QucsA TutorialModelling Operational Ampliers

Mike BrinsonCopyright c 2006, 2007 Mike Brinson Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation. A copy of the license is included in the section entitled GNU Free Documentation License.

IntroductionOperation ampliers (OP AMP) are a fundamental building block of linear electronics. They have been widely employed in linear circuit design since they were rst introduced over thirty years ago. The use of operational amplier models for circuit simulation using SPICE and other popular circuit simulators is widespread, and many manufacturers provide models for their devices. In most cases, these models do not attempt to simulate the internal circuitry at device level, but use macromodelling to represent amplier behaviour as observed at the terminals of a device. The purpose of this tutorial note is to explain how macromodels can be used to simulate a range of the operational amplier properties and to show how macromodel parameters can be obtained from manufacturers data sheets. This tutorial concentrates on models that can be simulated using Qucs release 0.0.9.

The Qucs built-in operational amplier modelQucs includes a model for an ideal operational amplier. Its symbol can be found in the nonlinear components list. This model represents an operational amplier as an ideal device with dierential gain and output voltage limiting. The model is intended for use as a simple gain block and should not be used in circuit simulations where operational amplier properties are crucial to overall circuit performance. Fig. 1 shows a basic inverting amplier with a gain of ten, based on the Qucs OP AMP model. The simulated AC performance of this circuit is shown in Fig. 2. From Fig. 2 it is observed that the circuit gain and phase shift are constant and do not change as the frequency of the input signal is increased. This, of course, is an ideal situation which practical operational ampliers do not reproduce. Let us compare the performance of the same circuit with the operational amplier represented by a device level circuit. Shown in Fig. 3 is a transistor circuit diagram for the well known UA741 operational amplier1 . The gain and phase results for the circuit shown in Fig. 1, where the OP AMP is modelled by the UA741 transistor level model, are given in Fig. 4. The curves in this gure clearly illustrate the dierences between the two simulation models. When simulating circuits that include operational ampliers the quality of the OP AMP model can often be a limiting factor in the accuracy of the overall simulation results. Accurate OP AMP models normally include a range of the following device characteristics: (1) DC and AC dierential gain, (2) input bias current, (3) input current and voltage osets, (4) input impedance, (5) common mode eects, (6) slew rate eects, (7) output impedance, (8) power supply rejection eects, (9) noise, (10) output voltage limiting, (11) output current limiting and (12) signal overload recovery eects. The exact mix of selected properties largely depends on the purpose for which the model is being used; for example,The UA741 operational amplier is one of the most studied devices. It is almost unique in that a transistor level model has been constructed for the device. Details of the circuit operation and modelling of this device can be found in (1) Paul R. Grey et. al., Analysis and Design of Analog Integrated Circuits, Fourth Edition, 2001, John Wiley and Sons INC., ISBN 0-471-32168-0, and (2) Andrei Vladimirescu, The SPICE book, 1994, John Wiley and Sons, ISBN 0-471-60926-9.1

1

Equation R3 R=47 k Vin V1 U=1 V R4 R=4.7 k OP1 G=1e6 Eqn1 d1=dB(Vout.v) d2=phase(Vout.v)

Vout

dc simulationDC1

ac simulationAC1 Type=log Start=1 Hz Stop=100 MHz Points=801

Figure 1: Qucs schematic for a basic OP AMP inverting amplier:Qucs OP AMP has G=1e6 and Umax=15V. if a model is only required for small signal AC transfer function simulation then including the output voltage limiting section of an OP AMP model is not necessary or indeed may be considered inappropriate. In the following sections of this tutorial article macromodels for a number of the OP AMP parameters listed above are developed and in each case the necessary techniques are outlined showing how to derive macromodel parameters from manufacturers data sheets.

2

0

Vout.v

-10

-20

1

10

100

1e3

1e4 Frequency Hz

1e5

1e6

1e7

1e8

40

dB(Vout.v)

20

0

1

10

100

1e3

1e4 Frequency Hz

1e5

1e6

1e7

1e8

400 phase(Vout.v) Degrees

200

0

1

10

100

1e3

1e4 Frequency Hz

1e5

1e6

1e7

1e8

Figure 2: Gain and phase curves for a basic OP AMP inverting amplier.

3

T6 P_VCC

T7

T14

T15

T24 T27

R10 R=50 VIN_N R5 R=39k C1 C=30pF T18 R11 R=25

T3 VIN_P

T4

T21 T26

VOUT

T2

T5 T17 T13

R9 R=40k T23

T11

T12

T8

T10 T16 R1 R=3k R6 R=50k

T19

R4 R=1k

R3 R=50k

R2 R=1k

R7 R=50

R8 R=50k

T20

P_VEE

Figure 3: Transistor level circuit for the UA741 operational amplier.

4

10

Vout.v

5

0 1 10 100 1e3 1e4 Frequency Hz 1e5 1e6 1e7

20 dB(Vout.v)

0

-20 1 200 10 100 1e3 Frequency Hz 1e4 1e5 1e6 1e7

phas(Vout.v)

100

0

1

10

100

1e3 1e4 Frequency Hz

1e5

1e6

1e7

Figure 4: Gain and phase curves for a times 10 inverting amplier with the OP AMP represented by a transistor level UA741 model.R2 R=47k Ohm Vin V1 U=1 V R3 R=4.7k Ohm SRC1 G=1 S T=0 R1 R=200k Ohm C1 C=159.15nF OP1 G=1 Equation Eqn1 d2=phase(Vout.v) d1=dB(Vout.v) Vout

dc simulationDC1

ac simulationAC1 Type=lin Start=1 Hz Stop=10 MHz Points=1800

Figure 5: Modied Qucs OP AMP model to include single pole frequency response. 5

10

Vout.v

5

0 1 10 100 1e3 Frequency Hz 1e4 1e5 1e6 1e7

20 dB(Vout.v)

0

-20 1 200 phase(Vout.v) Degrees 10 100 1e3 Frequency Hz 1e4 1e5 1e6 1e7

150

100 1 10 100 1e3 Frequency Hz 1e4 1e5 1e6 1e7

Figure 6: Gain and phase curves for the circuit shown in Fig. 5.

6

Adding features to the Qucs OP AMP modelIn the previous section it was shown that the Qucs OP AMP model had a frequency response that is independent of frequency. By adding external components to the Qucs OP AMP model the functionality of the model can be improved. The UA741 dierential open loop gain has a pole at roughly 5Hz and a frequency response that decreases at 20 dB per frequency decade from the rst pole frequency up to a second pole frequency at roughly 3 MHz. The circuit shown in Fig. 5 models the dierential frequency characteristics of a UA741 from DC to around 1 MHz. Figure 6 illustrates the closed loop frequency response for the modied Qucs OP AMP model.

Modular operational amplier macromodelsMacromodelling is a term given to the process of modelling an electronic device as a black box where individual device characteristics are specied in terms of the signals, and other properties, observed at the input and output terminals of the black box. Such models operate at a functional level rather than at the more detailed transistor circuit level, oering considerable gain in computational eciency.2 Macromodels are normally derived directly from manufacturers data sheets. For the majority of operational ampliers, transistor level models are not normally provided by manufacturers. One notable exception being the UA741 operational amplier shown in Fig. 3. A block diagram of a modular3 general purpose OP AMP macromodel is illustrated in Fig. 7. In this diagram the blocks represent specic amplier characteristics modelled by electrical networks composed of components found in all the popular circuit simulators4 . Each block consists of one or more components which model a single amplier parameter or a group of related parameters such as the input oset current and voltage. This ensures that changes to one particular parameter do not indirectly change other parameters. Local nodes and scaling are also employed in the macromodel blocks. Furthermore, because each block operates separately, scaled voltages do not propagate outside individual blocks. Each block can be modelled with a Qucs subcircuit that has the required specication and buering from other blocks. Moreover, all subcircuits are self contained entities where the internal circuit details are hidden from other blocks. Such an approach is similar to structured high-level computer programming where the internal details of functions are hidden from users. Since the device characteristics specied by each block are separate from all otherComputational eciency is increased mainly due to the fact that operational amplier macromodels have, on average, about one sixth of the number of nodes and branches when compared to a transistor level model. Furthermore, the number of non-linear p-n junctions included in a macromodel is often less than ten which compares favorable with the forty to fty needed to model an amplier at transistor level. 3 Brinson M. E. and Faulkner D. J., Modular SPICE macromodel for operational ampliers, IEE Proc.Circuits Devices Syst., Vol. 141, No. 5, October 1994, pp. 417-420. 4 Models employing non-linear controlled sources, for example the SPICE B voltage and current sources, are not allowed in Qucs release 0.0.9. Non-linear controlled sources are one of the features on the Qucs to-do list.2

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device characteristics only those amplier characteristics which are needed are included in a given macromodel. This approach leads to a genuinely structured macromodel. The following sections present the detail and derivation of the electrical networks forming the blocks drawn in Fig. 7. To illustrate the operation of the modular OP AMP macromodel the values of the block parameters are calculated for the UA741 OP AMP and used in a series of example simulations. Towards the end of this tutorial note data are presented for a number of other popular general purpose operational ampliers.

A basic AC OP AMP macromodel.A minimum set of blocks is required for the modular macromodel to function as an amplier: an input stage, a gain stage and an output stage. These form the core modules of all macromodels.

The input stage.The input stage includes amplier oset voltage, bias and oset currents, and the dierential input impedance components. The circuit for the input stage is shown in Fig. 8, where 1. R1 = R2 = Half of the amplier dierential input resistance (RD). 2. Cin = The amplier dierential input capacitance (CD). 3. Ib1 = Ib2 = The amplier input bias current (IB ). 4. Io = Half the amplier input oset current (IOFF ). 5. Vo1 = Vo2 = Half the input oset voltage ( VOFF ). Typical values for the UA741 OP AMP are: 1. RD = 2 M and R1 = R2 = 1M 2. CD = Cin1 = 1.4 pF. 3. IB = Ib1 = Ib2 = 80 nA. 4. IOFF = 20 nA and Io1 = 10 nA. 5. VOFF = 0.7 mV and Vo1 = Vo2 = 0.35 mV.

8

In+

In-

Common mode stage

Input Stage

Signal adder

Slew rate limiting stage

Voltage gain stage 1 Overdrive limiting stageOut

Vcc In+

Voltage gain stage 2

InVee

RPD Vee Vcc

Output stage

Vcc

Vee

Current limiting stage

Voltage limiting stage

Figure 7: Block diagram of an operational amplier macromodel. 9

The dierential output signal (VD) is given by V D P 1 V D N 1 and the common mode output signal (VCM ) by (V D P 1 + V D N 1)/2.

Voff1 U=0.35mV Ib1 I=80nA Ioff1 I=10nA Voff2 U=0.35mV Ib2 I=80nA R1 R=1M Ohm VCM1 Input stage InVd-

IN_N1

VD_N1 Cin1 C=1.4 pF

Vcm

R2 R=1M Ohm

In+

Vd+

IN_P1

VD_P1

SUB1 File=input_stage.sch

Figure 8: Modular OP AMP input stage block. 10

Voltage gain stage 1.The circuit for voltage gain stage 1 is shown in Fig. 9, where 1. RD1 = 100 M = A dummy input resistor - added to ensure nodes IN P 1 and IN N 1 are connected by a DC path. 2. GMP1 = 1 S = Unity gain voltage controlled current generator. 3. RADO = The DC open loop dierential gain ( AOL(DC) ) of the OP AMP. 4. CP1 = 1/(2**GBP), where GBP = the OP AMP gain bandwidth product. Typical values for the UA741 OP AMP are: 1. RADO = 200k. (AOL(DC) = 106 dB) 2. CP1 = 159.15 nF (The typical value for UA741 GBP is 1 MHz).

Derivation of voltage gain stage 1 transfer functionMost general purpose operational ampliers have an open loop dierential voltage gain which has (1) a very high value at DC (2) a dominant pole (fp1 ) at a low frequency typically below 100 Hz, and (3) a gain response characteristic that rolls-o at 20 dB per decade up to a unity gain frequency which is often in the MHz region. This form of response has a constant gain bandwidth product (GBP ) over the frequency range from fp1 to GBP. A typical OP AMP dierential open loop response is shown in Fig. 10. The voltage gain transfer function for this type of characteristic can be modelled with the electrical network given in Fig. 9, where the the AC voltage transfer function is vout(P OLE 1 OU T 1) = Where fP 1 = GM P 1 (V (IN P 1) V (IN N 1)) RADO 1 + j( RADO CP 1) 1 2 RADO CP 1POLE1 IN+ IN_P1 RD1 R=100M RADC1 R=200k Ohm CP1 C=159.15 nF POLE_1_OUT1 OUT INSUB1 File=pole1.sch

(1)

(2)

IN_N1

GMP1 G=1 S T=0

Figure 9: Modular OP AMP rst voltage gain stage. 11

Aol Aol(DC)

1

fp1

GBP

f Hz

Figure 10: OP AMP open loop dierential voltage gain as a function of frequency. Let RADC = Aol(DC) and GMP1 = 1 S. Then, because fp1*AOL(DC) = GBP, CP 1 = 1 2 GBP (3)

12

OUTSTG_OUT1 IN_P1 RD1 R=100M ROS1 R=75 Ohm

Output stage In+ Out InSUB1 File=out_stage.sch

IN_N1

EOS1 G=1 T=0

Figure 11: Modular macromodel output stage.

Output stage.The electrical network representing a basic output stage is given in Fig. 11, where 1. RD1 = 100 M = A dummy input resistor - added to ensure nodes IN P 1 and IN N 1 are connected by a DC path. 2. EOS1 G = 1 = Unity gain voltage controlled voltage generator. 3. ROS1 = OP AMP output resistance. A typical value for the UA741 OP AMP output resistance is ROS1 = 75.

A subcircuit model for the basic AC OP AMP macromodelThe model for the basic AC OP AMP macromodel is shown in Fig. 12. The input stage common mode voltage (V cm) is not used in this macromodel and has been left oating. To test the performance of the AC macromodel its operation was compared to the transistor level UA741 model. Figure 13 shows a schematic circuit for two inverting ampliers, each with a gain of ten, driven from a common AC source. One of the ampliers uses the simple AC macromodel and the other the transistor level UA741 model. Figure 14 illustrates the output gain and phase curves for both ampliers. In general the plotted curves are very similar. However, at frequencies above the GBP frequency the basic AC macromodel does not correctly model actual OP AMP performance. This is to be expected because the simple AC macromodel does not include any high frequency modelling components. Notice also that the DC output voltages for vout and vout3 are very similar, see the DC tabular results given in Fig. 13.

13

SUB2 File=input_stage.sch

In+ IN_P1

Vd+

POLE1 IN+ Output stage In+ Out InIn+ SUB3 File=out_stage.sch SUB5 File=op_amp_ac_IP1O.sch InOUT1

Vcm ININ_N1 In- VdInput stage

OUT

OP AMP IP1O

SUB4 File=pole1.sch

Figure 12: Simple AC OP AMP macromodel.

14

dc simulationDC1 number 1 vout.V 0.0068 vout3.V 0.0069

ac simulationAC1 Type=lin Start=1 Hz Stop=10 MHz Points=1801 R1 R=10k Ohm

Equation Eqn1 yp=phase(vout.v) yp3=phase(vout3.v) ydb=dB(vout.v) ydb3=dB(vout3.v)

vin InV1 R2 U=1 V R=1k Ohm vout

OP AMP IP1OSUB5 In+

R3 R=10k Ohm SUB6

R8 R=1k Ohm VEE UA741_tran VCC

vout3

V2 U=15 V

+

V3 U=15 V

Figure 13: Test circuit for an inverting amplier. Output signals: (1) vout for AC macromodel, (2) vout3 for UA741 transistor model.

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phase(vout.v) in degrees 1 10 100 1e3 1e4 1e5 Frequency Hz 1e6 1e7

20 dB(vout.v)

200

150

0

100

-20

1

10

100

1e3 1e4 1e5 Frequency Hz

1e6

1e7

0

phase(vout3.v) in degrees 1 10 100 1e3 1e4 1e5 Frequency Hz 1e6 1e7

20 db(vout3.v)

200

100

-20

0

1

10

100

1e3 1e4 1e5 Frequency Hz

1e6

1e7

Figure 14: Simulation test results for the circuit shown in Fig. 13.

16

A more accurate OP AMP AC macromodelMost general purpose OP AMPs have a high frequency pole in their dierential open loop gain characteristics. By adding a second gain stage to the simple AC macromodel the discrepancy in the high frequency response can be corrected. The model for the second gain stage is shown in Fig. 15. This additional gain stage has a structure similar to the rst gain stage, where 1. RD2 = 100 M = A dummy input resistor - added to ensure nodes IN_P2 and IN_N2 are connected by a DC path. 2. GMP2 = 1 S = Unity gain voltage controlled current generator. 3. RP2 = 1. 4. CP2 = 1/(2*fp2), where f p2 = the second pole frequency in Hz. A typical value for the UA741 OP AMP high frequency pole is fp2 = 3M Hz

Derivation of voltage gain stage 2 transfer function.The dierential voltage gain transfer function for voltage gain stage 2 is given by vout(P OLE 2 OU T 1) = GM P 2 (V (IN P 2) V (IN N 2)) RP 2 1 + j( RP 2 CP 2) (4)

Let RP2 = 1 and GMP2 = 1 S. Then vout(P OLE 2 OU T 1) = and CP 2 = 1 2 f p2 (6) V (IN P 2) V (IN N 2) 1 + j( CP 2) (5)

POLE2 IN+ IN_P2 RD2 R=100M RP2 R=1 Ohm CP2 C=53.05nF POLE_2_OUT1 OUT INSUB1 File=pole2.sch

IN_N2

GMP2 G=1 S T=0

Figure 15: Modular OP AMP second voltage gain stage. 17

ac simulationR1 R=10M Ohm C1 C=100mF vin V1 U=1 VIn-

AC1 Type=log Start=1 Hz Stop=100MHz Points=241 vout Equation Eqn1 y4=rad2deg(unwrap(angle(vout3.v))) y=dB(vout.v) y3=dB(vout3.v) y2=phase(vout.v)

OP AMP IP1P2OIn+

SUB1 File=op_amp_ac_IP1P2O.sch R2 R=10M Ohm

dc simulationDC1 V2 U=15 V

SUB2

C2 C=100 mF

VEE UA741_tran VCC

vout3

+

V3 U=15 V

Figure 16: Test circuit for simulating OP AMP open loop dierential gain.

Simulating OP AMP open loop dierential gainThe circuit shown in Fig. 16 allows the open loop dierential gain (Aol(f )) to be simulated. This circuit employes a feedback resistor to ensure DC stability. Fig. 16 illustrates two test circuits driven from a common AC source. This allows the performance of the AC macromodel and the UA741 transistor level model to be compared. The AC voltage transfer function for the test circuit is vout(f ) = Aol(f ) vin(f ) Aol(f ) 1+ 1 + j R C vout(f ) 1 + j R C (7)

where vout(f ) = (V + V ) Aol(f ), V + = vin(f ), and V = Provided

Aol(f )