MEMS1082
Chapter 5-4 Operational amplifier
Department of Mechanical Engineering Instrumentation amplifier
Very high input impedance Large common mode rejection ratio (CMRR) Capability to amplify low level signals Consistent bandwidth Department of Mechanical Engineering Integrator Lossless capacitor
1 t ∴VO = − vindt RC ∫o
1 t t ∴ y(t) = − x(t)dt = −k x(t)dt RC ∫o ∫o
Department of Mechanical Engineering Differentiation and integration
1 t ∴VO = − vindt RC ∫o
Department of Mechanical Engineering Improved Integrator
Lossy capacitor
Rs is shunt resistor
Rs=10R1
The use of shunt resistor is to limit the low frequency gain of the circuit. Otherwise, a small DC offset at the input would be integrated over time, eventually saturating the op amp.
DC offset due to the input bias current can be minimized by R2. Choose: R R R = 1 s 2 + R1 Rs Department of Mechanical Engineering Integrator
Department of Mechanical Engineering Example
−2t vs =15e u(t)V
vo (0) =10V
Find vo(t) for t>0
Department of Mechanical Engineering Summing Integrator
v1(t) v2 (t) x1(t) x2 (t) ic (t) = i1(t)+ i2 (t) = + = + R1 R2 R1 R2 1 t vc (t) = ic (t)dt ∫o t C x1(t) x2 (t) y(t) = − + dt t ∫0 1 v1(t) v2 (t) R1C R2C vc (t) = + dt ∫0 t C R1 R2 y(t) = − [k1x1(t)+ k2 x2 (t)]dt ∫0
Department of Mechanical Engineering Differentiator
Department of Mechanical Engineering Differentiator
The input signal to the differentiator is applied to the capacitor. The capacitor blocks any DC content so there is no current flow to the amplifier summing point, resulting in zero output voltage. The capacitor only allows AC type input voltage changes to pass through and whose frequency is dependant on the rate of change of the input signal. At low frequencies the reactance of the capacitor is "High" resulting in a low gain (Rf/Xc) and low output voltage from the op-amp. At higher frequencies the reactance of the capacitor is much lower resulting in a higher gain and higher output voltage from the differentiator amplifier. Department of Mechanical Engineering Differentiator
Department of Mechanical Engineering Improved Differentiator Amplifier
Department of Mechanical Engineering Example: Integrator and Switch
An integrator and a voltage-controlled switch ─ An integrator is a circuit that can perform the mathematical
operation of integration. The output voltage, vo(t), is related the input of the integrator, vs(t).
t = ⋅ 2 + v0 ( t) K ∫ vs (t)dt vo (t1 ) t1
─ Integrators have many applications. One is to
measure a time interval t2-t1.
v0 (t2 ) = K ⋅(t2 − t1 )⋅Vs + vo (t1 )
Department of Mechanical Engineering Example: Integrator and Switch
Voltage controlled switch
- Switch can be controlled electronically. The node voltage, vc(t), is called the control voltage.
Department of Mechanical Engineering Example: Integrator and Switch To design an integrator circuit
vo (t2 ) = K ⋅(t2 − t1 )⋅Vs + vo (t1 )
10V v (t ) = ⋅(t − t ) o 2 200ms 2 1
10V K ⋅V = s 200ms 1 K =10 s Department of Mechanical Engineering Example
1 t2 = − vo ∫ vsdt RC t1 1 = K RC C =1µF 1 R = =100kΩ v (t) = −50(t − t ) 10(V s)⋅1µF o 1 Department of Mechanical Engineering Op-Amp Open Loop Gain In Op-amp, practically, the gain is so high that the output will be driven to +Vcc or –Vcc for any appreciable difference between
V1 and V2. This is useful in the comparator. For practical applications other than the comparator, negative feedback is used to control the device gain.
Department of Mechanical Engineering Comparator
The extremely large open- loop gain of an op-amp makes it an extremely sensitive device for comparing its input with zero.
For practical purposes, if V2 >V1, the output is driven to the positive supply voltage; If
V2 Department of Mechanical Engineering Comparator The basic comparator will swing its output to at the slightest difference between its inputs. But there are many variations where the output is designed to switch between two other voltage values VH and VL. Also, the input may be tailored to make a comparison to an input voltage VT other than zero. Department of Mechanical Engineering Comparator: application Often it is desirable that switching occurs at prescribed voltage . Comparator can be used for this purpose. VH if v+ > v− vo (t) = VL if v+ < v− VH and VL are the only two values that vo(t) can have. Department of Mechanical Engineering Comparator: Example For the following circuit, suppose VA >VT > vc(0), VT is a threshold voltage; and the input voltages for the comparator are v+ = vc (t) v− = VT V if v (t) > V v (t) = H c T The output voltage o VL if vc (t) < VT −t (RC ) The capacitor voltage vc (t) = VA + [vc (0)−VA ]e At t1, the comparator voltage switches from VL to VH −t1 (RC ) VT = VA + [vc (0)−VA ]e v (0)−V = c A t1 RC ln Department of Mechanical Engineering VT −VA Comparator: Example For the shown circuit, the initial value of the capacitor voltage is vc(0) =1.667V. What value of the resistance, R, is required if the comparator is to switch from VL to VH at time t1=1ms? Department of Mechanical Engineering Comparator: Example For the shown circuit, the comparator is used to compare the resistor voltage vR(t) with a threshold voltage VT. Suppose VA >VT > RiL(0). At time t1, the comparator output voltage switches from VL to VH R V V − t = A + − A L iL (t) iL (0) e R R L Ri (0)−V R t = ln L A − t1 1 L R VT −VA VT = RiL (t) = VA + [RiL (0)−VA ]e Department of Mechanical Engineering Comparator: Example For the shown circuit, the initial value of the inductor is iL(0) =1mA. What value of the inductance, L, is required if the comparator is to switch from VL to VH at time t1=10ms? Department of Mechanical Engineering The Sample and Hold circuit When The Sample and Hold circuit uses two buffers to keep a voltage level stored in a capacitor. Pressing Ssample will charge the capacitor to the present signal level, while the input buffer ensures the signal won't be changed by the charging process. From there, the output buffer will make sure that the voltage level across the storage cap won't decrease over time. Pressing Sclear will short out the storage cap, discharging it and setting the output to 0V. We can build the Sample and Hold circuit with mechanical pushbutton switches to see it in action. In actual practice, the switches used are various forms of transistor switch, which provides cleaner switching and also allows another circuit to control the sample and clearing operations. Department of Mechanical Engineering Uses of Linear Op-Amp Circuits Amplifiers – Making signals bigger! Summing Amp – Mixing (e.g. audio applications) – Analogue arithmetic Integrator – Analogue computing – Active filter design Difference Amp – Balanced line receivers Department of Mechanical Engineering Ideal Op-Amp Linear Circuit Analysis 1. I+ = 0 and I− = 0 2. V− = V+ 3. VOUT = anything ! When negative feedback is applied, the use of these assumptions makes design/analysis much easier (although not necessarily simple!). Of course, they aren’t really true Department of Mechanical Engineering