Amplifiers: Op Amps Texas Instruments Incorporated Operational amplifier gain stability, Part 2: DC gain-error analysis By Henry Surtihadi, Analog Design Engineer, and Miroslav Oljaca, Senior Applications Engineer Introduction Figure 1. Non-inverting op amp configuration with ideal The goal of this three-part series of articles is closed-loop gain of +200 to provide readers with an in-depth under- stand ing of gain accuracy in closed-loop Network circuits using two of the most common opera- Feedback tional amplifier (op amp) configurations: non- Network 199 x R inverting and inverting. Often, the effects of VOUT V various op amp param eters on the accuracy of FB the circuit’s closed-loop gain are overlooked R and cause an unexpected gain error both in 199 x R VOUT the DC and AC domains. V This article, Part 2, focuses on DC gain error, FB which is primarily caused by the finite DC open- V loop gain of the op amp as well as its tempera- IN R ture dependency. This article builds upon the results obtained in Part 1 (see Reference 1), in which two separate equations were derived for calculating the transfer functions of non- inverting and inverting op amps. Part 2 pre- sents a step-by-step example of how to calculate the worst-case gain error, starting with finding the pertinent Also derived in the same article was the equation for data from the product data sheet. It then shows how to calculating the magnitude of the inverting configuration’s use the data in conjunction with the two aforementioned closed-loop gain. The result is repeated in Equation 3: equations to perform the gain-error calculation. AOL _ DC In Part 3, the gain error for AC input signals will be a 1A+b× OL_ DC calculated. In the AC domain, the closed-loop gain error A ( f )= 20 log (3) CL dB is affected by the AC open-loop response of the op amp. f12 Part 3 will discuss one of the most common mistakes that 1+× f22 (1+b× A ) occur when the AC gain response is calculated. 0 OL_ DC Transfer functions of non-inverting and Equation 3 uses the same variable b defined by Equation inverting op amps 2. Additionally, the variable a is defined by Equation 4: VR In Part 1 (Reference 1), the closed-loop transfer function a=FB = F (4) V RR+ of the non-inverting op amp configuration in the frequency IN I F domain was calculated. Specifically, the transfer function At this point, the closed-loop gain for non-inverting and was derived with the assumption that the op amp had a inverting amplifiers is represented by Equations 1 and 3, first-order open-loop response. For calculating gain error, respectively. These equations will be used for subsequent the magnitude response is of interest. For convenience, analysis. The analysis of DC closed-loop circuits has been the result is repeated in Equation 1: treated in slightly different ways in References 2 to 7; AOL _ DC however, the results agree with this analysis. 1A+b× OL _ DC A ( f )= 20 log , (1) DC gain error for non-inverting configuration CL dB f12 To illustrate the impact of an op amp’s finite open-loop gain 1+× on the accuracy of DC closed-loop gain in a non-inverting f22 (1+b× A ) 0 OL _ DC configuration, a step-by-step example will be presented on where b is defined as how to calculate the gain error when the op amp is set in an ideal closed-loop gain. An ideal closed-loop gain of 200 VR b=FB = I . (2) (1/b = 200), as shown in Figure 1, will be used. This V RR+ OUT I F example focuses on using only the Texas Instruments (TI) 24 High-Performance Analog Products www.ti.com/aaj 2Q 2010 Analog Applications Journal Texas Instruments Incorporated Amplifiers: Op Amps OPA211 op amp, but circuit designers can repeat the cal cu lation with similar values from the data Figure 2. OPA211’s simplified open-loop and closed- loop gain curves sheet of any other op amp they choose. To calculate the DC closed-loop-gain error of a non-inverting op amp, Equation 1 is evaluated 140 A for zero frequency (f = 0 Hz): OL_DC f0 120 A OL_ DC Open-Loop Gain, A ACL _ DC== A CL (0 Hz) (5) OL 1A+b× OL_ DC 100 In the case of an ideal op amp with infinite open- Loop 80 loop gain, the DC closed-loop gain of the non- Gain, × AOL inverting configuration is reduced to 60 Gain = 200 V/V or +46 dB A1OL _ DC ACL _ DC(ideal) ==lim . (6) 40 A →∞ 1A+b× b OL _ DC OL _ DC Gain (dB) Voltage Closed-Loop Gain, In other words, the DC closed-loop gain is entirely 20 ACL_DC determined by the external feedback network. From the closed-loop models of non- 0 inverting and inverting amplifiers in Figures 3 and 6, respectively, in Part 1 (see Reference 1), –20 it can be seen that the open-loop gain of the op 10 100 1 k 10 k 100 k1 M 10 M 100 M Frequency (Hz) amp is the ratio of VOUT to the input-error volt- age, VERR. VERR is the voltage difference between the inverting and non-inverting op amp inputs. It can also be seen as input offset voltage. In a product data configuration. The difference between these two curves is sheet, the open-loop gain is typically expressed in decibels. the loop gain, b × AOL. Because the focus of this example In this case, the number represents the ratio of VOUT to is DC gain error, only the loop gain at low frequency VERR in the logarithmic domain. For future calculation, (b × AOL_DC) is of interest. AOL_DC must always be converted from decibels to V/V. As When using the data from the typical curves, designers an example, an op amp with an open-loop gain of 106 dB should consider possible variations. To calculate worst-case can be written in terms of V/V as values, the open-loop-gain data provided in the product A 106 dB data sheet should be used. Such data are shown in Table 1 OL _ DC dB VV 20 20 OUT for the TI OPA211/2211 op amps. As the table shows, when AOL _ DC =10 = 10 ==199,526 . (7) V/ V VVERR the output signal is more than 200 mV from the supply rails Figure 2 shows the simplified open-loop gain of the and has a 10-kΩ load, the typical value for the DC open- OPA211 along with the closed-loop gain in a non-inverting loop gain is 130 dB, while the minimum ensured gain is 114 dB. To calculate the typical and the worst-case DC gain Table 1. Excerpt from TI OPA211/2211 data sheet ELECTRICAL CHARACTERISTICS: VS = ±2.25V to ±18V BOLDFACE limits apply over the specified temperature range, TA = –40ºC to +125ºC. At TA = +25ºC, RL = 10kΩ connected to midsupply, VCM = VOUT = midsupply, unless otherwise noted. Standard Grade High Grade OPA211AI, OPA2211AI OPA211I PARAMETER CONDITIONS MIN TYP MAX MIN TYP MAX UNIT OPEN-LOOP GAIN Open-Loop Voltage Gain AOL (V–) + 0.2V ≤ VO ≤ (V+) – 0.2V, 114 130 114 130 dB RL = 10kΩ AOL (V–) + 0.6V ≤ VO ≤ (V+) – 0.6V, 110 114 110 114 dB RL = 600Ω Over Temperature OPA211 AOL (V–) + 0.6V ≤ VO ≤ (V+) – 0.6V, 110 110 dB IO ≤ 15mA OPA211 AOL (V–) + 0.6V ≤ VO ≤ (V+) – 0.6V, 103 103 dB 15mA ≤ IO ≤ 30mA OPA2211 (per channel) AOL (V–) + 0.6V ≤ VO ≤ (V+) – 0.6V, 100 dB IO ≤ 15mA 25 Analog Applications Journal 2Q 2010 www.ti.com/aaj High-Performance Analog Products Amplifiers: Op Amps Texas Instruments Incorporated errors at room temperature, the minimum AOL_DC from the Over temperature, the OPA211 is characterized to data sheet should be substituted into Equation 5. Note that ensure that AOL_DC is higher than 110 dB over the speci- in the OPA211 data sheet, “AOL_DC” is written as “AOL.” fied temperature range and when loaded with less than The first step in this process is to convert AOL_DC from 15-mA output current, which is the absolute worst case. decibels to V/V: For this value, in terms of V/V, 110 dB is equivalent to 130 dB 110 dB V V A ==1020 3,162,278 (8) A ==1020 316,228 . (14) OL _ DC V/ V OL _ DC V/ V V V 114 dB This number can be substituted into Equation 5 to find the V A ==1020 501,187 (9) absolute worst-case condition for the DC closed-loop gain: OL _ DC V/ V V 316,228 A value for b of 1/200 (the ideal closed-loop gain of 200) A ==199.8736 (15) CL _ DC 110 dB 1 can be used in Equation 5 to find the typical DC gain: 1+ 316,228 200 A OL _ DC The gain error for this result, 0.063%, represents a slight ACL _ DC = 130 dB 1A+b× OL _ DC degradation from the room-temperature case of 0.0399% previously calculated in Equation 13. 3,162,278 (10) ==199.98735 1 DC gain error for inverting configuration 1+ 3,162,278 200 To illustrate the impact of the op amp’s finite open-loop The actual minimum ensured DC gain can be found in the gain on the accuracy of DC closed-loop gain in an invert- same manner: ing configuration, another step-by-step example will be presented of calculating the gain error when the op amp is 501,187 A ==199.92022 (11) set in an ideal closed-loop gain.