RC Circuit System Investigation Dr. Kevin Craig Greenheck Chair in Engineering Design & Professor of Mechanical Engineering Marquette University RC Circuit System Investigation K. Craig 1 Engineering System Investigation Process RC Circuit Electrical System Engineering System Investigation Process START HERE Physical System System Measurement Parameter Identification Physical Model Mathematical Model Measurement Analysis Mathematical Analysis Comparison: Predicted vs. Measured Design Changes YES Is The Comparison Adequate ? NO RC Circuit System Investigation K. Craig 2 Physical System for Investigation RC Circuit Electrical System RC Circuit System Investigation K. Craig 3 Physical Modeling • Simplifying Assumptions – Resistors and Capacitors are pure and ideal – Voltage sources are ideal and supply the intended voltage to the circuit no matter how much current (and thus power) this might require – Measuring devices are ideal and do not load the circuits by drawing any current out R in RC Circuit System Investigation C out K. Craig 4 Model Parameter Identification • Measure component values using the DMM by connecting the components as shown. • RC Circuit – 15 kΩ nominal (5% tolerance), 14.986 kΩ measured – 0.01 μF nominal, 9.85 nF = 0.00985 μF measured RC Circuit System Investigation K. Craig 5 Resistors In Series Resistors in Parallel Voltage Divider RC Circuit Components Ground RC Circuit System Investigation K. Craig 6 Resistance Measurement RC Circuit System Investigation K. Craig 7 Capacitance Measurement RC Circuit System Investigation K. Craig 8 Mathematical Modeling: General Circuit Laws • We focus on basic analysis techniques which apply to all electrical circuits. • Kirchhoff’s Laws are basic to electrical circuits. • One needs to know how to use these laws and combine this with knowledge of the current/voltage behavior of the basic circuit elements to analyze a circuit model. RC Circuit System Investigation K. Craig 9 • Kirchhoff’s Voltage Loop Law (KVL) – It is merely a statement of an intuitive truth; it requires no mathematical or physical proof. – This law can be stated in several forms: • The summation of voltage drops around a closed loop must be zero at every instant. • The summation of voltage rises around a closed loop must be zero at every instant. • The summation of the voltage drops around a closed loop must equal the summation of the voltage rises at every instant. RC Circuit System Investigation K. Craig 10 • Kirchhoff’s Current Node Law (KCL) – It is based on the physical fact that at any point (node) in a circuit there can be no accumulation of electric charge. In circuit diagrams we connect elements (R, L, C, etc.) with wires which are considered perfect conductors. – This law can be stated in several forms: • The summation of currents into a node must be zero. • The summation of currents out of a node must be zero. • The summation of currents into a node must equal the summation of currents out. RC Circuit System Investigation K. Craig 11 • Sign Conventions – In electrical systems we need sign conventions for voltages and currents. – If the assumed positive direction of a current has not been specified at the beginning of a problem, an orderly analysis is quite impossible. – For voltages, the sign conventions consist of + and – signs at the terminals where the voltage exists. – Once sign conventions for all the voltages and currents have been chosen, combination of Kirchhoff’s Laws with the known voltage/current relations which describe the circuit elements leads us directly to the system mathematical model. RC Circuit System Investigation K. Craig 12 Mathematical Modeling of System out R in C out i R = i C + i out KCL iR = iC + 0 iR = iC ein − eout deout =C R dt RC Circuit System Investigation ein − eout = iR Basic Component Equations deout (Constitutive Equations) i = C dt deout RC + eout = ein dt deout τ + eout = Kein dt K =1 τ = RC K. Craig 13 pS = constant = ρgH (bottom of reservoir) Analogies A C= ρg ρ = fluid density g = acceleration due to gravity B df o + fo = fi K dt RC Circuit System Investigation p tank = ρgh dh RC + h = H dt deout τ + eout = Kein dt K. Craig 14 RC Circuit System Investigation K. Craig 15 • Numerical Solution – How do we solve this differential equation numerically? deout τ + eout = Kein dt – We use a numerical approximation: deout τ + eout = Kein dt Δeout τ + eout = Kein Δt 1 Δeout = ⎡⎢ ( Kein − eout ) ⎤⎥ Δt ⎣τ ⎦ RC Circuit System Investigation K. Craig 16 • Algorithm for Solving this Equation – Step 1: Initialize Variables τ ein t start = 0 t end Δt ( eout )initial – Step 2: Increment time and stop when done t = t + Δt If t = t end then stop – Step 3: Compute ein(t) – Step 4: Solve 1 Δeout = ⎡⎢ ( Kein − eout ) ⎤⎥ Δt ⎣τ ⎦ – Step 5: Determine new eout ( eout )new = ( eout )old + Δeout – Step 6: Go back to Step 2 RC Circuit System Investigation K. Craig 17 • Plotting Numerical Data – Engineers are well known for their ability to plot many curves of experimental data and to extract all sorts of significant facts from these curves. – The better one understands the physical phenomena involved in a certain experiment, the better is one able to extract a wide variety of information from graphical displays of experimental data. – Understand The Physical Processes Behind The Data! – When data may be approximated by a straight line, the analytical relation is easy to obtain; but when almost any other functional variation (e.g., exponential, polynomial, complex logarithmic) is present, difficulties are usually encountered. – It is convenient to try to plot data in such a form that a straight line will be obtained for certain types of functional relationships. RC Circuit System Investigation K. Craig 18 • Analytical Solution to a Step Input eout t − ⎞ ⎛ = Kein ⎜1 − e τ ⎟ ⎝ ⎠ t − eout = 1− e τ Kein t − eout =e τ 1− Kein t − Kein − eout =e τ Kein t Kein = eτ Kein − eout RC Circuit System Investigation t Kein = eτ Kein − eout deout + eout = Kein dt t − ⎞ ⎛ eout = Kein ⎜1 − e τ ⎟ ⎝ ⎠ τ ⎡ Kein ⎤ t log10 ⎢ = log10 ( e ) ⎥ ⎣ Kein − eout ⎦ τ ⎡ Kein ⎤ ⎡ 1 ⎤t log10 ⎢ log e = ( ) ⎥ ⎢ τ 10 ⎥ Ke e − ⎦ ⎣ in out ⎦ ⎣ ⎡ Kein ⎤ log10 ⎢ ⎥ Ke e − ⎣ in out ⎦ 1 slope = log10 ( e ) τ t K. Craig 19 Another Approach: Impedance + Voltage Divider Impedance e/i Allows us to use Algebra in 1 CD out e = iR e =R i e 1 = i CD de i = C = CDe dt d D= Differential Operator dt 1 eout 1 Transfer CD = = ein R + 1 RCD + 1 Function CD ( RCD + 1) eout = (1) ein ( RCD ) eout + eout = ein RC Circuit System Investigation deout RC + eout = ein dt K. Craig 20 Mathematical Analysis and Prediction RC Circuit Unit Step Response de RC out + eout = ein dt RC ( Deout ) + eout = ein 1 0.9 0.8 ( RCD + 1) eout = ein eout 1 K = = ein RCD + 1 τD + 1 Amplitude 0.63 0.7 0.6 R = 15 KΩ C = 0.01 μF 0.5 0.4 0.3 0.2 K = 1 = Steady-State Gain τ = RC = Time Constant MatLab Commands RC Circuit System Investigation 0.1 0 0 1 K = 1; tau = 0.15E − 3; RC _ System = tf (K,[tau 1]) step(RC _ System) 2 3 4 5 Time (sec) τ = RC = 0.15 m sec K. Craig 21 • Time Constant τ – Time it takes the step response to reach 63% of the steady-state value, Kein. • Rise Time Tr = 2.2 τ – Time it takes the step response to go from 10% to 90% of the steady-state value, Kein. • Delay Time Td = 0.69 τ – Time it takes the step response to reach 50% of the steady-state value, Kein. • Steady-State Value – The steady-state value of the response is Kein and at 4τ seconds (4 time constants), the response has reached 98% of the steady-state value; for all practical purposes, this is steady state. RC Circuit System Investigation K. Craig 22 Semi-Log Paper 1.0 1.1 1.2 1.3 10 1.0 1.4 1.5 1.6 1.7 50 20 30 40 1.301 1.477 1.602 RC Circuit System Investigation log10 1.8 60 1.9 2.0 70 80 90 100 1.699 1.778 1.845 1.903 2.0 1.954 K. Craig 23 RC Circuit Frequency Response DC Value = K ( x ) dB = 20log10 ( x ) Slope = -20 dB/decade -3 dB 1 1 rad = = 6666.7 τ RC sec = 1061 Hz = Bandwidth MatLab Commands K = 1; tau = 0.15E − 3; RC _ System = tf (K,[tau 1]) bode(RC _ System) RC Circuit System Investigation Semi-log plots: Magnitude (dB) vs. log10 f Phase (deg) vs. log10 f K. Craig 24 RC Circuit Amplitude Ratio = 0.707 = -3 dB Response to Input 1061 Hz Sine Wave 1 Input 1061 Hz Sine Wave Phase Angle = -45° 0.8 0.6 0.4 amplitude 0.2 0 -0.2 R = 15 KΩ C = 0.01 μF Output -0.4 -0.6 -0.8 -1 0 0.5 RC Circuit System Investigation 1 1.5 2 time (sec) 2.5 3 3.5 4 x 10 -3 K. Craig 25 • Bandwidth – The bandwidth is the frequency where the amplitude ratio drops by a factor of 0.707 = -3dB of its gain at zero or low-frequency. – For a 1st-order system, the bandwidth is equal to 1/ τ. – The larger (smaller) the bandwidth, the faster (slower) the step response. – Bandwidth is a direct measure of system susceptibility to noise, as well as an indicator of the system speed of response. RC Circuit System Investigation K. Craig 26 Measurements • RC Circuit – Time Response (Step and Sine) – Frequency Response RC Circuit System Investigation K. Craig 27 Time Response Function Generator FGEN Analog Input Channel AI 6+ DC Power Ground RC Circuit System Investigation • • • • ELVIS Connections Circuit Input to FGEN Circuit Input to Channel AI 6+ Measured Signal to Channel AI 7+ Channels AI 6- and AI 7- and Circuit Ground to DC Power Ground Analog Input Channel AI 7+ Analog Input Channels AI 6- and AI 7K. Craig 28 RC Circuit Time Response RC Circuit System Investigation K. Craig 29 Why 500 Hz ? 5τ = ( 5 )( 0.15ms ) = 0.75ms Let’s Use 1.00 ms Square Wave Period = 2.00 ms Square Wave Frequency = 500 Hz RC Circuit System Investigation K. Craig 30 RC Circuit System Investigation K. Craig 31 63% τ = 0.118ms RC Circuit System Investigation K. Craig 32 RC Circuit System Investigation K. Craig 33 RC Circuit System Investigation K. Craig 34 RC Circuit System Investigation K. Craig 35 RC Circuit System Investigation K. Craig 36 f = 1061 Hz T = 0.943 ms Magnitude Ratio: 0.798 Phase Angle: 40.5 deg RC Circuit System Investigation K. Craig 37 RC Circuit System Investigation K. Craig 38 RC Circuit System Investigation K. Craig 39 RC Circuit System Investigation K. Craig 40 RC Circuit System Investigation K. Craig 41 RC Circuit System Investigation K. Craig 42 f = 10 KHz T = 0.100 ms Magnitude Ratio: 0.156 Phase Angle: 79.2 deg RC Circuit System Investigation K. Craig 43 Frequency Response Function Generator FGEN Analog Input Channel AI 1+ Power Ground RC Circuit System Investigation • • • • ELVIS Connections FGEN to Circuit Input Circuit Input to Channel AI 1+ Measured Signal to Channel AI 0+ DC Power Ground to Channels AI 0- and AI1- and Circuit Ground Analog Input Channel AI 0+ Analog Input Channels AI 0- and AI 1K. Craig 44 RC Circuit Frequency Response RC Circuit System Investigation K. Craig 45 RC Circuit System Investigation K. Craig 46 RC Circuit System Investigation K. Craig 47