Operational amplifiers (Op amps) v+ Ri v– + vi – + Av i – Ro + vo – View it as an ideal amp. Take the properties to the extreme: Ri → ∞, Ro = 0, A → ∞. ?!?!?!?! v+ v– + vi – + + Av i – vo – A→∞ Consequences: No voltage dividers at input or output. (That’s good.) No current flows into the input. (That’s good.) The gain is infinite. (Is that good?) EE 201 op amps – 1 How do we handle this infinite gain business? vo = Avi In order to keep vo finite, then as A → ∞ , vi → 0. In other words, we must force the difference signal at the input to go to zero. How do we do that? With feedback, of course. Recall that the difference signal in a feedback arrangement must become very small if the gain is very big. = + → 0, as A → ∞ One input is connected to the source voltage in some fashion. The other input is connected to the feedback network. If the feedback is working properly, then vi = v+ – v_ = 0. The condition of v+ = v_ is called a virtual short at the input. This should be the case, if negative feedback is working in the circuit. EE 201 op amps – 2 Ideal op amp v+ v– + – + vo – v+ → non-inverting input v_ → inverting input When using an op amp in a circuit: v+ ≈ v_ → virtual short (assuming a proper negative feedback configuration.) i+ = i_ = 0 → due to infinite input resistance No voltage divider effects at output. This means that we can connect anything to the output without worrying about loading effects. EE 201 op amps – 3 Non-inverting amplifier + – vS + – R1 + vo – R2 Write a node equation at the inverting terminal. = + v– = v+ = vs (virtual short due to feedback) i– = 0 (infinite input resistance) = = EE 201 = + = + (Worth memorizing.) op amps – 4 Inverting amplifier R2 R1 – + vS + – Write a node equation at the inverting terminal. + vo – = + v– = v+ = 0 (virtual ground!) i– = 0 (infinite input resistance) = = = Note the negative sign! (Also worth memorizing.) EE 201 op amps – 5 Summing amp (weighted summer) RF R3 vS3 R2 + – + – vS2 = – + R1 + v – S1 At the inverting terminal: v– = v+ = 0 (virtual + ground). Write a node vo equation there. (Or use – superposition.) + + + + = Use another inverter if you don’t like the negative sign. The “virtually grounded” inverting terminal becomes a summing node. EE 201 op amps – 6 Difference amp We would like to amplify only the difference between va and vb. Anything this applied in common to both, will not be amplified. R2 At the inverting input: R1 vb va – + R3 = R4 = + At the non-inverting input: + + if EE 201 = = + v+ = v– = + = then = ( + ) + Difference only! (Check it.) op amps – 7 Unity gain buffer vs + – vo Non-inverting amp with R2 = 0 and R1 → ∞ . So G = 1, meaning vo = vs. What good is that? v1 + – Ro1 Ri2 Connecting two circuits. If Ro1> Ri2, then much of the voltage is lost in the voltage divider, vi2 << v1. EE 201 v1 + – Ro1 + – Ri2 High input resistance of op amp makes v+ = v1. Zero output resistance of op amp makes vi2 = vo. Since vo = v+, then vi2 = v1. The op amp served as a buffer between the two circuits, eliminated the voltage divider problem. op amps – 8 Digital-to-analog converter (DAC) VREF = -1 V Data is stored as digital information on your phone or computer. Some of that data needs to be converted to analog form in order to use it, i.e. music. S vo (V) 0000 0 0001 0.125 0010 0.25 0011 0.375 0100 0.5 0101 0.625 0110 0.75 0111 0.875 1000 1 1001 1.125 1010 1.25 1011 1.375 1100 1.5 1101 1.625 So only closed: vo = –VREF/8. S1 only closed: vo = –VREF/4. 1110 1.75 S2 only closed: vo = –VREF/2. S3 only closed: vo = –VREF. 1111 2 4-bit DAC VREF S3 R3 = RF R2 = RF / 2 R1 = RF / 4 Ro = RF / 8 R3 S2 RF R2 S1 R1 S0 Ro – + + vo – The digital bits control the operation of the switches – one bit per switch. Si = 0, switch is open, Si = 1 switch is closed. EE 201 op amps – 9 Cascading amps va The various types of circuits can serve as building blocks for more complicated circuits. R4 = R3 R5 v R7 – o1 + inverting summing – vo3 + non-inverting R vb 6 + – vo2 = R1 R2 + = = EE 201 + op amps – 10 20 k! R5 10 k! R2 R1 vS – + 1 k! vo1 + – vo2 8 k! summing 2 k! = = = + = . want vo2 / vS. = . . Feedback loop around feedback loops !! = = EE 201 R3 R4 non-inverting . op amps – 11