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Berkeley ELENG 100 - Op-Amps Experiment Theory

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EE 43/100 Operational Amplifiers 1Op-Amps Experiment Theory 1. Objective The purpose of these experiments is to introduce the most important of all analog building blocks, the operational amplifier (“op-amp” for short). This handout gives an introduction to these amplifiers and a smattering of the various configurations that they can be used in. Apart from their most common use as amplifiers (both inverting and non-inverting), they also find applications as buffers (load isolators), adders, subtractors, integrators, logarithmic amplifiers, impedance converters, filters (low-pass, high-pass, band-pass, band-reject or notch), and differential amplifiers. So let’s get set for a fun-filled adventure with op-amps! 2. Introduction: Amplifier Circuit Before jumping into op-amps, let’s first go over some amplifier fundamentals. An amplifier has an input port and an output port. (A port consists of two terminals, one of which is usually connected to the ground node.) In a linear amplifier, the output signal = A × input signal, where A is the amplification factor or “gain.” Depending on the nature of the input and output signals, we can have four types of amplifier gain: voltage gain (voltage out / voltage in), current gain (current out / current in), transresistance (voltage out / current in) and transconductance (current out / voltage in). Since most op-amps are voltage/voltage amplifiers, we will limit the discussion here to this type of amplifier. The circuit model of an amplifier is shown in Figure 1 (center dashed box, with an input port and an output port). The input port plays a passive role, producing no voltage of its own, and is modelled by a resistive element Ri called the input resistance. The output port is modeled by a dependent voltage source AVi in series with the output resistance Ro, where Vi is the potential difference between the input port terminals. Figure 1 shows a complete amplifier circuit, which consists of an input voltage source Vs in series with the source resistance Rs, and an output “load” resistance RL. From this figure, it can be seen that we have voltage-divider circuits at both the input port and the output port of the amplifier. This requires us to re-calculate Vi and Vo whenever a different source and/or load is used: sisiiVRRRV⎟⎟⎠⎞⎜⎜⎝⎛+= (1)EE 43/100 Operational Amplifiers 2 iLoLoAVRRRV⎟⎟⎠⎞⎜⎜⎝⎛+= (2) RSVSViRiAViRoVoRLSOURCE AMPLIFIER LOAD+_+_INPUT PORTOUTPUT PORT Figure 1: Circuit model of an amplifier circuit. 3. The Operational Amplifier: Ideal Op-Amp Model The amplifier model shown in Figure 1 is redrawn in Figure 2 showing the standard op-amp notation. An op-amp is a “differential to single-ended” amplifier, i.e. it amplifies the voltage difference Vp – Vn = Vi at the input port and produces a voltage Vo at the output port that is referenced to the ground node of the circuit in which the op-amp is used. ViRiAViRoVo+_+_+_VpVnipin+_ ViAViVo+_+_+_VpVn+_ Figure 2: Standard op-amp Figure 3: Ideal op-amp The ideal op-amp model was derived to simplify circuit analysis and is commonly used by engineers for first-order approximation calculations. The ideal model makes three simplifying assumptions: Gain is infinite: A = ∞ (3) Input resistance is infinite: Ri = ∞ (4)EE 43/100 Operational Amplifiers 3 Output resistance is zero: Ro= 0 (5) Applying these assumptions to the standard op-amp model results in the ideal op-amp model shown in Figure 3. Because Ri = ∞ and the voltage difference Vp – Vn = Vi at the input port is finite, the input currents are zero for an ideal op-amp: in = ip = 0 (6) Hence there is no loading effect at the input port of an ideal op-amp: siVV= (7) In addition, because Ro = 0, there is no loading effect at the output port of an ideal op-amp: Vo = A × Vi (8) Finally, because A = ∞ and Vo must be finite, Vi = Vp – Vn = 0, or Vp = Vn (9) Note: Although Equations 3-5 constitute the ideal op-amp assumptions, Equations 6 and 9 are used most often in solving op-amp circuits. VinR2R1Vout+_VpVnI VinVout+_VpVn VinR2R1Vout+_VnVpI Figure 4a: Non-inverting amplifier Figure 5a: Voltage follower Figure 6a: Inverting amplifier VinVoutA>=1 VinVoutA=1 VinVoutA<0 Figure 4b: Voltage transfer curve Figure 5b: Voltage transfer curve Figure 6b: Voltage transfer curve of non-inverting amplifier of voltage follower of inverting amplifierEE 43/100 Operational Amplifiers 4 VinVoutA>=1-Vpower+Vpower VinVoutA=1-Vpower+VpowerVinVoutA<0-Vpower+Vpower Figure 4c: Realistic transfer curve Figure 5c: Realistic transfer curve Figure 6c: Realistic transfer curve of non-inverting amplifier of voltage follower of inverting amplifier 4. Non-Inverting Amplifier An ideal op-amp by itself is not a very useful device, since any finite non-zero input signal would result in infinite output. (For a real op-amp, the range of the output signal is limited by the positive and negative power-supply voltages.) However, by connecting external components to the ideal op-amp, we can construct useful amplifier circuits. Figure 4a shows a basic op-amp circuit, the non-inverting amplifier. The triangular block symbol is used to represent an ideal op-amp. The input terminal marked with a “+” (corresponding to Vp) is called the non-inverting input; the input terminal marked with a “–” (corresponding to Vn) is called the inverting input. To understand how the non-inverting amplifier circuit works, we need to derive a relationship between the input voltage Vin and the output voltage Vout. For an ideal op-amp, there is no loading effect at the input, so Vp = Vi (10) Since the current flowing into the inverting input of an ideal op-amp is zero, the current flowing through R1 is equal to the current flowing through R2 (by Kirchhoff’s Current Law -- which states that the algebraic sum of currents flowing into a node is zero -- to the inverting input node). We can therefore apply the voltage-divider formula find Vn: outnVRRRV⎟⎟⎠⎞⎜⎜⎝⎛+=211 (11)EE 43/100 Operational Amplifiers 5From Equation 9, we know that Vin = Vp = Vn, so inoutVRRV⎟⎟⎠⎞⎜⎜⎝⎛+=121 (12) The


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Berkeley ELENG 100 - Op-Amps Experiment Theory

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