Fall 2011 Lab38a3.doc ET 438a Automatic Control Systems Technology Laboratory 3 Practical Integrator Response Objective: Design a practical integrator circuit using common OP AMP circuits. Test the frequency response and phase shift of the integrator with a variable frequency sine wave signal. Compare the lab measurements to the theoretical calculations for the circuit to check the design. Observe the integrator output signals for various types of input signals commonly used in lab. Theoretical Background The mathematical operation of integration can be simulated by replacing the feedback resistor in an inverting OP AMP circuit and inserting a capacitor. This ideal integrator 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 and Iin = Vi/Rin (1) Figure 1. Ideal Integrator Circuit.Fall 2011 Lab38a3.doc 2 The current in the capacitor is derived from: Substituting Equation 1 into the integral equation above gives the input-output relationship of the ideal integrator circuit. The constant, KI in Equation 2b is the integrator's gain. This integrator circuit sums current Iin into the feedback capacitor as long as a voltage is applied to the input. This current produces the output voltage of the circuit. In an ideal OP AMP, the output voltage will remain constant until a negative voltage is applied to the input. This will cause the voltage at the output to decrease. If the input voltage remains connected long enough, a practical OP AMP circuit will reach its power supply limits and saturate. 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 integrator circuit, the first step is to take the Lap7lace transform of the input-output relationship of Equation 2a. In the Laplace domain, integration in time converts to division by the complex variable s. (s represents the complex frequency - transient and sine response of a system.) Taking the Laplace transform of 2a gives dt (t)i C1 - = Vdtdv C - = icoc∫ (1) (b) K = RC1 -(a) dt (t)V C R1 - = (t) VIo ∈∫ (2) (c) RCs1 - = (s)V(s)V(b) (s)V s1 RC1 - = (s)V(a) (s)V = (t))vL( (s)V = (t))vL(oooo∈∈∈∈ (3)Fall 2011 Lab38a3.doc 3 Equation 3c is the transfer function of the ideal integrator 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 in circuit gain as frequency changes, and then 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 integrator circuit. The equations in (5) show that the gain of this circuit increases as the frequency decreases. In fact, the circuit has an infinite gain to dc 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)] The plot in Figure 2 shows the gain response of the ideal integrator circuit. The phase shift is a constant 90 degree over the entire range of frequency. Notice that the gain of the ideal integrator decreases with a constant rate over the range of the plot. The gain goes down 20 db for every decade (power of 10) in frequency increase. The value of 20 db is 1/10 of the initial gain value. As frequency continues to increase the gain will continue to diminish at the same rate. As frequency decrease, the gain will continue to increase. This increasing gain to lower frequencies produces a practical limit for using this circuit. (z) Re(z) Im = )(z Im + )(z Re = |z |1-22tanφ (4) °90 = 90 - 180 = RC1 = |)( A |RCj 1 - = )(j V)(j V = )(j Avinnovφωωωωωω (5)Fall 2011 Lab38a3.doc 4 1 10 100 1 1031 1040204060Frequency (rad/S)Gain (DB)db ωkωk Figure 2. Frequency Response of an Idea Integrator. The ideal integrator is not a practical circuit. The infinite gain to dc makes it impossible to construct because practical OP AMP require bias currents to flow in the inverting and non-inverting leads. These currents cause the capacitor to charge to the amplifier maximum output voltage even when on input is connect to the circuit. A practical OP AMP integrator approximates the characteristics of the ideal circuit, but has a fixed gain to dc. Bias currents also produce 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/sFall 2011 Lab38a3.doc 5 Practical OP AMP Integrators 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 low pass 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,