Fall 2011 Lab38a4.doc ET 438a Automatic Control Systems Technology Laboratory 4 Practical Differentiator Response Objective: Design a practical differentiator circuit using common OP AMP circuits. Test the frequency response and phase shift of the differentiator with a variable frequency sine wave signal. Compare the lab measurements to the theoretical calculations for the circuit to check the design. Observe the differentiator output signals for various types of input signals commonly used in lab. Theoretical Background The mathematical operation of differentiation can be simulated by removing the input resistor in an inverting OP AMP circuit and inserting a capacitor. This ideal differentiator circuit is show in Figure 1. If ideal OP AMP circuit operation is assumed, no current will flow into the inverting terminal of the amplifier due to the infinite input impedance. Also, the voltage between the inverting and non-inverting terminal is equal due to the effects of the negative feedback. This means that the voltage at the inverting terminal is at ground potential. So: -If = IC CRfViVo Figure 1. Ideal Differentiator Circuit.Fall 2011 Lab38a4.doc 2 The current in the capacitor is given by: Combining the last three equations above gives the input-output relationship of the ideal differentiator circuit. The constant, Kd define in Equation 2b is the differentiators gain. This differentiator circuit only has current flowing in the input when there is change in Vi(t). When there is no change in the input voltage, no current will flow and the output voltage Vo(t) will be zero. The ideal differentiator circuit only produces an output when ever there is a change in the input signal. This is useful in control circuits where rapid response to a change in the control variable is necessary. Another way of examining the circuit is to check its output gain response to sine waves of different frequencies. When the gain of these tests is represented in db and the frequency is plotted on a logarithmic scale, a Bode plot is produced. Bode plots are used to determine the stability of control systems and the frequency response of filter circuits. To find the Bode plot of the ideal differentiator circuit, the first step is to take the Laplace transform of the input-output relationship of Equation 2a. In the Laplace domain, differentiation in time converts to multiplication by the complex variable s. (s represents the complex frequency - transient and sine response of a system.) R i - = Vdtdv C = iffoc• (1) (b) K = RC -(a) dt(t) V d C R - = (t) Vdio (2)Fall 2011 Lab38a4.doc 3 Taking the Laplace transform of 2a gives: Equation 3c is the transfer function of the ideal differentiator circuit of Figure 1. To convert this to a Bode plot, we must replace the complex variable s with its imaginary part to find the change of the circuit's gain as frequency changes, the magnitude and phase shift of the transfer function can be found. The magnitude and phase of any complex quantity can be found from the following relationships: Where z = a complex value Re(z) = the real part of z Im(z) = the imaginary part of z φ = the phase angle of z The equations below show this theory applied to the ideal differentiator circuit. The equations in (5) show that the gain of this circuit increases as the frequency increases. In fact, the circuit has an infinite gain to high frequency signals. The phase shift is a constant -90 degrees. This includes the 180 degree shift due to the inverting OP AMP configuration. To construct the Bode plot the gain must be converted to db by using the formula db(ω) = 20 log[Av(ω)] (c) RCs - = (s)V(s)V(b) (s)V sRC - = (s)V(a) (s)V = (t))vL( (s)V = (t))vL(ioioiioo (3) (z) Re(z) Im = )(z Im + )(z Re = |z |1-22tanφ (4) °90- = 270 = 90 + 180 = RC = |)( A |RCj - = )(j V)(j V = )(j Aviovφωωωωωω (5)Fall 2011 Lab38a4.doc 4 The plots below show the gain response of the ideal differentiator circuit. The phase shift is a constant -90 degree over the entire range of frequency. Notice that the gain of the ideal differentiator increases at a constant rate over the range of the plot. The gain goes up 20 db for every decade (power of 10) in frequency increase. The value of 20 db is x10 that of the initial gain value. The ideal differentiator is not a practical circuit. The infinite gain to high frequencies makes it impossible to construct because most noise signals are at high frequencies. Using the configuration shown is Figure 1 will cause the OP AMP circuit to go to saturation due to the high gain amplification of this electrical noise. The bias current flowing in Rf also produces offset voltage error in the output. This voltage error can be minimized by adding an appropriately sized resistor in the non-inverting input of the OP AMP. The bias currents flowing through these resistors will develop a common mode voltage (same magnitude and phase) at the inputs to the OP AMP. The common mode voltage will not be amplified. Note that the gain of the circuit reaches 0 db (1) at the frequency given by the value ωc = 1/RC Where ωc = the cutoff frequency of the device in rad/s 10 100 1 1031 1041 1050204060Frequency (rad/S)Gain (DB)db ωkωk Figure 2. Ideal Differentiator Frequency Response.Fall 2011 Lab38a4.doc 5 Practical OP AMP Differentiators Figure 3 shows a practical integrator circuit that overcomes the limitations of the ideal circuit and still simulates the integrator action that is useful in control applications. This circuit is also known as an active highpass filter. The value of resistor Rb is given by the parallel combination of the input and feed back resistances. In equation form this is: Rb = Ri || Rf = Rf(Ri)/(Rf+Ri) If the transfer characteristic of an inverting OP AMP circuit is written as the ratio of two impedances that have been converted using the rules of the Laplace transform, then the elements in the input branch can be combined using the rules of series impedances. The resulting value can then be substituted into the inverting gain formula and the transfer function written without a large amount of computation. The following equations sketch out the mathematics used to find the transfer function for the practical integrator circuit. Taking the Laplace of 6a