Experiment 9: First and Second Order Circuit Transients | ECE 225, Lab Reports of Electrical Circuit Analysis

Download Experiment 9: First and Second Order Circuit Transients | ECE 225 and more Lab Reports Electrical Circuit Analysis in PDF only on Docsity! Boise State University Department of Electrical and Computer Engineering ECE225L – Circuit Analysis and Design Lab Experiment #9: First and Second-Order Circuit Transients 1 Objectives The objectives of this laboratory experiment are: • To investigate the time response of a first-order circuit to a step input and to determine its characteristics. • To investigate the time response of an underdamped second-order circuit to a step input and to determine its characteristics. 2 Theory K ωs+ o V (s)V (s)i o Figure 1: First-Order Transfer Function Consider the transfer function of a first-order circuit represented by the block diagram shown in Figure 1. This transfer function is said to be normalized if K = ωo. Otherwise, it is said to be unnormalized. For a unit step change in the input voltage, vi(t) = u(t) =⇒ Vi(s) = 1 s (1) the circuit output response is Vo(s) = H(s)Vi(s) = K s + ωo × 1 s = K s(s + ωo) Expanding this Laplace fraction into a partial expansion of elementary terms, we obtain Vo(s) = K/ωo s − K/ωo s + ωo (2) whose inverse Laplace transform is vo(t) = K ωo (1− e−ωot) = K ωo (1− e−t/τ ) (3) where τ = 1 ωo (4) 1 is the time constant of the circuit. This time constant can be interpreted as the time it takes for the output voltage vo(t) to increase from zero to vo(τ) = K ωo (1− e−1) = 0.632 K ωo (5) or about 63.2% of its final or steady-state value Vo,ss = K ωo (6) The steady error of the output with the unit step input is Ess = Vo,ss − Vi = K ωo − 1 oiV (s) V (s)2 2 n nωs + 2 s +ξω K Figure 2: Second-Order Transfer Function Consider the transfer function of a second-order circuit represented by the block diagram shown in Figure 2. This transfer function is said to be normalized if K = ω2n. Otherwise, it is said to be unnormalized. In the following discussion, assume a normalized transfer function with K = ω2n and expressed in a standard notation as H(s) = Vo(s) Vi(s) = ω2n s2 + 2ζωns + ω2n (7) where ωn is called the natural frequency and has units of rad/s and ζ is called the damping factor and is unitless. The physical interpretation of ωn and ζ will be explained shortly. For given values of ωn and ζ and a step change in the input voltage vi(t), the transient performance of the output voltage will depend on the poles of the transfer function H(s), that is, on the roots of the denominator of H(s). These roots will be either real or complex depending on the sign of the discriminant ∆ = (2ζωn)2 − 4ω2n = 4ω2n(ζ2 − 1) (8) The second-order circuit is said to be overdamped if the discrimant ∆ is positive, that is, if ζ > 1. In this case, the transfer function H(s) has two real negative poles given by s1,2 = −σ1,2 = − ζωn ± ωn √ ζ2 − 1 (9) The second-order RLC circuit is said to be critically-damped if the discrimant ∆ is zero, that is, if ζ = 1. In this case, the transfer function H(s) has a double real negative pole given by s1,2 = −σ, − σ = − ωn, − ωn (10) The second-order circuit is said to be underdamped if the discrimant ∆ is negative, that is, if 0 ≤ ζ < 1. In this case, the transfer function H(s) has two complex conjugate poles with negative real parts given by s1,2 = −σ ± jω = − ζωn ± jωn √ 1− ζ2 (11) 2 Figure 4(b) shows the response of vo(t) for a unit step input vi(t) = u(t). The final (steady-state) value of vo(t) is Vo,ss = 1. Notice that vo(t) is oscillatory in shape because of the complex nature of the poles of H(s). The transient frequency ω is defined as ω = ωn √ 1− ζ2 (24) This frequency can be found in Figure 4(b) by measuring the transient period T between two suc- cessive peaks. Other performance measures of the underdamped response are as follows: The maxima and minima (peaks) of the underdamped circuit response occur when dvo dt = ωn√ 1− ζ2 sin(ωn √ 1− ζ2t) = 0 =⇒ ωn √ 1− ζ2t = 0, π, 2π, . . . (25) The peak time Tp is defined as the time of the first maximum peak and occurs when ωn √ 1− ζ2Tp = π =⇒ Tp = π ωn √ 1− ζ2 (26) The peak value at time Tp is equal to Mp = (1 + e−ζωnTp) = (1 + e−πζ/ √ 1−ζ2) (27) Thus the maximum percent overshoot PO is equal to PO(%) = (Mp − 1)× 100% = 100e−πζ/ √ 1−ζ2 (28) and it depends only on the damping factor ζ in the circuit. 3 Equipment • HP/Agilent 54810A Infinium Oscilloscope • HP/Agilent 33120A Function/Arbitrary Waveform Generator • Fluke 111 True RMS Multimeter 4 Procedure Part A: Transient Response of a First-Order Circuit 1. Build the op-amp circuit of Figure 5. Measure the capacitor value C4 using an RLC meter and use precision resistors. f (kHz) C4 (nF) Nominal 1 10 Measured 1 2. Apply a 20-Hz, 1-V peak-to-peak square wave with 0.5 V DC offset to the op-amp circuit. Using the cursors on the infinium scope, measure the time constant τ and the steady-state value Vo,ss of the first-order circuit. 5 V (s) i V (s) − + o v (t) i v (t) − + − + +++ (c) −1 − s+K K V (s) −−− R=10k R =100k 4 R =100k 4 C =10n 3 R =100k 1 2 R =100k R=10k R=10k R=10k o oi V (s) s+2K K − + (b)(a) Figure 5: Closed-Loop First-Order Circuit τ (ms) Vo,ss (V) Nominal 0.5 0.5 Measured Part B: Transient Response of an Underdamped Second-Order Circuit 1. Build the op-amp circuit of Figure 6. Measure the capacitor value C2 using an RLC meter and use precision resistors. f (kHz) C2 (nF) Nominal 1 10 Measured 1 2. Apply a 20-Hz, 1-V peak-to-peak square wave with 0.5 V DC offset to the op-amp circuit. Observe the underdamped circuit response. Notice that the square wave allows us to charge and discharge the capacitor voltage so that we can continuously display the transient response on the oscilloscope. 3. Measure the peak time Tp, peak value Mp, and rise time Tr. Tp (ms) Mp (V) Tr (ms) Nominal 3.63 1.16 - Measured 6 i V (s)V (s) i V (s) − + o v (t) i v (t) − + − + + − + − + − (c) − o 2 2 s+Ks+K K o V (s) s K − 2 C =10n 4 4 R =100k C =10n 3 R =100k 1 R =100k R=10k R=10k R=10k R=10k s+K K − + (b)(a) Figure 6: Closed-Loop Second-Order Circuit 5 Report Questions 1. In Part A, compute the measured time constant τ and compare it with the expected (theo- retical) value of τ = R4C4/2. 2. In Part B, compute ζ using Equation (27) and the measured Mp. 3. Substitute ζ into Equation (26) and solve for ωn using the measured Tp. Compare this value of ωn with the expected one of ωn = K = 1/(R4C4). 4. Compute the poles of the underdamped transfer function by solving the quadratic equation D(s) = s2 + Ks + K2 = 0 (29) where K = 1/(R4C4) = 1/(R1C2) and compare these values to the measured ones given by s1,2 = −ζωn ± jωn √ 1− ζ2 (30) 7