MCE63: Interfacing and Control of Mechatronic Systems Chapter 1: Impedance Analysis for Electromechanical Interfacing Part B: Input and Output Impedance Cleveland State University Mechanical Engineering Hanz Richter, PhD MCE63 p.1/14 Input and Output Impedances The input impedance of a network is the phasor ratio of the applied voltage to the resulting current at the terminals. A high input impedance does not draw too much current (power) from the source and viceversa. It is commonly said that the input impedance of a network is the impedance seen by the signal connected at the input of the network. Accordingly, one can determine the input impedance from direct measurement of amplitude and phase of current and voltage at input terminals, over a frequency range. The output impedance of a network or source is the Thévenin impedance. A low output impedance is desirable in power supplies and signal sources, so that the current drawn by the load does not change the voltage at the terminals. In situations when input or output impedances are inappropriate, an impedance matching device must be inserted between source and load. More later. MCE63 p.2/14
Power in AC Circuits Suppose a network has terminal voltage V φ+θ, while the current is I φ, that is, θ is the angle between voltage and current. The instantaneous power flowing across the terminals is p(t) = VI. The average value over a period is given by P = T VIsin(wt+φ+θ)sin(wt+φ)dt Using the trig identity sinasinb = 1 2 cos(a B) 1 2 cos(a+b) we obtain P = VI [ T T ] cosθdt cos(2wt+2φ+θ)dt 2T The second integral vanishes, while the second gives Power in AC Circuits... P = VI 2 cosθ MCE63 p.3/14 Only a fraction of VI is power dissipated at the load. The term cosθ is known as power factor. The quantity VI is known as apparent power, measured in VA, while VI 2 cosθ is the active power, measured in Watts. Resistive networks draw only active power (θ = ), while capacitors and inductors introduce reactive power. Reactive power is only exchanged between the source and reactive elements in the load Exercise: Prove that if Z = R+Xi, with tanθ = X/R then P = V 2 2 R R 2 +X 2 (1) MCE63 p.4/14
Power in AC Circuits... Resistive networks draw only active power (θ = ), while capacitors and inductors introduce reactive power. Reactive power is only exchanged between the source and reactive elements in the load. Active power can be also expressed in terms of rms voltage and current: 1 I rms = T T I 2 1 dt, V rms = T For sinusoids, I rms /I = V rms /V = 1/ 2, giving P = V rms I rms cosθ T V 2 dt The rms values are also called effective, since the heat produced by a resistor is I rms V rms and not IV. Is the 11 V at the electrical outlet the amplitude or the rms value? Although the power required to drive a generator matches the one consumed actively (assuming 1% efficiency), the utilities charge by the Impedance Matching and Bridging MCE63 p.5/14 Two typical situations demand impedance adjustements: 1: A measurement instrument has a low input impedance and/or the measured object has a high output impedance. In this case, the instrument draws too much current, which creates a significant voltage drop due to the high output impedance, altering the measurement. In measurement, one wants high input impedance for the instrument and low output impedance for the measurand. Some transducers (piezoelectric accelerometers, for instance), possess a very high ouput impedance, making voltmeter or oscilloscope readings impossible without a charge amplifier, providing impedance adjustment. Ideally, the amplifier must present a high impedance to the transducer (comparable to the transducer s output impedance) and a very low impedance to the rest of the circuit. This is called buffering, impedance bridging. MCE63 p.6/14
Impedance Matching... 2: A driven device has an input impedance that is different than the driver s output impedance. In this case, part of the power is reflected back from the load to the source. For maximum power delivered from a source to a load, impedances must be matched (more next). Waves in a rope example. Maximum Power Theorem - Variable Load Consider a source represented by its electrical Thévenin equivalent and connected to a variable load Z b = R b +X b i. The power transferred to the load is given by (show it by using Eq. 1) P = 1 2 V 2 R b Z b +Z th 2 Consider this expression as a function of two real variables P = P(R b,x b ), maximize and show that Z b = Zth ( denotes complex conjugate) results in maximum power. Find the percentage of total power that is dissipated in each of R th and R b. MCE63 p.7/14 MCE63 p.8/14
Maximum Power Theorem - Variable Internal Impedance If Z b is fixed and Z th is adjustable, it s easy to see that maximum power occurs at Z th = Edison s intuition: See interesting article at http://www.du.edu/ jcalvert/tech/jacobi.htm This case makes sense for the design of generators and power supplies For sensor interfacing, we typically care about impedance bridging A simple way to adjust impedances is by using a transformer. MCE63 p.9/14 Characteristic Impedance of a Cable A cable can be modeled as follows R R R R SE 1 1 I C I C If the cable is connected to a DC supply on one end and it is infinitely long (or very long with the opposite end open), a constant current is theoretically calculated (or measured), even when the leakage resistance is zero (or negligible). This is because energy travels endlessly through the cable (no reflection). Viewed from the battery, this is equivalent to a resistor dissipating the energy. This resistance is the characteristic impedance of the cable. MCE63 p.1/14
Characteristic Impedance of a Cable If a short cable is terminated with a resistance equal to its characteristic impedance, the input impedance of the combination is independent of length and equal to the characteristic impedance. Typical coaxial cables are 5 or 75 Ω. When a signal generator is connected to a receiving instrument with a cable, the impedances of the three elements should be as matched as possible for maximum power transfer, especially over long cables. The characteristic impedance of a coaxial cable can be approximated as Z o = 138 ǫ log 1 d 1 d 2 where d 1 is the inner diameter of the outer conductor, d 2 is the outer diameter of the inner conductor and ǫ is the relative permittivity of the insulation between the conductors (range from 2 to 4 from common cable insulators like Nylon, Polyethylene, etc.). Impedance Control in Tele-manipulation MCE63 p.11/14 MCE63 p.12/14
Impedance Control in Tele-manipulation MCE63 p.13/14 Mandatory Reading Applications of Micromechatronics in Minimally Invasive Surgery, Tendick,F., Sastry, S., Fearing, R. and Cohn, M.,IEEE/ASME Transactions on Mechatronics, v3 N.1, 1998, pp.34-42. The article is available online through the CSU Library (use IEEE Xplore). This material may be part of the course evaluation (any exam). MCE63 p.14/14