fT (9)

fTorT ) is a widely quoted parameter used to compare or “benchmark” active devices.

Sometimes fTorT ) is interpreted as a measure of the maximum speed a device can

drive a replica of itself. It is easy to compute and historically has been easy to measure

with bridges and later using S-parameter test equipment. However, fT does have

interpretative limitations because it is defined as current into a short-circuit output.

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Therefore, it ignores the effect of both input resistance and output capacitance upon

actual circuit performance.

Likewise, voltage and power gain expressions can be derived. It is necessary to

define the output impedance before either can be quantified however. Let Ro be the

effective output resistance at the output terminal of the active device. We shall make the

assumption that input and output RC time constants are identical, that is, RiCi = RoCo.

That may not be true in general, but we must assume something to define these output

parameters and it not too far from reality in many applications. The voltage gain Gv can

be expressed in terms of Gi,

,ov i i

i

RG G G

R i

o

C

C (10)

where Ro is the parallel equivalent output resistance from all resistances at the output

node.

The power gain Gp is computed from the product of GiGv along with the power

gain-bandwidth product. These results are listed in Table 1 as summarized from

Johnson and Rose [1]. These simple expressions are valid for all devices as interpreted

from the charge control perspective. They provide for a first-order comparison, in terms

of a few simple parameters, among the active devices commonly available. From an

examination of Table 1 it is evident that maximizing Ci and minimizing r leads to higher

transconductance, higher parametric gains and greater frequency response. This is an

important observation in understanding how to improve upon the performance of any

active device.

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Whereas fThas limitations, the frequency at which the maximum power gain

extrapolates to unity, denoted by max, is a more useful indicator of the frequency limit of

the device’s active region. The primary limitation of max is that it is very difficult to

measure directly and is therefore usually extrapolated from S-parameter measurements in

which the extrapolation is an approximation.

The simple charge control model is useful because of the physical insight it gives

in understanding active devices. First, all active devices have internal capacitance Ci

from the presence of charge on the controlling electrode and in the transit region. Every

active device experiences a -20 dB/decade gain falloff because of the existence of

capacitance Ci. In a field-effect transistor, Ci is established by the coupling of the charge

on the gate electrode to the charge in the channel. In contrast, in a bipolar junction

transistor Ci consists of both the controlling charge and the controlled charge (i.e., the

minority carrier charge in transit) coexisting simultaneously in the base region. For this

reason Ci is generally much greater in a bipolar transistor than in a field-effect transistor –

this is the principle reason bipolar transistors area capable of achieving much greater

transconductance gm than field-effect devices. Transistor designers try to maximize Ci as

much as possible while still achieving the values needed for other transistor parameters.

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References

[1] E. O. Johnson and A. Rose, “Simple general analysis of amplifier devices with

emitter, control, and collector functions,” Proceedings of the IRE, 47, 407, 1959.

[2] E. M. Cherry, and D. E. Hooper, Amplifying Devices and Low-Pass Amplifier

Design, Wiley, New York, 1968, Chapters 2 and 5.

[3] R. Beaufoy, and J. J. Sparkes, “The junction transistor as a charge-controlled device,”

ATE Journal, 13, 310, 1957.

Table 1. After Johnson & Rose [1]

Parameter Symbol Expression

Transconductance gm CCi

r

T i

Current Amplification Gi 1

r

T

Voltage Amplification Gv 1

r

i

o

T i

o

C

C

C

C

Power Amplification Gp = GiGv 12 2

2

2

r

i

o

T i

o

C

C

C

C

Current Gain-Bandwidth Product Gif 1

r

T

Voltage Gain-Bandwidth Product Gvf 1

r

i

o

T

i

o

C

C

C

C

Power Gain-Bandwidth Product Gpf2 1

2

2

r

i

o

T

i

o

C

C

C

C

Note: Table assumes RiCi = RoCo.

These notes were taken from D. B. Estreich, “Compound Semiconductor Devices for

Analog and Digital Circuits,” Chapter 72, in The VLSI Handbook, 2nd

edition, edited by

Wai-Kai Chen, CRC Press (Taylor & Francis Group), Boca Raton, FL, 2007; pages 72-4

to 72-9. ISBN 978-0-8493-4199-1