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The Differential

Asst. Prof. MONTREE SIRIPRUCHYANUN, D. Eng. Dept. of Teacher Training in Electrical Engineering, Faculty of Technical Education King Mongkut’s Institute of Technology North Bangkok http://www.te.kmitnb.ac.th/msn [email protected]

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BJT Differential Pair

• Differential pair circuits are one of the most V widely used circuit building blocks. The CC input stage of every op amp is a differential αI/2 αI/2 amplifier RC RC • Basic Characteristics

– Two matched with emitters VCC-αIRC/2 VCC-αIRC/2 shorted together and connected to a I/2 I/2 – Devices must always be in active mode – Amplifies the difference between the two input voltages, but there is also a v CM I v - 0.7 common mode amplification in the non- CM ideal case

• Let’s first qualitatively understand how this circuit works. – NOTE: This qualitative analysis also applies for MOSFET differential pair circuits

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 2 Case 1

• Assume the inputs are shorted together to a common VCC voltage, vCM, called the common mode voltage αI/2 αI/2 – equal currents flow through Q1 and Q2 RC RC – emitter voltages equal and at vCM-0.7 in order for the devices to be in active mode VCC-αIRC/2 VCC-αIRC/2 – collector currents are equal and so collector Q Q voltages are also equal for equal load 1 2 – difference between collector voltages = 0 I/2 I/2

• What happens when we vary v ? CM v CM I v - 0.7 – As long as devices are in active mode, equal CM

currents flow through Q1 and Q2

– Note: current through Q1 and Q2 always add up to I, current through the current source – So, collector voltages do not change and difference is still zero…. – Differential pair circuits thus reject common mode signals

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Case 2 & 3

VCC • Q2 base grounded and Q1 base at +1 V αI 0 – All current flows through Q1 RC RC – No current flows through Q2

– Emitter voltage at 0.3V and Q2’s EBJ not FB VCC-αIRC Q1 Q2 – vC1 = VCC-αIRC +1V

– vC2 = VCC I 0

VCC 0αI I 0.3V RC RC

• Q2 base grounded and Q1 base at -1 V VCC VCC-αIRC Q Q – All current flows through Q -1V 1 2 2 0 I – No current flows through Q1 – Emitter voltage at -0.7V and Q1’s EBJ not FB

– vC2 = VCC-αIRC I -0.7V – vC1 = VCC

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 3 Case 4

• Apply a small signal vi – Causes a small positive ΔI to flow in Q1 – Requires small negative ΔI in Q2

•since IE1+IE2 = I – Can be used as a for small signals (ΔI is a function of vi) • Differential pair responds to differences in the input voltage – Can entirely steer current from one side of the diff pair to the other with a relatively small voltage

• Let’s now take a quantitative look at the large-signal operation of the differential pair

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BJT Diff Pair – Large-Signal Operation

• First look at the emitter currents when the emitters are tied together

• Some manipulations can lead to the following equations

• and there is the constraint:

• Given the exponential relationship, small differences in vB1,2 can cause all of the current to flow through one side

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 4

• Notice vB1-vB2 ~= 4VT enough to switch all of current from one side to the other • For small-signal analysis, we are interested in the region we can approximate to be linear

– small-signal condition: vB1-vB2 < VT/2

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BJT Diff Pair – Small-Signal Operation

• Look at the small-signal operation: small

differential signal vd is applied

multiply top and bottom by

– expand the exponential and keep the first two terms

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 5 Differential Voltage Gain

• For small differential input signals, vd << 2VT, the collector currents are…

• We can now find the differential gain to be…

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BJT Diff Pair – Differential Half Circuit

• We can break apart the differential pair circuit into two half circuits – which then

looks like two circuits driven by +vd/2 and –vd/2

Virtual Ground

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 6 Small-Signal Model of Diff Half Circuit

• We can then analyze the small-signal operation with the half circuit, but must remember

– parameters rπ,gm, and ro are biased at I / 2

– input signal to the differential half circuit is vd/2

vc1 gmvπ vd/2 rπ vπ ro RC

– voltage gain of the (output taken differentially) is equal to the voltage gain of the half circuit

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Common-Mode Gain

• When we drive the differential pair with a common-mode signal, vCM, the incremental resistance of the bias current effects circuit operation and results in some gain (assumed to be 0 when R was infinite)

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 7 Common Mode Rejection Ratio

• If the output is taken differentially, the output is zero since both sides move together. However, if taken single-endedly, the common-mode gain is finite

• If we look at the differential gain on one side (single-ended), we get…

• Then, the common rejection ratio (CMRR) will be

– which is often expressed in dB

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CM and Differential Gain Equation

• Input signals to a differential pair usually consists of two components: common

mode (vCM) and differential(vd)

• Thus, the differential output signal will be in general…

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 8 MOS Diff Pair

• The same basic analysis can be applied to a MOS differential pair

– and the differential input voltage is

– With some algebra…

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• We get full switching of the current when…

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 9 Another Way to Analyze MOS Differential Pairs

• Let’s investigate another technique for analyzing the RD RD RD RD MOS differential pair V XYV V XYV • For the differential pair circuit out1 out2 out1 out2 on the left (driven by two independent signals), compute V V V the output using superposition in1 in2 in1 I I – Start with Vin1, set Vin2=0 and first solve for X w.r.t.

Vin1 – Reduces to a degenerated common-source amp – neglecting channel-length RD and body- RD RD

effect, RS = 1/gm2 V X V XYV out1 –so… out1 out2

M1 M 2 Vin1 Vin1 RS

RS

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Cont’d

• Now, solve for Y w.r.t. Vin1

• Replace circuit within box with a Thevenin equivalent RD RD – M is a source follower with V =V 1 T in1 XY Vout1 Vout2 – RT=1/gm1

• The circuit reduces to a common-gate amplifier where… M1 M2

Vin1

• So, overall (assuming gm1 = gm2)

RD

Y by symmetry Vout2

RT

VT

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 10 Differential Pair with MOS loads

• Can use load resistors or MOS devices as loads

– Diode-connected nMOS loads = 1/gm load resistance • Load resistance looking into the source

– Diode-connected pMOS loads = 1/gm load resistance • Load resistance looking into diode connected drain

– pMOS current source loads = ro load resistance • Has higher gain than diode-connected loads – pMOS • Differential input and single-ended output

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Differential Pair with MOS Loads

Vb

Vout Vout

Vin Vin

I I

• Consider the above two MOS loads used in place of resistors • Left:

– a diode connected pMOS has an effective resistance of 1/gmP

• Right:

– pMOS devices in saturation have effective resistance of roP

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 11 Active-Loaded CMOS Differential Amplifier

• A commonly used amplifier topology in CMOS technologies • Output is taken single-endedly for a differential input M3 M4 –with a vid/2 at the gate of M1, i1 flows

i1 v i o – i1 is also mirrored through the M3-M4 current mirror 1 i2 –a –vid/2 at the gate of M2 causes i2 to also flow through M2 M1 M2

vid

• Given that ID= I / 2 (nominally) I

• The voltage at the output then is given by…

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Differential Amp with Linearized Gain

• Use source generation to make the gain linear with respect to the

differential input and independent of gm – Can build in two ways…

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 12

• Assuming a virtual ground at node X, we can draw the following small-signal half circuit.

gmvπ

vid vπ ro RD vο

i R s S vS

– Assume ro is very large (simplifies the math)

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Offsets in MOS Differential Pair

• There are 3 main sources of offset that affect the performance of MOS differential pair circuits – Mismatch in load resistors – Mismatch in W/L of differential pair devices

– Mismatch in Vth of differential pair devices • Let’s investigate each individually

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 13 Mismatch

• For the differential pair circuit shown, consider the case where RD1 RD2 – Load resistors are mismatched by ΔRD VO

– All other device parameters are perfectly matched • With both inputs grounded, I = I = I/2, but V is not zero 1 2 O I1 I2 due to differences in the voltages across the load resistors I

– It is common to find the input-referred offset which is calculated as

– since Ad = gmRD

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W/L Mismatch

• Now consider what happens when device sizes W/L are mismatched for the two differential pair MOS devices M1 and M2

• This mismatch causes mismatch in the currents that flow through M1 and M2

– This mismatch results in VO

– So in the input referred offset is…

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 14

Vt Mismatch

• Lastly, consider mismatches in the threshold voltage

• Again, currents I1 and I2 will differ according to the following saturation current equation

– For small ΔVt << 2(VGS-Vth)

– Again, using VOS=VO/Ad (Ad = gmRD and VO =2ΔIRD) we get…

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Mismatch Summary

• The 3 sources of mismatch can be combined into one equation:

– arising from Vt, RD, and W/L mismatches

• Notice that offsets due to ΔRD and ΔW/L are functions of the overdrive voltage VGS –Vt

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Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN