Lab 4: Differential and common-mode in a differential-pair

1. Introduction

1.1. Differential signal transmission

Differential signaling uses a pair of wires to send signals between circuits, especially when they are widely separated. Implied in this system is a third node that is common to both the sender and receiver. It makes it possible to remove several types of noise and interference added to the signal along its path from transmitter to receiver. USB, Ethernet, HDMI, DisplayPort, and PCI Express connections are common examples of differential signaling in practice.

The desired signal is driven as the difference between the two wires and, ideally, the noise voltages/currents are added equally and in-phase to each conductor of the pair. The receiver simply takes the voltage difference of the pair, therefore removing all of the noise.

Figure 1. Differential signal transmission.

Twisting the two signal wires together helps arrange that external magnetic and electric fields interacting them along the length affects both wires equally. The subtraction at the receiver, helped by the impedance balance, removes nearly all of this added noise.

Figure 2. Impedances in a balanced system.

Figure 2 shows the impedances associated with a differential connection. The v_noise term will be completely removed by the receivers subtraction only if and when the impedances between each wire and the common reference are equal to each other. Cancellation depends on having the two voltage divider ratios Z_a : Z_ina and Z_b : Z_inb equal.

1.2. Differential and common-mode

A differential pair involves three nodes: a common reference best defined at the receiver’s end of the circuit, and the two signal nodes A and B (or other labels like [+in, -in], or [in_p, in_m]). The two node voltages \(v_A\) and \(v_B\) (measured with respect to the circuit’s reference node) include terms from all three independent sources (V_outa, V_outb, and V_Noise). Though these completely describe the situation, it is much more useful to instead use the average (read: common-mode) and difference of the two nodes:

\[\begin{cases} \begin{aligned} v_D &= v_A - v_B \\ v_{CM} &= (v_A + v_B) / 2 \end{aligned} \end{cases}\]

If you have the set of differential and common-mode voltages for a pair of nodes, it is a small matter of algebra to find the individual node voltages again:

\[\begin{cases} \begin{aligned} v_A &= v_{CM} + v_D / 2 \\ v_B &= v_{CM} - v_D / 2 \\ \end{aligned} \end{cases}\]

Figure 3 is a graphical version of how the A and B voltages can be specified in two different ways.

Figure 3. Two ways of specifying the voltage at a pair of nodes, red and blue sets.

Another way of showing the two different schemes for specifying two voltages is as a coordinate transformation. Here, the axes are rotated about the origin by 45°. Any {v_A, v_B} coordinate has a corresponding {v_D, v_CM} coordinate.

Figure 4. Two coordinate systems for the same plane

Some additional points to remember

In any system, two terms are required to completely specify the potentials at the two nodes.
The equations use terms in lower_UPPER notation, the total or physical quantity, to emphasize that these two systems are always valid.
… meaning they are also valid for DC quantities (UPPER_UPPER notation) and for small-signal quantities (lower_lower notation), and even for phasor quantities (UPPER_lower).

The noise cancellation feature of the differential signaling idea depends on the relative symmetry of impedances. It has no requirement that the two signal waveforms are equal and opposite.

2. Lab goals

Gain experience with differential and common-mode signal quantities and measurements — converting between them and A, B node voltage “coordinate systems.”

Goals

Gain experience converting between differential and single-ended quantities.

Objectives

Build differential pair circuits with different tail current circuits.

We are using the AD2 specifically because it:

is able to output two phase-locked waveforms and
the input channels are fully differential (the "−" pin can connect anywhere).

There is no need to use “Math” to subtract channel 2 from channel 1 → just directly attach the two terminals of a channel to the two nodes that are to be subtracted.

3. Procedure

Figure 5. Differential-pair schematic

3.1. DC conditions

Build the circuit of Figure 5. Use the datasheets for the CA3086 and the MC3346 to check that the pin numbers labeled in the schematic are correct for the model number you have.
Apply 0 V to both inputs inA and inB. Either by attaching to the 0 V node or actively outputting 0 V with the Wavegen channels.
Compute what you expect the voltages and currents to be using hand analysis.
Use the nice benchtop multimeters to measure the node voltages and the currents through the resistors to 3 digits of precision. A trick is to use Ohm’s law to compute the resistor currents instead of using an ammeter.

Table 1. DC Measurements
Name	Expected (hand analysis)	Measured Value
V_outA
V_outB
V_E
I_R1
I_R2
I_Rtail

4. Amplifier measurements

Configure both of the AD2’s function generators (W1 and W2) to output a pair of signals which yield a pure common-mode signal of: ^[1]

\[\begin{align} v_{IN,\, cm}(t) &= 2\,\sin(2\pi 1000 t) \;\mathrm{V}\\ v_{IN,\, d}(t) &= 0 \;\mathrm{V} \end{align}\]

Translate these cm, d quantities into the “a, b” quantities needed for the two generator outputs.

Be sure to change the Wavegen settings within the AD2’s Waveforms software from "No synchronization" to "Synchronized". This ensures that both signal generators start at exactly the same time. It is critical that the two output signals have the same relative phase, otherwise your results will be bogus.

Measure the three small-signal common-mode voltage gains in this circuit.
- These are the signal amplitudes, in volts peak-to-peak or volts RMS (preferred),
  → do NOT use "Math" to divide one oscilloscope channel by the other.
- You can predict the first two single-ended gains by viewing the circuit as two separate common-emitter amplifiers with emitter resistor values of 2R_E (gain ≈ − R_c / 2R_E).
- Predict the common to differential mode gains by thinking about the circuit’s symmetry.

Common-mode to single-ended gains

\[\begin{align} A_{cm\,A} &= v_{outA} / v_{in,\,cm} \\ A_{cm\,B} &= v_{outB} / v_{in,\,cm} \\ \end{align}\]

Common-mode to differential mode conversion gain

\[A_{cm} = v_{out,\,d} / v_{in,\,cm} = (v_{outA} - v_{outB}) / v_{in,\,cm}\]

Go back and review the lab’s goals and objectives.

Measuring the differential output \(v_{out,\,d}\) is where you can use the fact that the AD2’s input channels are differential. There is no need, like there is with a benchtop oscilloscope, to use two probes, one for each node A and B, and then subtract the channels using the Math function. Simply connect the + and − leads of the AD2 channel directly across nodes outA and outB.

Configure the AD2’s function generators to output a pure differential triangular signal of:

\[\begin{align} v_{in,\, cm}(t) &= 0 \;\mathrm{V} \\ v_{in,\, d}(t) &= 0.3\, \operatorname{tri}(2\pi 1000 t) \;\mathrm{V} \end{align}\]

Plot the large-signal differential-input to differential-output transfer function using an X-Y display (in Waveforms: Scope → View → Add XY), \((V_{outA} - V_{outB}) \text{ versus } V_{in,\,d}\).
From this X-Y plot, determine the maximum differential input amplitude that still gives an un-distorted differential output signal. (hint: it will be within a small multiple of the thermal voltage \(V_T\))
Reduce the differential input amplitude to this value and measure the following gains. They will be around 90 or 180 V/V.

\[\begin{align} A_{d,\,A} &= v_{outA} / v_{in,\,d}\\ A_{d,\,B} &= v_{outB} / v_{in,\,d}\\ A_{d} &= (v_{outA} - v_{outB}) / v_{in,\,d} \end{align}\]

Extension work.

Complete the previous work first.

Figure 6. Differential-pair with tail current source schematic

Build the circuit of Figure 6.
Determine the closest E12 value for R3 to give the same current as was through R_E. Verify this current by measurement. Hint: what is the voltage across R3?
Measure the same six gains on this circuit as on Figure 5 (three \(A_{cm}\) variations and three \(A_d\) variations).

Measure input currents and the base-emitter voltages with as much precision as you can. Place a 10k resistor in series with the bases of each of Q1 and Q2. Set \(v_{i,\,cm} = 0\) and \(v_{i,\,d} = 0\) using the function generators (and keep them "running"). Measure the base currents of Q1 and Q2 by measuring the DC voltage drop across the series resistors. This is a great opportunity to use the nice, new, Keysight meters!

Table 2. Diff-pair DC measurements
Name	Value
I_B1
I_B2
V_BE1
V_BE2

1. Convert the (cm, d) voltages into single-ended (a, b) node voltages to get the settings needed for the Wavegen outputs. Remember the Lab goals