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- 1. Signals and Electronic System Design Limitations of Operational Amplifiers © Linear Mario Mata mario.mata@gcu.ac.uk M204
- 2. 2 Signals and electronic systems design M Mata Contents • Input currents • Closed-loop input and output impedances • Slew rate • OpAmps open-loop and closed-loop bandwidths – GBW product • Dynamic performance of amplifiers – Effects of capacitive coupling – Effects of parasitic series L and shunt C – Effect on pulse response • Linear and nonlinear distortion – Total Harmonic Distortion (THD) • Stability: Phase margin. Compensation • Driving capacitive loads
- 3. 3 Signals and electronic systems design M Mata Input currents • The input terminals of OpAmps are connected to the base (or gate) of the input stage transistors (thus the large input impedance Rin) • But this also means that some input current must be present to set up the bias point of the internal transistors i+ i- i- i+ • Assuming i+,i- ~ 0 in calculations implies that the feedback current iF needs to be iF >> i+,i- : iF ~mA for BJT-based OpAmps, or nA for FET- based ones. This limits practical values for R1, RF • The OpAmp datasheet gives values for the average of both input currents (input bias current) and also for the unbalance (difference) between them (input offset current): BJT-based OpAmps: iB~80nA, ioffs~20nA FET-based OpAmps: iB~50pA, ioffs~20pA
- 4. 4 Signals and electronic systems design M Mata Input currents compensation • The effect of these currents is relevant when working near Vin=0V. Then ideally VO=0V, and feedback current iF~0A as well. As VO~0, input current i- is flowing by the shunt of R1 and RF Vo~0V R1 Vin=0V RF Rin + - i+ i- R1 RF Vo~0V i- V- This unbalances the input: Which in turn makes VO≠0V: • This is easily compensated by causing a similar unbalance in V+ by inserting a compensation resistor RCOMP=R1||RF at this input: R1 Vin RF Rin + - i+ i- RCOMP This which works as long as i+ ≈ i- In practice it compensates for the input bias current iB, but is still affected by the (smaller) unbalance ioffs = |i+- i-|
- 5. 5 Signals and electronic systems design M Mata Design Example Add the corresponding input current compensation resistors to the following amplifier stages: 10kW 10kW VO VIN VIN VO 10kW 100kW
- 6. 6 Signals and electronic systems design M Mata Input and output impedances • By design, OpAmps show: • A very large input impedance Rin • Rin~107 for BJT-based OpAmps • Rin~109 for FET-based OpAmps • A low output impedance RO • Typically RO~50W • When used for linear applications (in closed-loop circuits) the feedback loop can also improve the overall input and output impedances
- 7. 7 Signals and electronic systems design M Mata Closed-Loop Input Impedance Closed-loop input impedance for the non-inverting amplifier: Vo R1 Vin Rf Rin Zin + - Ro iin Vin RF + - R1 Vo AOL Feedback ratio As AOL>>1: ACL=1/ Closed-loop impedance is larger than Rin! As AOL = AOL/ACL >> 1:
- 8. 8 Signals and electronic systems design M Mata Closed-Loop Input Impedance Closed-loop input impedance for the inverting amplifier Vin + - R1 Vo RF AOL iin iF Rin i- as long as we keep iF>>i- (R1,RF<<Rin) As AOLR1>>R1+RF: ACL = 1 - 1/ As AOLR1>>R1+RF: Closed-loop impedance is just R1
- 9. 9 Signals and electronic systems design M Mata Closed-Loop Output Impedance Closed-loop output impedance for non-inverting amp Typ AOL~106, R0 ~50 to 100 W Notice that output impedance is the same for the inverting amplifier Closed-loop output impedance is reduced below R0 R1 RF Rin Ro + - io VO 𝑉 = 𝐴 𝑉 − 𝑉 O O CL O i V Z ) ( Feedback ratio As RO<<(1+ AOL)(R1+RF):
- 10. 10 Signals and electronic systems design M Mata Design example Design an inverting amplifier with a closed-loop gain of -10, having an input impedance of at least 10kW Vs Vo Rf R1 RC + - The closed-loop voltage gain is given by: As the input impedance is Zin(CL)~R1: We may then take for instance and then solve for RF:
- 11. 11 Signals and electronic systems design M Mata Design Example An op-amp non-inverting amplifier is used as a pre-amp having a closed- loop gain ACL = 10. To match impedances to those required by the previous and posterior circuits, the amplifier must present input and output impedances of 50W. Suggest what additional components will be required to achieve this. Draw the resulting circuit.
- 12. 12 Signals and electronic systems design M Mata Slew rate The output of an operational amplifier cannot change instantly. This is especially relevant for step-like and square signals If the input changes suddenly, the output will change “as fast as possible”. The parameter measuring the maximum rate of change of the output is the Slew Rate (SR), usually measured in V/ms. Example: Typical SR values are around 0.5V/ms t Vin VO=5Vin 1ms SR V Ideal output For instance, if Vin goes from 0V to 1V and the gain ACL is 5: • The output final value will be VO = 5Vin = 5V • But it will take some time for the output to reach this final value: 𝑡 → = Δ𝑉 𝑆𝑅 = 5 − 0 𝑉 0.5 𝑉/𝜇𝑠 = 10𝜇𝑠 If the period of the signal is small, this effect is a limiting factor. I.e. f=50kHz T=20ms the rise time is already half the period! Real output
- 13. 13 Signals and electronic systems design M Mata OpAmp bandwidth – internal compensation • OpAmps are designed to present a huge open-loop gain (usually represented as AV or AOL). Typically AOL~106 – This causes instability: the open-loop output saturates – Closed-loop circuits can stabilize the OpAmp, allowing linear applications (such as amplifiers) to be implemented • To improve stability, an internal capacitor is often added in the design of the OpAmp (inserting a pole at s = -wCol) – Open-loop cut-off frequency wCol is chosen in the order of 10 rad/s – This overwhelms the effect of any internal parasitic capacitances. It’s said to be an internally compensated OpAmp. Most OpAmps are! this is a DC model Frequency model for an internally compensated OpAmp
- 14. 14 Signals and electronic systems design M Mata OpAmp Open-loop frequency response Typical open-loop response of a compensated amplifier The constant -20dB/dec rate of change of gain with frequency implies that there is only one dominant pole (the internal compensation capacitor) Having a single dominant pole makes it always stable in closed-loop AOL(dB) Open-loop critical frequency fCol fT Midrange f (Hz) 106 0 ~10Hz 1Hz ~1MHz -20dB/dec Open-loop unity-gain frequency The frequency at which the OpAmp shows unity open- loop gain AOL=1, fT, is a relevant parameter
- 15. 15 Signals and electronic systems design M Mata VO VIN OpAmp closed-loop bandwidth. GBP • Example: non-inverting amplifier • Closed-loop gain is • Closed-loop cut-off frequency is: • Both wCol and AOL are internal design parameters of the OpAmp. The product wColAOL is a constant, usually named GBP or GBWP (gain- bandwidth product). Typ. GBP ~ 1MHz to 10MHz • Closed-loop bandwidth wCL is then given by: = feedback ratio 𝑉 𝑠 = 𝐴 𝑉 𝑠 − 𝑉 𝑠 𝛽 𝑠/𝜔 + 1 𝑉 𝑠 𝑠/𝜔 + 1 = 𝐴 𝑉 𝑠 − 𝑉 𝑠 𝛽 𝑉 𝑠 𝑠/𝜔 + 1 + 𝐴 𝛽 = 𝐴 𝑉 𝑠 As AOL>>1: wCL ACL 𝑪𝑳 𝑪𝒐𝒍 𝑶𝑳 𝑪𝑳 𝑪𝑳
- 16. 16 Signals and electronic systems design M Mata OpAmp closed-loop bandwidth. GBP, fT For open-loop response curves with -20dB/dec roll-off (compensated OpAmps), the closed-loop -3dB bandwidth limit (cut-off frequency wCL) occurs at the ACL intercept with the open-loop gain curve AOL fCol fT fCL ACL=100 Open-loop cut-off frequency -20dB/dec -3dB 𝑪𝑳 𝑪𝑳 𝑪𝒐𝒍 𝑶𝑳 Datasheets show either GBP (gain bandwidth product) or fT (unity gain bandwidth) fT = Open-loop Unity-gain frequency Then the open-loop unity-gain frequency fT is coincident with the GBP value: aka Unity-gain bandwidth ACL=10 ACL=1
- 17. 17 Signals and electronic systems design M Mata OpAmp closed-loop bandwidth. GBP A certain compensated op-amp has an open-loop gain AOL=106 and a unity gain bandwidth fT=2MHz. For the following situations, discuss whether this gain can be sustained at a signal frequency of 300kHz. If not, suggest what value of unity- gain bandwidth the op-amp should have. a) Non-inverting amp. having a DC closed-loop gain ACL=10. b) Inverting amplifier having a DC closed-loop gain ACL= -10. • For inverting stages, the BW is calculated from the GBP assuming the NON-INVERTING gain (aka NOISE gain). • Example:
- 18. 18 Signals and electronic systems design M Mata Example popular Op-amps Parameter LM741 LF411 AOL fT SR Zin(OL) Zout(OL) Voff IB Ioff (Bipolar) (FET) 2 x 105 1.5MHz 0.5V/ms 2MW 75W 1.0mV 80nA 20nA 2 x 105 4.0MHz 15V/ms 1012 W 75W 0.8mV 50pA 25pA DC open-loop gain Unity-gain bandwidth Output offset voltage (at V+=V-) Open-loop input impedance Open-loop output impedance Input bias current IB=(i++i-)/2 Input offset current Ioff =|i+-i-| Slew rate CMRR 95dB 100dB Common-mode rejection ratio
- 19. 19 Signals and electronic systems design M Mata Dynamic Performance of Amplifiers • The Slew Rate models some limitations for dynamic signals in time domain (specially relevant for square signals) • The GBP models these limitations in frequency domain • In a wider context, no real amplifier has equal gain at all frequencies, and will exhibit a tail-off in performance both at high and low frequencies • This non-ideal behavior is a result of: - Limitations of the active devices in the circuit (i.e. OpAmps) - Series and shunt capacitances in the circuit, added to achieve the required operation - Parasitic impedances due to wiring and traces in the circuit layout: how components are arranged and interconnected in the printed circuit board (PCB)
- 20. 20 Signals and electronic systems design M Mata Effect of capacitive coupling • When input signals contain an unwanted DC component, it is usually blocked by using a series coupling capacitor It blocks the DC component of the signal while letting higher frequencies through • As a side effect, this also leads to a drop of gain at lower frequencies (as the series C behaves as a high-pass filter): C R Vin Vo C C s s s H w w 1 ) ( RC C 1 w C w C w Phase lead
- 21. 21 Signals and electronic systems design M Mata Effect of parasitic series L / shunt C • Stray wiring or parasitic capacitances and inductances are often integral to the components, or to how they are wired on a PCB • As a side effect, this causes the gain of the amplifier to drop at high frequencies (as the series L and/or shunt C behaves as a low-pass filter): R Vin C Vo C s s H w 1 1 ) ( RC C 1 w C w C w Phase lag
- 22. 22 Signals and electronic systems design M Mata Combined effect: band-pass behaviour fL fH Lower cut-off frequency Upper cut-off frequency Phase lead Phase lag Bandwidth (BW) Intended operation ACL=
- 23. 23 Signals and electronic systems design M Mata t90 t10 tr 90% 10% 100% Pulse / square signal response The described effects may end up producing a poor amplifier response to a pulsed input: High freq Low freq (~DC) Low freq (~DC) Rise time High freq Many parasitic poles may show oscillations (if underdamped) vi(t) t Slew rate Limited by slew rate and bandwidth BW: 𝒓
- 24. 24 Signals and electronic systems design M Mata Avoiding linear distortion in amplification constant gain over the range of frequencies contained in the input signal To avoid linear waveform distortion, an amplifier should have: f f A phase delay that is always 0 (usually not feasible), or linear versus frequency Then all frequency components will be delayed by the same amount of time: All frequency components delayed by the same time k 0º
- 25. 25 Signals and electronic systems design M Mata Non-linear Distortion: saturation • Any amplifier’s output voltage range (output voltage swing) is limited by its supply voltages +VCC and –VCC: • vOmax < +VCC • vOmin > -VCC • If vO = ACLvin exceeds these limits, saturation (clipping) occurs this is a non-linear distortion vOmax vOmin vi vO slope = ACL • Any DC component in the input makes clipping more likely to occur, as the DC offset is also amplified
- 26. 26 Signals and electronic systems design M Mata Measuring distortion • The Total Harmonic Distortion (THD) is a frequency-based measure of how much distortion is introduced by these undesirable effects Fundamental harmonic Harmonic distortion components
- 27. 27 Signals and electronic systems design M Mata Total Harmonic Distortion • THD is defined as the amplitude ratio of the geometric average of all harmonic distortion terms to the fundamental harmonic: • A well-designed audio amplifier might have a THD of about 0.01% at rated power output % 100 ..... 2 4 2 3 2 2 f V f V f V f V THD Vf is amplitude of fundamental frequency V2f is amplitude of second harmonic Vnf is amplitude of nth harmonic • Harmonic amplitudes Vnf can be analytically calculated using the Fourier transform, or measured using a spectrum analyser It is a digital instrument that digitizes the signal with a very high sampling ratio, then calculates its FFT (Fast Fourier Transform)
- 28. 28 Signals and electronic systems design M Mata Total Harmonic Distortion Example: Calculate the THD of an audio power amplifier driving an 8W load from the following measurements using a 1kHz test signal. The measured rms voltage of the 1kHz signal at the load is 2.826V. THD V 2 f 2 V 3f 2 V 4 f 2 ..... V f 100% THD 0.01132 0.06332 0.00142 ..... 2.8258 100% THD 2.38% Harmonic Voltage (Vrms) 2 3 4 5 6 7 8 9 0.0113 0.0633 0.0014 0.02 0.0005 0.0005 0.0003 0.0006
- 29. 29 Signals and electronic systems design M Mata Stability: Gain and Phase Margin o V + - AOL(f) Vin Vo Instability sets in if, at any given frequency, it may happen that 1 + AOL(f) = 0 Single dominant pole systems can only reach a phase of -90º always stable That’s why most OpAmps are internally compensated this way Open-loop gain depends on frequency: AOL=AOL(f) 𝟐𝟎 𝐥𝐨𝐠 𝜷𝑨𝑶𝑳 𝒇 𝒅𝑩 𝑶 𝒊𝒏 𝑶𝑳 𝑶𝑳 Gain margin and phase margin measure stability (how far from the above condition) Phase margin Gain margin occurring at the same frequency
- 30. 30 Signals and electronic systems design M Mata Stability Criteria Gain(dB) fC1 fT f (Hz) fC2 fC3 AOL(mid f) -20dB/dec -40dB/dec -60dB/dec • If the closed-loop gain ACL= intersects the open-loop response on a -40db/dec or greater region (across more than 1 dominant pole), instability can occur as phase may be -180º for |AOL(f)| = 1 However multiple-pole systems can reach a phase of -180º or more (-90nº for n poles) may be unstable!. For instance, for an uncompensated OpAmp: 1st pole 2nd pole 3rd pole -90º phase -180º phase -270º phase • To ensure stability, any OpAmp must be operated at a closed-loop gain ACL such that the roll-off rate at the corresponding BW does not exceed -20dB/dec (fC2 in the example) BW ACL fC1, fC2, fC3, … internal parasitic poles
- 31. 31 Signals and electronic systems design M Mata Phase compensation AOL fT f (Hz) AOL(mid f) Uncompensated open-loop gain (showing 3-poles behaviour) Effect of phase compensation on OpAmp response With some compensation BW1 ACL With more compensation (2-poles behaviour) BW2 Inserting one additional dominant pole (external capacitor) makes the system approach the single-pole behaviour (-20dB/dec) -20dB/dec -60dB/dec Gain (dB) -40dB/dec With no (or too little) compensation, operating at ACL the bandwidth is BW1. Phase can reach -180º (2-pole gain roll-off) instable Adding enough compensation the bandwidth decreases to BW2. But now phase is under -180º stable
- 32. 32 Signals and electronic systems design M Mata Phase Compensation example LM101A uncompensated OpAmp IC pins 1, 8: compensation terminals C1: external compensation capacitor AOL (dB) AOL (dB) ACL Uncompensated C1=3 pF C1=30 pF
- 33. 33 Signals and electronic systems design M Mata Driving Capacitive Loads • Amplifier load capacitance is a potential problem - A capacitive load CL inserts an additional pole at s = -ROCL, decreasing phase margin (increasing overall phase lag, potentially over -180º) • ZL can be parasitic: PCB capacitance can build up quickly - For instance on a typical 0.03” G10 dielectric 2-sided PCB, a 0.025” trace running over a ground plane will create a capacitance of about 22pF/foot • Even a relatively small load capacitance (i.e., <100 pF) can be troublesome at frequencies near the BW limit; even if not causing outright oscillation, it can still stretch amplifier settling time
- 34. 34 Signals and electronic systems design M Mata External Compensation – Series Resistor • At high frequencies, CL is dominant over any load resistance RL: • Then RSERIES keeps the OpAmp output resistance and feedback network separated from the capacitive load ZL • Usually RSERIES values from 5Ω to 15Ω are sufficient SERIES RESISTOR COMPENSATION TO STABILIZE AN OPAMP DRIVING CAPACITIVE LOADS ZL=ZCL||RL 𝑺𝑬𝑹𝑰𝑬𝑺 𝑪𝑳 𝑳 𝑺𝑬𝑹𝑰𝑬𝑺 𝑳 𝑳 𝑳 𝝎→ 𝑺𝑬𝑹𝑰𝑬𝑺 𝑳 𝑳 𝑳 𝑺𝑬𝑹𝑰𝑬𝑺 𝑳