3.4 - BJT DIFFERENTIAL AMPLIFIERS

3.4 - BJT DIFFERENTIAL AMPLIFIERS

INTRODUCTION

Objective

The objective of this presentation is:

1.) Define and characterize the differential amplifier 2.) Show the large-signal and small-signal performance

3.) Show alternate implementations of the differential amplifier

Outline

• Characterization and definitions • Large-signal transconductance • Large-signal voltage transfer • Small-signal performance

• Other characteristics of the differential amplifier • Summary

CHARACTERIZATION AND DEFINITIONS

What is a Differential Amplifier?

A differential amplifier is an amplifier that amplifies the difference between two voltages and rejects the average or common mode value of the two voltages.

Symbol for a differential amplifier:

+ - + vOUT -+ v₁ -+ -v₂ Fig. 5.2-1A

Differential and common mode voltages:

v₁ and v₂ are called single-ended voltages. They are voltages referenced to ac ground. The differential-mode input voltage, v_ID, is the voltage difference between v₁ and v₂. The common-mode input voltage, v_IC, is the average value of v₁ and v_{2 .}

∴ v_ID = v₁ - v₂ and v_IC = v1+v2

2 ⇒ v1 = vIC + 0.5vID and v2 = vIC - 0.5vID

v_OUT = A_VDv_ID ± A_VCv_IC = A_VD(v₁ - v₂) ± A_VC _v1 + v₂ 2_ where

A_VD = differential-mode voltage gain A_VC = common-mode voltage gain + - + v_OUT -vIC v_ID 2 v_ID 2 Fig. 5.2-1B

Differential Amplifier Definitions

• Common mode rejection rato (CMRR)

CMRR =       A_VD A_VC

CMRR is a measure of how well the differential amplifier rejects the common-mode input voltage in

favor of the differential-input voltage. • Input common-mode range (ICMR)

The input common-mode range is the range of common-mode voltages over which the differential amplifier continues to sense and amplify the difference signal with the same gain.

Typically, the ICMR is defined by the common-mode voltage range over which all MOSFETs remain in the saturation region and all BJTs remain in the active region.

• Output offset voltage (V_OS(out))

The output offset voltage is the voltage which appears at the output of the differential amplifier when the input terminals are connected together.

• Input offset voltage (V_OS(in) = V_OS)

The input offset voltage is equal to the output offset voltage divided by the differential voltage gain.

V_OS = VOS(out)

LARGE-SIGNAL TRANSCONDUCTANCE

Transconductance Characteristic of the Differential Amplifier

Consider the following NPN-BJT differential amplifier (sometimes called an emitter-coupled pair):

V_EE v_I1 v_I2 Q1 Q2 I_EE BJTDA01 i_C1 i_C2 + -v_BE1 + -vBE2 Large-Signal Analysis:

1.) Input loop eq.: v_I1 - v_BE1 + v_BE2 - v_I2 = v_I1 - v_I2 - v_BE1 + v_BE2 = v_ID - v_BE1 + v_BE2 = 0 2.) Foward-active region: v_BE1= V_t ln

      i_C1 I_S1 and vBE2= Vt ln_     i_C2 I_S2 3.) If I_S1 = I_S2 then iC1 i_C2 = exp_     v_I1-v_I2 V_t = exp_     v_ID V_t

4.) Nodal current equation at the emitters: -(i_E1+i_E2) = I_EE = _α1

F (iC1 + iC2)

5.) Combining the above equations gives: i_C1 = αFIEE 1 + exp       -v_ID V_t and i_C2 = αFIEE 1 + exp      v_ID V_t

Transconductance Characteristic of the Differential Amplifier - Continued

Plotting the collect current as a function of v_ID:

BJTDA00 0 0.2 0.4 0.6 0.8 1 v_ID V_t -4 -2 0 2 4 -5 -3 -1 1 3 5 i_C αI_EE i_C2 i_C1 Transconductance: g_m1 = ∂_∂iC1 v_ID | Q = α_FI_EE       1+exp       -v_ID V_t 2 V_t = αFIEE 4V_t = I_C1 2V_t when VID = 0 g_m2 = ∂_∂iC2 v_ID | Q = -α_FI_EE       1+exp       v_ID V_t 2 V_t = -αFIEE 4V_t = -I_C2 2V_t when VID = 0

VOLTAGE TRANSFER CHARACTERISTICS

Large-Signal Voltage Transfer Function

Assume load resistors as shown:

∴ v_OD = v_O1-v_O2 = V_CC-i_C1R_C - V_CC + i_C2R_C

= R_C(i_C2-i_C1)

Substituting in the previous expressions and using hyperbolic trig identities gives

v_OD = α_FI_EER_C tanh       -v_ID 2V_t Illustration → V_CC V_EE v_i1 v_i2 Q1 Q2 I_EE BJTDA03 i_{C1 iC2} + -v_BE1 + -vBE2 +vOD -+ -v_O1 + -v_O1 R_C R_C -1 -0.5 0 0.5 1 -4 -2 0 2 4 -5 -3 -1 1 3 5 αFIEERC v_OD v_ID V_t BJTDA04

Emitter Degeneration of the BJT Differential Amplifier

Increases the range over which the emitter-coupled pair behaves as a linear amplifier with lower gain at the cost of lower gain.

V_CC V_EE v_i1 v_i2 Q2 I_EE BJTDA05 i_{C1 iC2} + -v_BE1 + -vBE2 +vOD -R_{C RC} R_E R_E -25 -20 -15 -10 -5 5 10 15 20 25 1 0.5 -0.5 -1 Increasing R_E R_E=0 αFIEERC v_OD v_ID V_t

We know that, i_D = i_C1 - i_C2 = α_FI_EE tanh_ _ v_ID - (R_Ei_D/α_F) 2Vt ≈ αFIEE_     v_ID - (R_Ei_D/α_F) 2Vt

Solving for i_D gives, i_D = i_C1 - i_C2 = αF

I_EEv_ID 2V_t + R_EI_{E E} ∴ g_m(DC) = diD dv_ID = (α_FI_EE) 2V_t + R_EI_{E E} = (α_FI_EE)/2V_t 1+ R_EI_EE/2V_t

SMALL-SIGNAL PERFORMANCE

Differential and Common-mode Small-Signal Performance

The small-signal performance of a differential amplifier can be separated into a differential mode and common mode analysis. This separation allows us to take advantage of the following simplifications.

V_CC V_EE v_{i1 v} i2 Q1 Q2 I_EE BJTDA06 i_{c1 ic2} + -v_be1 + -vbe2 +vod -+ -v_o1 + -v_o2 R_C R_C R_EE V_EE V_CC v_{id v} id Q1 Q2 i_{c1 ic2} + -v_be1 + -vbe2 +vod -+ -v_o1 + -v_o2 R_C R_C 2 ₂ V_CC Differential Mode Analysis V_CC V_EE v_{ic v} ic Q1 Q2 I_EE i_{c1 ic2} + -v_be1 + -vbe2 +vod -+ -v_o1 + -v_o2 R_C R_C 2R_EE V_EE 2 V_EE I_EE 2R_EE V_EE 2 V_CC Common Mode Analysis Half-Circuit Concept:

SMALL-SIGNAL PERFORMANCE

Differential and Common-mode Small-Signal Performance

The small-signal performance of a differential amplifier can be separated into a differential mode and common mode analysis. This separation allows us to take advantage of the following simplifications.

V_CC V_EE v_{i1 v} i2 Q1 Q2 I_EE BJTDA06 i_{c1 ic2} + -v_be1 + -vbe2 +vod -+ -v_o1 + -v_o2 R_C R_C R_EE V_EE V_CC v_{id v} id Q1 Q2 i_{c1 ic2} + -v_be1 + -vbe2 +vod -+ -v_o1 + -v_o2 R_C R_C 2 ₂ V_CC Differential Mode Analysis V_CC V_EE v_{ic v} ic Q1 Q2 I_EE i_{c1 ic2} + -v_be1 + -vbe2 +vod -+ -v_o1 + -v_o2 R_C R_C 2R_EE V_EE 2 V_EE I_EE 2R_EE V_EE 2 V_CC Common Mode Analysis Half-Circuit Concept:

Small-Signal, Common Mode Performance of the BJT Differential Amplifier

Circuit and small-signal model:

V_CC V_EE v_ic Q1 I_EE i_c1 + -v_be1 + -v_o1 R_C 2R_EE V_EE 2 v_ic v_out R_C 2R_EE + -+ -r_π v_π g_mv_π r_o BJTDA08 R_{in Rout} + -Half-circuit performance: R_in1 = r_π1 +(1+β_o1)2R_EE, R_out1= r_o       1+ βο1 2R_EE 2R_EE+r_π1 + 2REE||rπ1 , and v_o1 v_ic = - β_o1R_C r_π1+(1+β_o1)2R_EE Common mode performance:

R_ic = r_π1+(1+β_o1)2R_EE 2 , Roc= ro       1+ β_ο12R_EE 2R_EE+r_π1 + 2REE||rπ1 , and v_oc v_ic = - β_o1R_C r_π1+(1+β_o1)2R_EE where g_m1 = g_m2, r_π1 = r_π2 and β_o1= β_o2.

Common Mode Rejection Ratio (CMRR)

The common mode rejection ratio is a measure of the differential amplifier’s ability to reject the common mode signal and amplify the differential mode signal.

CMRR =       A_{d m} A_cm = _     v_o1/v_id v_o1/v_ic = g_m1R_C 2 β_o1R_C r_π1+(1+β_o1)2R_EE ≈ g_m1R_EE = IEEREE 2V_t

OTHER CHARACTERISTICS OF THE DIFFERENTIAL AMPLIFIER Input Common Mode Voltage Range

The input common mode voltage range (ICMR) is the range of common mode input voltages over which the differential amplifier amplifies the differential signal without significant change.

Consider the following:

V_CC V_EE v_ic(max) Q1 Q2 I_EE BJTDA03 + -V_BE1 + -vBE2 R_{C RC} Q3 V_Bias I_EE 2 I_EE 2 + -I_EER_C 2 + -v_CE(sat) V_CC V_EE v_ic(min) Q1 Q2 I_EE + -V_BE1 + -vBE2 R_{C RC} Q3 V_Bias I_EE 2 I_EE 2 + -v_CE(sat) + -v_CE(sat)

Maximum Input Common Mode Voltage:

v_ic(max) = V_CC - 0.5I_EER_C -v_CE1(sat)+V_BE1

Minimum Input Common Mode Voltage: v_ic(min) = V_EE+v_CE3(sat)+V_BE1

Slew Rate of the BJT Differential Amplifier

Slew rate is a voltage rate limit due to the fact that the current available to charge a capacitor is constant. Slew rate = SR = dvC

dt = i_C

C

where i_C and v_C are the current through and voltage across a capacitor C.

A differential amplifier that has a capacitive load will experience slew rate which is seen as follows:

V_CC V_EE v_i1<<0 v_i2>>0 Q1 Q2 I_EE BJTDA09A i_C1 i_C2 + -v_BE1 + -vBE2 + -v_OD R_C R_C C_s C_s V_EE v_i1<<0 v_i2>>0 Q1 Q2 I_EE + -v_BE1 + -vBE2 + -v_C1 R_C R_C C_o V_CC i_C1=0 C_s Co C_s I_EE 2 i_C2=I_EE I_EE 2 i_Cs i_Cs i_Co Situation at v_OD ≈ 0 v_OD ≈ 0 v_C2

Note that the current in Q1 is 0 and Q2 is I_EE. Therefore, the initial value of i_Co is i_Co = 0.5I_EE - i_Cs. If we define the slew rate across C_o as SR =

i_Co C_o , then iCs = 0.5SR·Cs = i_Co 2C_oCs , i.e. dv_C1 dt = 0.5 dv_OD dt ∴ SR = IEE 2C_o - i_Cs C_o = I_EE 2C_o - SR·C_s 2C_o ⇒ SR _     1 + Cs 2C_o = I_EE 2C_o ⇒ S R = 0.5I_EE C_o+0.5C_s = I_EE 2C_o+C_s

SUMMARY

• A differential amplifier amplifies the difference signal between two voltages and rejects the common mode signal

• The transconductance characteristics of the BJT differential amplifier switches from all of the current in one side to the other side within ±100mV

• The large-signal differential voltage transfer function has the form of hyperbolic tangent

• Emitter degeneration increases the range over which the differential amplifier behaves as a linear amplifier

• The half circuit concept is very useful for analyzing the small-signal differential and common mode performance

• The maximum and minimum input common mode range is:

v_ic(max) = V_CC - 0.5I_EER_C -v_CE1(sat)+V_BE1

v_ic(min) = V_EE+v_CE3(sat)+V_BE1

• The differential amplifier has a slew rate limit of I_EE/C_eq where C_eq is the capacitance seen to ground from either collector.