Gilbert Cells
A Gilbert cell is a crosscoupled differential amplifier,
similar to the topology in figure 1, where the gain is controlled by
modulating the emitter bias current. The amplitude of a differential input RF
signal, applied to pins 6 and 7 of the HFA3101, can be linearly controlled by
a differential ac voltage applied to pins 1 and 4. Because the gain control
is highly linear, Gilbert cells are often referred to as fourquadrant
multipliers and have common applications as mixers, AGC amplifiers, amplitude
modulators, double sideband (DSB) modulators, single sideband (SSB)
modulators, AM detectors, SSB and DSB detectors, frequency doublers, squaring
circuits, dividers, squareroot circuits, and rootmeansquare, r.m.s.,
measuring circuits. In order to understand how the Gilbert cell operates, it
is necessary to review some fundamental concepts of bipolar transistors. Bipolar Junction Transistor Models Figure 2 shows two equivalent small signal models for a
bipolar transistor. All ac components will be represented by a lower case
letter with vbe denoting the ac input voltage across the baseemitter
junction of the transistor, and ic will be the corresponding ac
collector current. The transconductance, gm, of a transistor is set by
it's dc quiscent collector current. In figure 3, the dc quiscent collector
current is denoted by Io/2, therefore (1) where VT is the thermal voltage and is taken at 0.025
volts at room temperature. Figure
1. Harris HFA3101 5 GHz Gilbert cell array. Once gm is established by setting the value of the
dc quiscent collector current, Io/2, the gain can be derived from gm vbe = ic = ib
(2) where is the current amplification of the transistor
and is dependent upon the transistor selected. The input impedance for an ac signal between the
baseemitter junction of a transistor will be denoted by and is given by
(3) Equations (19) are small signal approximations and they
are valid for vbe less than 10 millivolts. The Differential Amplifier Figure 3 shows a differential amplifier using a constant
current source. The following equations immediately provide the means to
analyze figure 3. The ac voltage across the baseemitter junction of Q1
is ,
(4) or .
(5) Figure
2. Equivalent small signal models for a bipolar transistor. The ac gain from vin to vout1 is then
(6) Notice here that if Rin << than , the total gain is
linearly dependent on the value of the constant current source, Io. In this
case
(7) The total voltage at vout1 is
(8) Similarly, at the collector of Q2, the output voltage, vout2
is
(9) with the assumption that both transistors are well
matched. Figure
3. Differential amplifier with constant current source. The Single Balanced Modulator In figure 4, the concept of a differential amplifier is
now extended to a single balanced modulator by modulating the constant
current source with a low frequency signal such that I = Io + k1 cos(wmt ) for k1 < Io.
(10) If vin is also represented as a sine wave such that
vin = k2 cos(wct)
(11) then substituting equations (10) and (11) into equations (8)
and (9) provide
(12) and
(13) Figure
4. Single balanced modulator It is seen that both (12) and (13) contain frequency
components at four distinct frequencies, including wc, wc  wm, wc + wm, and wm.
By taking the difference of vout1 and vout2 with a difference amplifier, the
baseband term at frequency, wm, is eliminated leaving
(14) or, by using a trigonometric identify,
(15) The balanced modulator with the difference amplifier that
implements equation (15) is shown in Figure 5. The corresponding voltages as
seen on an oscilloscope for vout1, vout2, and vout are shown in Figure 6. Figure
5. Single balanced modulator with difference amplifier Figure
6. Voltages at vout1, vout2, and vout, for a single
balanced modulator with difference amplifier The Double Balanced Modulator The double balanced modulator eliminates the carrier
frequency at wc and effectively implements a mixer that generates only
the sum and difference frequencies. An extension of the balanced modulator to
create a double balanced modulator is shown in figure 7. By analogy with the
balanced modulator, using equations (12) and (13), the equations for the
double balanced modulator in figure 7 become
(16) and
(17) The difference amplifier at the output of figure 7
eliminates the dc term leaving
(18) The output waveform, vout, for the double balanced
modulator is shown in figure 8. The double balanced mixer in figure 9 generates both a sum
and difference frequency. Further deriviatives of the double balanced mixer
can be used to generate only the upper, or lower, sideband during the
modulation process, or can be used to mix two signal and reject either the
lower, or the upper, image frequency. The analysis details are based on the
trigonometric identity
(19) Figure
7. Double balanced modulator with elimination of the dc component at vout Figure
8. Typical output waveform for a double balanced modulator shows a double
sideband (DSB) waveform and includes only the sum and difference frequencies,
. 




