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As should be evident from Section 354, the computation of matrix exponentials is important Various approaches are compared in [100] A key conclusion of that article is that there is no universally best algorithm An algorithm that often works very well combines the Pad approximae tion and scaling and squaring That algorithm is presented well in [60] This section presents two approaches that provide insight into the matrix exponential

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What is power, exactly Here is an all-encompassing definition: Power is the rate at which energy is expended, radiated, or dissipated This definition can be applied to mechanical motion, chemical effects, dc and ac electricity, sound waves, radio waves, sound, heat, infrared (IR), visible light, ultraviolet (UV), X rays, gamma rays, and high-speed subatomic particles In all cases, the energy is converted from one form into another form at a certain rate

(at)3 (at)2 + + 2! 3!

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Units of Power The standard unit of power is the watt, abbreviated W A watt is equivalent to a joule per second ( J/s) Sometimes power is given as kilowatts (kW or thousands of watts), megawatts (MW or millions of watts), or gigawatts (GW or billions of watts) It is also sometimes expressed as milliwatts (mW or thousandths of watts), microwatts ( W or millionths of watts), or nanowatts (nW or billionths of watts) Volt-Amperes In dc circuits, and also in ac circuits having no reactance, power can be defined this way: Power is the product of the voltage across a circuit or component and the current through that same circuit or component Mathematically this is written P = EI If E is in volts and I is in amperes, then P is in volt-amperes (VA) This translates into watts when there is no reactance in the circuit (Fig 17-1) The root-mean-square (rms) values for voltage and current are always used to derive the effective, or average, power Volt-amperes, also called VA power or apparent power, can take various forms A resistor converts electrical energy into heat energy, at a rate that depends on the value of the resistance and the current through it A light bulb converts electricity into light and heat A radio antenna converts high265

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for a, t R Extension of this power series to matrix arguments serves as a de nition of the matrix exponential eFt = I + Ft + (Ft) (Ft) + + 2! 3!

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reactance in an ac component, the power P is the product of the voltage E across the component and the current I through the component

where t R, F Rn n , and I is the identity matrix in Rn n Note that by the de nition of the matrix exponential, it is always true that a matrix commutes with its exponential: FeFt = eFt F Also, eFt t=0 = I Power series expansion is usually not the best numeric technique for the computation of matrix exponentials; however when the structure of the F matrix is appropriate, power series methods are one approach for determining closed form solutions for the matrix exponential of F An example of this is contained in Example 313

frequency ac into radio waves A speaker converts low-frequency ac into sound waves The power in these forms is a measure of the intensity of the heat, light, radio waves, or sound waves

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eFt = L 1 (sI F)

Instantaneous Power Usually, but not always, engineers think of power based on the rms, or effective, ac value But for VA power, peak values are sometimes used instead If the ac is a sine wave, the peak current is 1414 times the rms current, and the peak voltage is 1414 times the rms voltage If the current and the voltage are exactly in phase, the product of their peak values is twice the product of their rms values There are instants in time when the VA power in a reactance-free, sine-wave ac circuit is twice the effective power There are other instants in time when the VA power is zero; at still other moments, the VA power is somewhere between zero and twice the effective power level (Fig 17-2) This constantly changing power is called instantaneous power In some situations, such as with a voice-modulated radio signal or a fast-scan television signal, the instantaneous power varies in an extremely complicated fashion Have you ever seen the modulation envelope of such a signal displayed on an oscilloscope

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