Differential Amplifier

Differential Amplifier

Before getting into differential amplifiers, let's review small-signal transistor models, and introduce a new one that will be of service. In the figure below, the new model is on the right, and our familiar models on the left. The resistances r_π an

d r_e both are related to how much the output current changes with respect to input voltage, and the relations of them to the general parameter, the transconductance g_m, are shown. Once you know the bias collector current, you can calculate these important parameters.

In the new model, which we may call the alpha model, the direct connection of the base may worry you. However, note that the current-controlled current source in the collector makes sure that only the proper base current flows. In the (good) approximation that α = 1, there will be no base current at all. The connection to this internal, inaccessible node is only to put the base-emitter voltage across the resistor r_e to make the emitter current. The collector then takes what it likes, leaving the rest for the base. In fact, alpha is often taken as unity when you use the alpha model for quick insight, for which it is specially adapted.

There are two inputs and two outputs, and the circuit is assumed to be symmetrical, which it always is in practice, to a first approximation. If only one output is used, the other collector resistor can be eliminated, with no change to the functioning of the circuit. It is very convenient to express the inputs as the sum of a common-mode input v_cm, which is the average of the two input voltages, and the differential mode input v_d. In the common mode, the same voltage v_cm is applied to the two inputs. In the differential mode, opposite and equal voltages &plusmi;v_d are applied to the two inputs.
In the common mode, the amplifier acts as a single amplifier with collector resistance R_C/2 and emitter resistance R_S + (R_E + r_e)/2, so the gain (the ratio of the resistances), input impedance and output impedance can easily be found, as for the single-transistor amplifier. If the input is taken single-ended, the gain is R_E/(2R_S + R_E + r_e). R_S is usually made very large, so that the common-mode gain is small. In many cases, R_S is actually replaced by a current source that has a very high internal resistance, so the common-mode gain is practically zero. This means that the amplifier will not respond to any signal common to the two inputs, which is generally desirable.
In the differential mode, the symmetry of the inputs means that node "a" will not change in voltage, and i₁ = -i₂ = v_d/(2R_E + 2r_e). From this result, the gain at an output node is easily found: G = R_C/2R_E + 2r_e). This is called the single-ended gain. If the output is taken between the two collectors (double-ended) the gain is twice this value. Note that we have inverting and noninverting outputs as well as inputs. The differential input simply divides the same total current between the two sides of the amplifier. It can go no farther than to put all the current through one collector or the other, however.
Consider a circuit in which the input is to v₁, the other input grounded, and the output is taken at v_o2, with R_C1 removed. Then we have an emitter follower driving the collector of the second transistor, which forms a common-base (this input is grounded) amplifier. This circuit is good for very high frequencies, for reasons that we shall explore elsewhere. This circuit uses feedback, of course, though we did not use feedback analysis.
Construct and test a differential amplifier using R_C = 4.7k, R_E = 470Ω. Choose R_S so that the collector current in each transistor is about 1 mA, and use a ±12V supply. The single-ended differential gain should be about 5, the common-mode gain about 0.5 and the input impedance at one input about 100k. (How did I get these numbers?). The actual differential gain is a little smaller than 5, because the current source for the emitters is not a good one. The input source can be a buffered potentiometer.

 

A differential amplifier can be constructed using an op-amp, as shown at the right. An op-amp is a differential amplifier itself, but its gain is far too high and frequency dependent to be used as such. The inverting and noninverting inputs are shown by v- and v+. The formula for the differential gain is easily obtained by considering the noninverting input grounded, which means the op-amp tries to keep both its inputs at zero, so the feedback on the inverting side is like a lever with lengths R₁ and R₂ (since the same current flows in both resistors, of course). The common-mode gain is easily seen to be zero, since the inputs go up together if the output voltage is zero. Of course, the actual common-mode gain will not be zero because of various small inequities, both in the resistors and in the op-amp. I tested such a circuit with a 411 op-amp, using R₁ = 10k and R₂ = 100k, for a differential gain of 10 (20 dB). Using 5% resistors, the common-mode gain was 0.029, and with 1% resistors, it was 0.019. The common-mode rejection ratio, CMRR, was, therefore, 10/0.019 = 526 or 54 dB.
If you put a 100Ω resistor between the input terminals of the amplifier I tested, the result is a current sensor with a sensitivity of 1 V per mA that can be used to observe the current at any point with an oscilloscope, which must sense with ground as one terminal. The differential amplifier removes this limitation, which is often quite annoying.
http://mysite.du.edu/~etuttle/electron/elect8.htm
Rooselvet Ramirez EES