Physics 15b Lab 4: Responses to Time Dependent Voltages Chapter 8 in Purcell covers AC circuit. The chapter focuses on the response to sinusoidal signals V = Vo cos(wt). If a circuit containing only resistors, capacitors, and inductors is attached to a voltage sources that produces a sinusoidal voltage as a function of time with a characteristic frequency w, the voltage and current for each circuit element must also be have the same frequency, though it may be phase shifted with respect to the original signal. Thus, it must be of the form A sin (wt +phi), where A is an amplitude that may be larger or smaller than the applied voltage Vo, and phi is the phase shift between the applied voltage signal and the current or the voltage across a circuit element. Voltages with p/2 phase leads and phase lags are shown below. Providing only solution to sinusoidal input voltages may at first seem very limiting since there are not many truly sinusoidal signals in the world. In lab 2, you saw that even the voltage coming out of the wall in the science center is not completely sinusoidal. If there are no sinusoidal signals, why spend so much time on the response of circuits to signals they will never get? The answer is that any well behaved function of time can be expressed as a linear combination of Fourier components of the form A sin( wt) + B Cos( wt). If the function is periodic, then the linear combination is a sum over the fourier seris corresponding to the basic frequency of the function (w) , as well as all of its higher harmonics (nw) where n is an integer. The discussion below is copied from the WikipediaWikipedia uses the variable x in the discussion above, but you can replace it with the time t to get the expressions appropriate for the time dependent voltages being discussed here. The nth coefficient in the Fourier series correspond to the fraction of the signal that is at frequency n. The figure above shows a cosine wave as a function of time, its Fourier coefficients corresponding to the values n as defined in the Wikipedia discussion above, and the spectrum of the cosine wave. The cosine wave consists of one single cosine, so the amplitude of the n=1 coefficient is 1, and the amplitude for all other n is zero because the integral of the product of cos(w t) cos(n w t) =0 unless n=1. The spectrum,g(w), is a function of frequency where the value of g(w) corresponds to the amplitude of the corresponding Fourier coefficient. It again is a 1 for n=1, and a zero for all other n. This is consistent with the idea that the cosine wave contains only the single frequency w. A square wave as a function of time is the curve shown in the upper right. In contrast to the cosine wave, the cosine expansion for a square wave includes many frequencies, and allof those frequencies contribute to the square wave signal. The amplitude of the contribution of the higher frequency components decreases as 1/n, so leaving out very high n does not make a large difference to the final function. In the function generators there is always a cutoff frequency, so the square wave that comes out of a signal generator is only an approximation. The principle of superposition implies that the effect of applying several different voltages is simply the sum of the effects of each of the individual voltages. Thus, the response to an arbitrary time dependent signal is the sum of the signals due to the individual frequencies; therefore, there is no need to solve for every possible excitation, one only needs to determine the response of the system to an periodic excitation, which is done in chapter 8 of Purcell. Any arbitrary time dependent voltage can be expressed as the sum over such individual terms. The solutions to each of these excitations is determine in chapter 8. Given the individual current and voltage responses to each frequency in the arbitrary signal, one can obtain the current and voltage response to the arbitrary signal be adding up the known responses to each periodic voltage that appears in the decomposition of the arbitrary signal. This is much easier than explicitly solving for the current and voltage response of every time dependent voltage that is of interest. Given the discussion above, it is sensible to consider the response of an electronic circuit to a sinusoidal voltage at a particular frequency, and then to calculate the response to a general time dependent voltage by expressing that voltage in terms of Fourier components and using superposition to express the final result as the sum over the response to each of the individual frequency components. A very nice discussion of AC circuits excited by sinusoidal voltage is given at http://www.physclips.unsw.edu.au/jw/AC.html Pre-lab Questions: 1. Consider an RC series circuit. Let the drive voltage difference across the series combination be VS=VoCos[wt]. At what frequency is the RMS voltage across R equal to the RMS voltage across C? For w such that 1/(wC)>>R, the RC series combination can be replaced by a single circuit element. Is this element R or C? Answer the next few questions by using that singe element replacement. What is the phase shift, DF, between the voltage across this circuit element and the Vs? What is the phase shift between the current,I, and the Vs? For w such that 1/(wC)<<R, the RC series combination can be replaced by a single circuit element. Is this element R or C? Fill in the table below by replacing the series circuit with the appropriate single element, either R or C. (Hint: see the supplemental information section on driven RL and RC circuits). Frequency VR/VS VC/VS ΔΦ I and VS ΔΦ I and VR ΔΦ I and VC ΔΦ VS & VR ΔΦ VS & VCw<<1/(RC) w>>1/(RC)2. Consider an RL series circuit. Let the drive voltage difference across the series combination be VS=VoCos[wt]. At what frequency is the RMS voltage across R equal to the RMS voltage across L? For w such that wL>>R, the RL series combination can be replaced by a single circuit element. Is this element R or L? Answer the next few questions by using that singe element replacement. What is the phase shift, DF, between the voltage across this circuit element and the Vs? What is the phase shift between the current,I, and the Vs? For w such that wL<<R, the RL series combination can be replaced by a single circuit element. Is this element R or L? Fill in the table below by replacing the series circuit with the