Operational Amplifiers 1

A useful integrated circuit implementation of an almost idealized amplifier Figure 2.1 Circuit symbol for the op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

Often Vcc = Vee for balance with respect to ground (important as it maintains symmetry when overdriven) Figure 2.2 The op amp shown connected to dc power supplies. Microelectronic Circuits - Fifth Edition Sedra/Smith

Infinite input impedance (no source loading) Idealized OpAmp: Infinite input impedance (no source loading) Extremely high gain (A – watch out for oscillation) Zero output impedance (can deliver power to a load) Differential input (high degree of “common mode” rejection) Sometimes differential output Figure 2.3 Equivalent circuit of the ideal op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

- input + input Vid = v2 - v1 ; vicm = (v2 + v1 )/2 Figure 2.4 Representation of the signal sources v1 and v2 in terms of their differential and common-mode components. Microelectronic Circuits - Fifth Edition Sedra/Smith

v3 = GmR(v2-v1) ; Gm = 10 mA/V, R = 10k,  = 100 ; A = 104 V/V or 80 dB ; Input Impedance? Output Impedance? Figure E2.3 Microelectronic Circuits - Fifth Edition Sedra/Smith

Vout = -Vin*(R2/R1) ; hint: v1 = 0 since any voltage differential means vo is large and i1 = 0 so a simple voltage divider yields the result (next slides) Figure 2.5 The inverting closed-loop configuration. Microelectronic Circuits - Fifth Edition Sedra/Smith

Why is the negative input voltage equal to the positive input voltage (0, “virtual ground”)? Figure 2.6 Analysis of the inverting configuration. The circled numbers indicate the order of the analysis steps. Microelectronic Circuits - Fifth Edition Sedra/Smith

Finite gain: G = -( R2/R1 )/(1 + (1 + R2/R1 )/A ); hint: v1 = -vo / A Figure 2.7 Analysis of the inverting configuration taking into account the finite open-loop gain of the op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

v1=0, i2=i1, node eq. at x, A = -(R2/R1)*( 1 + R4/R2 + R4/R3 ) Figure 2.8 Circuit for Example 2.2. The circled numbers indicate the sequence of the steps in the analysis. Microelectronic Circuits - Fifth Edition Sedra/Smith

Figure 2. 9 A current amplifier based on the circuit of Fig. 2. 8 Figure 2.9 A current amplifier based on the circuit of Fig. 2.8. The amplifier delivers its output current to R4. It has a current gain of (1 + R2/R3), a zero input resistance, and an infinite output resistance. The load (R4), however, must be floating (i.e., neither of its two terminals can be connected to ground). Microelectronic Circuits - Fifth Edition Sedra/Smith

TransResistance Amplifier Ex2 TransResistance Amplifier Ex2.5; Ri=0 (vo=v1=0), Rm=-10k, Ro=0, vo=-5V; see table 1.1 for the definition of TransResistance Figure E2.5 Microelectronic Circuits - Fifth Edition Sedra/Smith

Ex 2.6: Vo doesn’t change with load – low output impedance Figure E2.6 Microelectronic Circuits - Fifth Edition Sedra/Smith

We have an adder! (use superposition – assumes linearity!) Figure 2.10 A weighted summer. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0211.jpg Figure 2.11 A weighted summer capable of implementing summing coefficients of both signs. Microelectronic Circuits - Fifth Edition Sedra/Smith

Now v1=vi ; A = 1 + R2/R1 ; see next slide Figure 2.12 The noninverting configuration. Microelectronic Circuits - Fifth Edition Sedra/Smith

Note the “1+” Figure 2.13 Analysis of the noninverting circuit. The sequence of the steps in the analysis is indicated by the circled numbers. Microelectronic Circuits - Fifth Edition Sedra/Smith

A simple and useful Op Amp circuit: can be used as a power amplifier can be used to isolate stages in a filter – allows an easy design methodology Figure 2.14 (a) The unity-gain buffer or follower amplifier. (b) Its equivalent circuit model. Microelectronic Circuits - Fifth Edition Sedra/Smith

Voltage divider on the positive input; A = 1+9 = 10; v+=0. 6v1+0 Voltage divider on the positive input; A = 1+9 = 10; v+=0.6v1+0.4v2 so vo=6v1+4v2 Figure E2.9 Microelectronic Circuits - Fifth Edition Sedra/Smith

See Ex 2.13 Figure E2.13 Microelectronic Circuits - Fifth Edition Sedra/Smith

As in Figure 2.4 Figure 2.15 Representing the input signals to a differential amplifier in terms of their differential and common-mode components. Microelectronic Circuits - Fifth Edition Sedra/Smith

See next two slides Figure 2.16 A difference amplifier. Microelectronic Circuits - Fifth Edition Sedra/Smith

Again assuming linearity. Figure 2.17 Application of superposition to the analysis of the circuit of Fig. 2.16. Microelectronic Circuits - Fifth Edition Sedra/Smith

If R3=R1 and R4=R2; Ad=R2/R1 then Acm=0 Figure 2.18 Analysis of the difference amplifier to determine its common-mode gain Acm ; vO / vIcm. Microelectronic Circuits - Fifth Edition Sedra/Smith

But the input impedance is low! 2*R1 – not good Figure 2.19 Finding the input resistance of the difference amplifier for the case R3 = R1 and R4 = R2. Microelectronic Circuits - Fifth Edition Sedra/Smith

Requires good resistor matching available in IC’s with careful design Common mode rejection in 2nd stage. First stage provides high input impedance. Requires good resistor matching available in IC’s with careful design Resistor value set by resistivity*length/(width*depth) Resistivity is variable chip to chip and with temperature, but consistent in each chip if temperature differences are minimized Length and width are accurately set by the “mask” Depth is variable chip to chip (diffusion) but again consistent on a particular chip Figure 2.20 A popular circuit for an instrumentation amplifier: (a) Initial approach to the circuit; (b) The circuit in (a) with the connection between node X and ground removed and the two resistors R1 and R1 lumped together. This simple wiring change dramatically improves performance; (c) Analysis of the circuit in‘ (b) assuming ideal op amps. Microelectronic Circuits - Fifth Edition Sedra/Smith

Watch out for “pot” noise, modern circuits use a voltage controlled resistor (FET). Figure 2.21 To make the gain of the circuit in Fig. 2.20(b) variable, 2R1 is implemented as the series combination of a fixed resistor R1f and a variable resistor R1v. Resistor R1f ensures that the maximum available gain is limited. Microelectronic Circuits - Fifth Edition Sedra/Smith

As long as the open loop gain is high compared to the feedback-determined gain, not a problem. See next slide Amplifier roll-off needs to be controlled to maintain stability (ch.8 – not in the first course). Figure 2.22 Open-loop gain of a typical general-purpose internally compensated op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

Flat response till the open-loop gain drops to near the ideal closed loop gain. Figure 2.23 Frequency response of an amplifier with a nominal gain of +10 V/V. Microelectronic Circuits - Fifth Edition Sedra/Smith

Same here Figure 2.24 Frequency response of an amplifier with a nominal gain of –10 V/V. Microelectronic Circuits - Fifth Edition Sedra/Smith

The output current limit restricts you to higher load impedances You want symmetrical clipping (only generate odd harmonics) so music sounds better The output current limit restricts you to higher load impedances Figure 2.25 (a) A noninverting amplifier with a nominal gain of 10 V/V designed using an op amp that saturates at ±13-V output voltage and has ±20-mA output current limits. (b) When the input sine wave has a peak of 1.5 V, the output is clipped off at ±13 V. Microelectronic Circuits - Fifth Edition Sedra/Smith

This can also cause signal distortion The slew rate is the fastest rate of change that the output can deliver. This can also cause signal distortion Figure 2.26 (a) Unity-gain follower. (b) Input step waveform. (c) Linearly rising output waveform obtained when the amplifier is slew-rate limited. (d) Exponentially rising output waveform obtained when V is sufficiently small so that the initial slope (vtV) is smaller than or equal to SR. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0227.jpg Figure 2.27 Effect of slew-rate limiting on output sinusoidal waveforms. Microelectronic Circuits - Fifth Edition Sedra/Smith

We need to compensate for this offset in real circuits. Real OpAmps: the devil is in the details. Here the text discussed the “offset voltage” (referred to the input) We need to compensate for this offset in real circuits. Figure 2.28 Circuit model for an op amp with input offset voltage VOS. Microelectronic Circuits - Fifth Edition Sedra/Smith

Ex 2.23 Figure E2.23 Transfer characteristic of an op amp with VOS = 5 mV. Microelectronic Circuits - Fifth Edition Sedra/Smith

The offset gets amplified! Figure 2.29 Evaluating the output dc offset voltage due to VOS in a closed-loop amplifier. Microelectronic Circuits - Fifth Edition Sedra/Smith

This is a manual operation, not great, but I’ve used it. Figure 2.30 The output dc offset voltage of an op amp can be trimmed to zero by connecting a potentiometer to the two offset-nulling terminals. The wiper of the potentiometer is connected to the negative supply of the op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

The DC offset gain is 2, the signal (AC) gain is –R2/R1 The offset causes a DC offset in the output and asymmetrical clipping. Figure 2.31 (a) A capacitively coupled inverting amplifier, and (b) the equivalent circuit for determining its dc output offset voltage VO. Microelectronic Circuits - Fifth Edition Sedra/Smith

Input bias current offset – another imperfection Figure 2.32 The op-amp input bias currents represented by two current sources IB1 and IB2. Microelectronic Circuits - Fifth Edition Sedra/Smith

The input resistor translates the offset current into an offset voltage. Figure 2.33 Analysis of the closed-loop amplifier, taking into account the input bias currents. Microelectronic Circuits - Fifth Edition Sedra/Smith

Balancing the input resistors helps. Figure 2.34 Reducing the effect of the input bias currents by introducing a resistor R3. Microelectronic Circuits - Fifth Edition Sedra/Smith

Balance in the input circuit is maintained in an AC coupled OpAmp by matching R2 Figure 2.35 In an ac-coupled amplifier the dc resistance seen by the inverting terminal is R2; hence R3 is chosen equal to R2. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0236.jpg Figure 2.36 Illustrating the need for a continuous dc path for each of the op-amp input terminals. Specifically, note that the amplifier will not work without resistor R3. Microelectronic Circuits - Fifth Edition Sedra/Smith

Same as with resistances, but now we have “s” (the Laplace transform complex frequency) ZL=sL; ZC=1/sC; ZR=R Figure 2.37 The inverting configuration with general impedances in the feedback and the feed-in paths. Microelectronic Circuits - Fifth Edition Sedra/Smith

Set s=j to get the complex frequency response A=R2||(1/sC)/R1; DC gain = -R2/R1 High frequency gain =0; 3dB point determined by R2C2 Set s=j to get the complex frequency response Figure 2.38 Circuit for Example 2.6. Microelectronic Circuits - Fifth Edition Sedra/Smith

Used to build “Analog Computers” (along with adders and multipliers) They were popular (and very useful) when I was an undergraduate – 1960’s Interestingly, Simulink (Matlab, a very useful tool) simulates an analog computer on a digital computer Figure 2.39 (a) The Miller or inverting integrator. (b) Frequency response of the integrator. Microelectronic Circuits - Fifth Edition Sedra/Smith

Details again. Put a large resistor across C to fix (see slide after next) Figure 2.40 Determining the effect of the op-amp input offset voltage VOS on the Miller integrator circuit. Note that since the output rises with time, the op amp eventually saturates. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0241.jpg Figure 2.41 Effect of the op-amp input bias and offset currents on the performance of the Miller integrator circuit. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0242.jpg Figure 2.42 The Miller integrator with a large resistance RF connected in parallel with C in order to provide negative feedback and hence finite gain at dc. Microelectronic Circuits - Fifth Edition Sedra/Smith

Yes, it does the integral Figure 2.43 Waveforms for Example 2.7: (a) Input pulse. (b) Output linear ramp of ideal integrator with time constant of 0.1 ms. (c) Output exponential ramp with resistor RF connected across integrator capacitor. Microelectronic Circuits - Fifth Edition Sedra/Smith

High frequency noise is a problem here Put a resistor across the input capacitor to limit HF gain And a small capacitor across the output resistor to provide a second HF “pole” Figure 2.44 (a) A differentiator. (b) Frequency response of a differentiator with a time-constant CR. Microelectronic Circuits - Fifth Edition Sedra/Smith

Spice models (used in Multisym, OrCad or LTspiceIV) You can also use a manufacturer supplied spice model Figure 2.45 A linear macromodel used to model the finite gain and bandwidth of an internally compensated op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0246.jpg Figure 2.46 A comprehensive linear macromodel of an internally compensated op amp. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0247.jpg Figure 2.47 Frequency response of the closed-loop amplifier in Example 2.8. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0248.jpg Figure 2.48 Step response of the closed-loop amplifier in Example 2.8. Microelectronic Circuits - Fifth Edition Sedra/Smith

Low noise (small signal amplification) The 741 was an early, practical, OpAmp and is still in use (available at low cost) Many better ones, specialized for particular characteristics are available: Low noise (small signal amplification) High power out/slew rate (Output stages) Low power drain (battery operation) High voltage operation (large voltage swings) Figure 2.49 Simulating the frequency response of the µA741 op-amp in Example 2.9. Microelectronic Circuits - Fifth Edition Sedra/Smith

Stability is almost guaranteed by the low phase shift (just over 90) at unity gain (0 dB) Figure 2.50 Frequency response of the µA741 op amp in Example 2.9. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0251.jpg Figure 2.51 Circuit for determining the slew rate of the µA741 op amp in Example 2.9. Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_0252.jpg Figure 2.52 Square-wave response of the µA741 op amp connected in the unity-gain configuration shown in Fig. 2.51. Microelectronic Circuits - Fifth Edition Sedra/Smith

The remaining 31 figures are from end-of-chapter problems. Figure P2.2 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02008.jpg Figure P2.8 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02016.jpg Figure P2.16 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02022.jpg Figure P2.22 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02025.jpg Figure P2.25 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02030.jpg Figure P2.30 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02031.jpg Figure P2.31 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02032.jpg Figure P2.32 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02033.jpg Figure P2.33 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02034.jpg Figure P2.34 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02035.jpg Figure P2.35 Microelectronic Circuits - Fifth Edition Sedra/Smith

D to A Converter Figure P2.43 Microelectronic Circuits - Fifth Edition Sedra/Smith

Analog Voltmeter (I used to use a VTVM) Figure P2.46 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02047.jpg Figure P2.47 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02049.jpg Figure P2.49 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02050.jpg Figure P2.50 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02051.jpg Figure P2.51 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02059.jpg Figure P2.59 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02062.jpg Figure P2.62 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02068.jpg Figure P2.68 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02069.jpg Figure P2.69 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02070.jpg Figure P2.70 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02071.jpg Figure P2.71 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02077.jpg Figure P2.77 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02078.jpg Figure P2.78 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02108.jpg Figure P2.108 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02117.jpg Figure P2.117 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02118.jpg Figure P2.118 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02119.jpg Figure P2.119 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02122.jpg Figure P2.122 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02125.jpg Figure P2.125 Microelectronic Circuits - Fifth Edition Sedra/Smith

sedr42021_p02126.jpg Figure P2.126 Microelectronic Circuits - Fifth Edition Sedra/Smith