L-2.7_2.8 and 2.9 Voltage Series Feedback Amplifier (2).pdf

Linear Integrated Circuits and Applications Voltage Series Feedback Amplifier (Non-inverting Feedback Amplifier) Resources Text Books: 1. Ramakant A. Gayakwad, ‘Op-Amps and Linear Integrated Circuits’, Pearson Education, 2000. 2. Sergio Franco, ‘Design with Operational Amplifiers and Analog...

Linear Integrated Circuits and Applications Voltage Series Feedback Amplifier (Non-inverting Feedback Amplifier) Resources Text Books: 1. Ramakant A. Gayakwad, ‘Op-Amps and Linear Integrated Circuits’, Pearson Education, 2000. 2. Sergio Franco, ‘Design with Operational Amplifiers and Analog Integrated Circuits’, McGraw-Hill, 2002. SLIDES OF THIS COURSE ARE BASED ON THESE BOOKS AND SOME OF THE FIGURES ARE TAKEN FROM THESE BOOK. Voltage Series Feedback Amplifier The circuit is commonly known as a noninverting amplifier with feedback because it uses feedback, and the input signal is applied to the noninverting input signal is applied to the noninverting input terminal of op-amp. The schematic diagram of the voltage-series feedback amplifier is shown ahead. The op-amp is represented by its schematic symbol, including its large-signal voltage gain A, and the feedback circuit is composed of two resistors, 𝑅1 and 𝑅𝐹. Voltage Series Feedback Amplifier: v1 +VCC vid A + v2 −VEE vin vo RL + RF vf R1 − − Feedback Circuit Voltage Series Feedback Amplifier (or noninverting amplifier with feedback). Voltage Series Feedback Amplifier: Before proceeding, it is necessary to define some important terms for the voltage series feedback amplifier. Specifically, the voltage gain of the op-amp with and without feedback, and the gain of the feedback circuit are defined as follows: 𝑣 Open‐loop voltage gain (or gain without feedback) 𝐴 = 𝑣𝑜 𝑖𝑑 𝑣 Closed-loop voltage gain (or gain with feedback) 𝐴𝐹 = 𝑣 𝑜 𝑖𝑛 𝑣𝑓 Gain of the feedback circuit. B=𝑣 𝑜 This is standard form for representing a system with feedback and also indicates the relationship between different variables of the system. The block‐diagram approach helps to simplify the analysis of complex closed‐loop networks, particularity if they are composed of nonresistive feedback circuits. Summing Junction + 𝑣i𝑑 𝑣𝑖𝑛 𝑣0 Σ A − 𝑣𝑓 B 𝑣0 Representation of non-inverting amplifier with feedback. Negative feedback: Referring to the circuit Kirchhoff’s voltage equation for the input loop is 𝑣𝑖𝑑 = 𝑣𝑖𝑛 − 𝑣𝑓 (1) Where 𝑣𝑖𝑛 = input voltage 𝑣𝑓 = feedback voltage 𝑣𝑖𝑑 = difference input voltage Recall, however, that an op-amp always amplifies the difference input voltage 𝑣𝑖𝑑. From Equation (1) this difference voltage is equal to the input voltage 𝑣𝑖𝑛 minus the feedback voltage 𝑣𝑓. In other words, the feedback voltage always opposes the input voltage (or is out of phase by 180° with respect to the input voltage) hence the feedback is said to be negative. Closed-Loop Voltage Gain: As defined previously, the closed loop voltage gain 𝑣𝑜 𝐴𝐹 = 𝑣𝑖𝑛 However, by using equation 𝑣𝑜 = 𝐴(𝑣1 − 𝑣2 ) Referring to Figure, we see that 𝑣1 = 𝑣in 𝑅 1 𝑣𝑜 𝑣2 = 𝑣𝑓 = 𝑅1 +𝑅𝐹 since 𝑅𝑖ሶ >> 𝑅1 therefore, 𝑅1 𝑣𝑜 𝑣𝑜 = 𝐴(𝑣in − ) 𝑅1 + 𝑅𝐹 Closed-Loop Voltage Gain: Rearranging we get 𝐴(𝑅1 + 𝑅𝐹 )𝑣𝑖𝑛ሶ 𝑣𝑜 = 𝑅1 + 𝑅𝐹 + 𝐴𝑅1 Thus 𝑣𝑜 𝐴(𝑅1 +𝑅𝐹 ) 𝐴𝐹 = = (exact) (2) 𝑣𝑖𝑛 𝑅1 +𝑅𝐹 +𝐴𝑅1 Generally, 𝐴 is very large. Therefore, 𝐴𝑅1 >> (𝑅1 + 𝑅𝐹 ) and (𝑅1 + 𝑅𝐹 + 𝐴𝑅1 ) ≡ 𝐴𝑅1 Thus 𝑣 𝑅 𝐴𝐹 = 𝑣 𝑜 = 1 + 𝑅𝐹 ideal (3) 𝑖𝑛 1 Closed-Loop Voltage Gain: Equation (3) is important because it shows that the gain of the voltage series feedback amplifier is determined by the ratio of two resistors, 𝑅1 and 𝑅𝐹. For instance, if a gain of 11 is desired, we can then choose 𝑅1 = 1 kΩ and 𝑅𝐹 = 10 kΩ or 𝑅1 = 100 Ω and 𝑅𝐹 = 1 kΩ. In other words, in setting the gain the ratio of 𝑅1 and 𝑅𝐹 is important and not the absolute values of these resistors. As a general rule, however, all external component values should be less than 1 MΩ so that they do not adversely affect the internal circuitry of the op-amp. This is especially true for older-generation ICs such as the 709 and 741.  Closed-Loop Voltage Gain: Another interesting result can be obtained from Equation. As defined previously, the gain of the feedback circuit (𝐵) is the ratio of 𝑣𝑓 and 𝑣𝑜. Referring to Figure this gain is 𝑣𝑓 𝐵= (4) 𝑣𝑜 𝑅1 = 𝑅1 + 𝑅𝐹 Comparing ideal equations and this last one, we can conclude that 1 𝐴𝐹 = (ideal) (5) 𝐵 This means that the gain of the feedback circuit is the reciprocal of the closed‐loop voltage gain. In other words, for given 𝑅1 and 𝑅𝐹 the values of 𝐴𝐹 and 𝐵 are fixed. Closed-Loop Voltage Gain: Finally, the closed‐loop voltage gain 𝐴𝐹 can be expressed in terms of open‐ loop gain 𝐴 and feedback circuit 𝑣 𝐴(𝑅1 +𝑅𝐹 ) gain 𝐵 as follows. Rearranging Equation (2), 𝐴𝐹 = 𝑣 𝑜 = 𝑅 ,we get 𝑖𝑛 1 +𝑅𝐹 +𝐴𝑅1 𝑅 + 𝑅 𝐴( 𝑅1 + 𝑅 𝐹 ) 1 𝐹 𝐴𝐹 = 𝑅1 + 𝑅𝐹 𝐴𝑅1 𝑅1 + 𝑅𝐹 + 𝑅1 + 𝑅𝐹 Using Equation (4) yields 𝐴 𝐴𝐹 = (6) 1+𝐴𝐵 where 𝐴𝐹 = closed‐loop voltage gain 𝐴= 𝑜𝑝𝑒𝑛‐loop voltage gain 𝐵 = gain of the feedback circuit 𝐴𝐵 = Loop gain A one line block diagram of Equation (6) is shown in figure. This block diagram illustrates a standard form for representing a system with feedback and also indicates the relationship between different variables of the system. The block‐diagram approach helps to simplify the analysis of complex closed‐loop networks, particularity if they are composed of nonresistive feedback circuits. The gain with negative feedback in terms of open-loop gain of op-amp, A and feedback path gain of B is: 𝐴 𝐴𝐹 = 1+𝐴𝐵 Summing Junction + 𝑣i𝑑 𝑣𝑖𝑛 𝑣0 Σ A − 𝑣𝑓 B 𝑣0 Representation of non-inverting amplifier with feedback. Difference Input Voltage Ideally Zero: Let us reconsider Equation, which can be rewritten as 𝑣𝑜 𝑣𝑖𝑑 = 𝐴 Since 𝐴 is very large (ideally infinite), 𝑣𝑖𝑑 ≡ 0 (7a) That is, 𝑣1 ≡ 𝑣2 (𝑖𝑑𝑒𝑎𝑙) (7b) Equation (7b) says that the voltage at the noninverting input terminal of an op‐ amp is approximately equal to that at the inverting input terminal provided that 𝐴 is very large.  This concept is useful in the analysis of closed‐loop op-amp circuits. For example, ideal closed‐loop voltage gain [Equation(3)] can be obtained using the preceding results as follows. 𝑣1 = 𝑣𝑖𝑛 𝑅1 𝑣𝑜 𝑣2 = 𝑣𝑓 = 𝑅1 +𝑅𝐹 Substituting these values of 𝑣1 and 𝑣2 in Equation (7b), we get 𝑅1 𝑣𝑜 𝑣𝑖𝑛ሶ = 𝑅1 + 𝑅𝐹 That is, 𝑣𝑜 𝑅𝐹 𝐴𝐹 = =1+ 𝑣𝑖𝑛 𝑅1  Voltage Series Feedback Amplifier (Noninverting amplifier with feedback) As we know, voltage series feedback amplifier is commonly known as a noninverting amplifier with feedback because it uses feedback, and the input signal is applied to the noninverting input signal is applied to the noninverting input terminal of op-amp. In the previous session we derived: S. No. Parameter Voltage Series Feedback Amplifier (Non-inverting amplifier) 1 Voltage gain 𝐴 𝑅1 + 𝑅𝐹 𝐴 𝐴𝐹 = exact ; 𝐴𝐹 = 𝑅1 + 𝑅𝐹 + 𝐴𝑅1 1 + 𝐴𝐵 𝑅𝐹 𝐴𝐹 = 1 + 𝑖𝑑𝑒𝑎𝑙 𝑅1 2 Gain of the feedback circuit 𝑅1 𝐵= 𝑅1 + 𝑅𝐹 Thevenin Theorem for Dependent sources The Thevenin approach reduces a complex circuit to one with a single voltage source and a single resistor. Independent sources must be turned on because the dependent source relies on the excitation due to an independent source. For circuit with dependent sources, we cannot directly obtain the Rth from simple circuit reduction. The procedure to get Rth Find open circuit voltage = 𝜈𝑜𝑐 Find the short-circuit current = 𝑖𝑠𝑐 𝜈𝑜𝑐 𝑅𝑡ℎ = 𝑖𝑠𝑐  Input Resistance With Feedback: +VCC v1 + Ro Figure shows a voltage-series feedback iin vid Ri amplifier with the op-amp equivalent circuit. Avid v2 + In this circuit 𝑅𝑖 is the input resistance (open − vin vo RL −VEE loop) of the op-amp, and 𝑅𝑖𝐹 is the input + − resistance of the amplifier with feedback. vf R1 − R i𝐹 Circuit for Derivation of Input Resistance With Feedback: Input Resistance With Feedback: The input resistance with feedback is defined as 𝜈𝑖𝑛 𝑅𝑖𝐹 = 𝑖𝑖𝑛 𝜈𝑖𝑛 = 𝜈𝑖𝑑 ∕ 𝑅𝑖 𝜈0 𝐴 However 𝜈𝑖𝑑 = 𝑎𝑛𝑑 𝑣0 = 1+𝐴𝐵 𝜈𝑖𝑛 𝐴 𝜈𝑖𝑛 𝑅𝑖𝐹 = 𝑅𝑖 𝜈0 ∕𝐴 Therefore 𝜈𝑖𝑛 𝑅𝑖𝐹 = 𝐴𝑅𝑖 𝐴𝜈𝑖𝑛 ∕ 1+𝐴𝐵 (8) 𝑅𝑖𝐹 =Ri( 1 + AB) Output resistance with feedback: Output resistance is the resistance determined looking back into the feedback amplifier from the output terminal as shown in Figure. This resistance can be obtained by using Thevenin's theorem for dependent sources. Specifically, to find output resistance with feedback 𝑅𝑜𝐹 , reduce independent source 𝜈𝑖𝑛 to zero, apply an external voltage 𝑣𝑜 , and then calculate the resulting current 𝑖0 ,. In short, 𝑅𝑜𝐹 is defined as follows: 𝜈0 𝑅𝑜𝐹 = (9a) 𝑖0 Output Resistance With Feedback: +VCC v1 + − Ro + vid RI Avid v2 − vin vo −VEE RF R1 R oF Derivation of resistance with feedback. Writing Kirchhoff’s current equation at output node N, we get 𝑖𝑜 = 𝑖𝑎 +𝑖𝑏 since [(𝑅𝐹 +𝑅1 )| 𝑅𝑖 ≫ 𝑅𝑜 and 𝑖𝑎 ≫ 𝑖𝑏. Therefore, 𝑖𝑜 ≅ 𝑖𝑎 Kirchhoff’s voltage equation for the output loop: 𝑣𝑜 – 𝑖𝑜 𝑅𝑜 – A𝑣𝑖𝑑 = 0 𝜈0 −𝐴𝜈𝑖𝑑 However, 𝑖𝑜 = 𝑅𝑜 𝜈𝑖𝑑 = 𝜈1 − 𝜈2 = 0 − 𝜈𝑓 𝑅 1 𝜈𝑜 𝜈𝑖𝑑 = − = − B𝑣𝑜 𝑅1 +𝑅𝐹  Therefore, 𝑣𝑜 +𝐴𝐵𝜈𝑜 𝑖𝑜 = 𝑅𝑜 Sustaining the value of Io, in equation , we get 𝜈0 𝑅𝑜𝐹 = (9b) 𝜈0 +𝐴𝐵𝜈0 ∕𝑅0 𝑅0 𝑅𝑜𝐹 = 1+𝐴𝐵 This result shows that the output resistance of the voltage-series feedback amplifier is 1/(1 + AB) times the output resistance 𝑅𝑜 of the op-amp. That is, the output resistance of the op-amp with feedback is much smaller than the output resistance without feedback. Voltage Series Feedback Amplifier: S. No. Parameter Voltage Series Feedback Amplifier (Non-inverting amplifier) 1 Voltage gain 𝐴 𝑅1 + 𝑅𝐹 𝐴𝐹 = exact 𝑅1 + 𝑅𝐹 + 𝐴𝑅1 𝐴 𝐴𝐹 = 1 + 𝐴𝐵 𝑅𝐹 𝐴𝐹 = 1 + (𝑖𝑑𝑒𝑎𝑙) 𝑅1 2 Gain of the feedback circuit 𝑅1 𝐵= 𝑅1 + 𝑅𝐹 3 Input resistance 𝑅𝑖𝐹 =Ri( 1 + AB) 4 Output resistance 𝑅0 𝑅𝑜𝐹 = 1 + 𝐴𝐵 Bandwidth with Feedback The bandwidth of an amplifier is defined as the band (range) of frequencies for which the gain remains constant. Manufacturers generally specify either the gain-bandwidth product or supply open-loop gain versus frequency curve for the op-amp. For the 741 op-amp the latter is typical. Bandwidth with Feedback: Figure shows the open-loop gain versus frequency curve of the 741C op-amp. Open-loop gain vs frequency curve of the741C Bandwidth with Feedback From this curve for a gain of 200,000, the bandwidth is approximately 5 Hz; or the gain-bandwidth product is (200,000 X 5 Hz) = 1 MHz. On the other extreme, the bandwidth is approximately 1 MHz when the gain is 1. Thus, the gain-bandwidth product is constant. However, this holds true only for op-amps, like the 741 that have just one break frequency below unity gain- bandwidth. For the 741, 5 Hz is the break frequency, the frequency at which the gain A is 3 dB down from its value at 0 Hz. We will denote it by 𝑓𝑜. Bandwidth with Feedback: The frequency at which the gain equals 1 is known as the unity gain-bandwidth (UGB). The relationship between the break frequency 𝑓𝑜 , open-loop voltage gain A, bandwidth with feedback 𝑓𝐹 , and the closed-loop gain 𝐴𝐹 can be established as follows. Since for an op-amp with a single break frequency 𝑓𝑜 , the gain-bandwidth product is constant, and equal to the unity gain-bandwidth (UGB), we can write, UGB = (𝐴)(𝑓0 ) (10a) Where A = open-loop voltage gain 𝑓0 = break frequency of an op-amp, or, alternatively, only single break frequency op-amp, UGB = (𝐴𝐹 )(𝑓𝐹 ) = (𝐴)(𝑓0 ) (10b) Where 𝐴𝐹 = closed-loop voltage gain 𝑓𝐹 = bandwidth with feedback Therefore equating Equations (10a) and (10b). Bandwidth with Feedback: (𝐴)(𝑓0 ) = (𝐴𝐹 )(𝑓𝐹 ) or (𝐴)(𝑓0 ) 𝑓𝐹 = (10c) 𝐴𝐹 However, for the noninverting amplifier with feedback, 𝐴 𝐴𝐹 = 1 + 𝐴𝐵 Therefore, substituting the value of 𝐴𝐹 in Equation (10c), we get (𝐴)(𝑓0 ) 𝑓𝐹 = 𝐴/(1 + 𝐴𝐵) or 𝑓𝐹 = 𝑓𝑜 (1 + 𝐴𝐵) (10d) Total output offset voltage with Feedback: In an op-amp when the input is zero, the output is also expected to be zero. However, because of the effect of input offset voltage and current, the output is significantly larger, a result in large part of very high open-loop gain. That is, the high gain aggravates the effect of input offset voltage and current at the output. We call this enhanced output voltage the total output offset voltage VooT. The saturation voltages are specified on the data sheets as output voltage swing. Since with feedback the gain of the noninverting amplifier changes from A to A/(1 + AB) [Equation (6)] in previous slides, the total output offset voltage with feedback must also be 1/(1 + AB) times the voltage without feedback. Total output offset voltage with Feedback: Total output offset voltage without feedback Total output offset voltage with feedback = 1 + 𝐴𝐵 or ±𝑉𝑠𝑎𝑡 VooT = (11) 1+𝐴𝐵 where 1/(1 + AB) is always less than 1 and 𝑉𝑠𝑎𝑡 = saturation voltages, the maximum voltage the output of an op-amp can reach. Remember that in an open-loop configuration even a very small voltage at the input of an op-amp can cause the output to reach maximum value (+𝑉𝑠𝑎𝑡 or - 𝑉𝑠𝑎𝑡 ) because of its very high voltage gain. Therefore, according to Equation (11) for a given op- amp circuit the VooT is either positive or negative voltage because 𝑉𝑠𝑎𝑡 can be either positive or negative. Effect of Negative feedback: Negative feedback can also be used to reduce significantly the effect of noise, variations in supply voltages, and changes in temperature on the output voltage of a non inverting amplifier. In fact, the higher the value of (1 + AB), the smaller is the effect of noise and of variations in supply voltages and changes in temperature on the output voltage of a non inverting amplifier. The Voltage Series Feedback Amplifier has very high input resistance, very low output resistance, stable voltage gain, large band width, and very little (ideally zero) output offset voltage. Voltage Follower (Special Case) The lowest gain that can be obtained from a non inverting amplifier with feedback is 1. When the noninverting amplifier is configured for unity gain, it is called a voltage follower because the output voltage is equal to and in phase with the input. In other words, in the voltage follower the output follows the input. It has much higher input resistance, and the output amplitude is exactly equal to the input which made it preferable over discrete emitter follower. Voltage Follower: To obtain the voltage follower from the noninverting amplifier simply open R1 and short R F. Voltage series feedback Amplifier Voltage Follower: The resulting circuit is shown below. In this figure all the output voltage is fed back into the inverting terminal of the op-amp: consequently, the gain of the feedback circuit is 1 (B = AF = 1).Since the voltage follower is a special case of the noninverting amplifier, all the formulas developed for the latter are indeed applicable to the former except that the gain of the feedback circuit is 1 (B = 1). The applicable for R o formulas are AF = 1 R iF = AR 𝑖 Ro R oF = 𝐴 fF = Af𝑜 ±Vsat VooT = 𝐴 Voltage Series Feedback Amplifier: S. No. Parameter Voltage Series Feedback Amplifier (Non-inverting amplifier) 1 Voltage gain 𝐴 𝑅1 + 𝑅𝐹 𝐴𝐹 = exact 𝑅1 + 𝑅𝐹 + 𝐴𝑅1 𝐴 𝐴𝐹 = 1 + 𝐴𝐵 𝑅𝐹 𝐴𝐹 = 1 + (𝑖𝑑𝑒𝑎𝑙) 𝑅1 2 Gain of the feedback circuit 𝑅1 𝐵= 𝑅1 + 𝑅𝐹 3 Input resistance 𝑅𝑖𝐹 =Ri( 1 + AB) 4 Output resistance 𝑅0 𝑅𝑜𝐹 = 1 + 𝐴𝐵 5 Bandwidth 𝑓𝐹 = 𝑓𝑜 ( 1 + AB) UGB 𝑓𝐹 = 𝐴𝐹 6 Total output offset voltage ±Vsat VooT = 1 + 𝐴𝐵