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Transcript
Section H2: Preliminary Material In this section, we are going to briefly review topics in s-domain analysis and transfer functions, as well as discussing the frequency characteristics of capacitive components. s-Domain Analysis Most of our work will be performed in the complex frequency domain, or s-domain, where s=jω and j = − 1 . In s-domain, capacitors are replaced by either their admittance, given by YC=sC, or their impedance, given ZC=1/sC. Inductors are replaced by an admittance of YL=1/sL or an impedance of ZL=sL. All circuit analysis techniques are still utilized and all circuit parameters and behaviors may be defined in terms of s-domain impedances. For example, in the s-domain, the frequency dependent voltage gain may be expressed as a transfer function, T(s), as T (s) = Vo (s) . Vi (s) Note: The discussion that follows holds for all transfer functions. The transfer function notation, T(s), is used above instead of our usual Av(s) to keep stuff as general as possible. Once the transfer function has been determined, it may be evaluated for physical frequencies by replacing s with jω. The resulting complex transfer function T(jω) may be defined in terms of a magnitude and an angle for all values of ω. The magnitude, |T(jω)|, provides the magnitude of the amplifier response with respect to frequency and the angle of T(jω) gives the phase response as a function of frequency. In general, we are going to be dealing with transfer functions of the form T (s) = am s m + am−1 s m−1 + K a1 s 1 + a0 s n + bn −1 s n −1 + K b1 s 1 + b0 , where (1) ¾ the order of the numerator (m) is less than or equal to the order of the denominator (n), and ¾ the coefficients (all a’s and b’s) are real numbers. In addition, if all the roots of the denominator have negative real parts, the system is stable… very important and we’ll get to it shortly. The transfer function of Equation 1 may also be expressed in the form: T (s) = am (s − Z1 )(s − Z 2 ) L (s − Z m ) , where (s − P1 )(s − P2 ) L (s − Pn ) (2) ¾ Z1, Z2,…,Zm are the roots of the numerator polynomial, called the zeros of the transfer function, ¾ P1, P2,…,Pn are the roots of the denominator polynomial, called the poles of the transfer function, and ¾ am is the multiplicative constant (the coefficient of sm in the original numerator polynomial). A transfer function is completely defined in terms of its poles, zeros and the multiplicative constant. Poles and zeros may be real or complex numbers. If complex, poles (or zeros) must occur in complex conjugate pairs. For example, if a root of the denominator (pole) occurs at 6+j3, there is automatically another pole at 6-j3 (note that the real part of a complex conjugate pair may be equal to zero; i.e., for this example with a zero real part the pair would consist of +j3,-j3). Since we are focused on linear amplification, the amplifiers we are interested in may be analyzed as linear systems where the complete frequency response is given by the magnitude and phase shift for all input frequencies. We will be concentrating on the Bode plot method, which will allow us to analyze and generate frequency plots almost by inspection after we define the poles and zeros of the system transfer function as shown in Equation 2 above. Capacitive Considerations Before we start talking about specific frequency characteristics and responses, let’s look at what actually happens to the frequency dependent components – specifically external capacitances - that we have previously considered as ideal. We have used bypass and coupling capacitors to couple stages, provide dc isolation, and reduce or eliminate the effects of certain resistors. In our studies to date, the capacitor was considered an open to dc and frequencies below the operational range and a short to frequencies in the operational range. This allowed complete blocking of dc bias voltages and any propagating dc signal while allowing all desired signal components to pass without attenuation, for a frequency response as illustrated in the figure to the right. Realistically, capacitors do not instantaneously switch from a short circuit condition to an open circuit condition. Recall from circuit theory that capacitive impedance is given by ZC = 1 1 = , jωC sC where ω is the radian frequency (in radians per second) and is related to the cyclical frequency, f (in cycles per second or Hertz) by ω=2πf. This means that the coupling and bypass (external) capacitances cannot respond as ideal, and that they gradually take on open circuit characteristics as the frequency gets smaller. Also, in addition to any capacitances that are intentionally placed in a circuit, unintentional capacitances also exist. Remember that a simple parallel plate capacitor is composed of two conducting plates with a non-conducting (dielectric) medium between them. Using this concept, we can see that internal capacitances exist within semiconductors in the junctions and depletion regions, between contacts, and between conductors in IC layouts. We’re going to be talking more about these internal capacitances when we discuss high frequency behavior but, usually, their overall effect is to cause a decrease in gain at high frequencies by effectively shorting the output signal. The frequency dependent behavior of capacitive impedance, regardless of the source of capacitance, forces a modification to the ideal gain versus frequency plot presented above. A generic gain versus frequency response plot for an RC coupled amplifier is presented to the right (Figure 10.1 of your text, slightly modified). As previously mentioned, high frequency behaviors are usually determined by the internal capacitances, while the external capacitances (coupling and bypass) are responsible for lowfrequency response. The frequency axis in the figure above indicates the two possible labeling schemes; i.e., linear frequencies in Hz (cycles per second) or radian frequencies in radians per second. Don’t let this confuse you, but be sure that you know which frequency representation is being used! Remember, ω=2πf and f=ω/2π. Note that there are three distinct regions defined in Figure 10.1. In the midfrequency range (also called midband) the gain is maximum and the frequency response is (ideally) flat. The corner frequencies, which are also called break frequencies, half-power frequencies, -3dB frequencies, or cutoff frequencies, are indicated in the figure above by fL and fH (or ωL and ωH) and define the lower and upper limits of the midband range. Whatever name is used to refer to these points, they are by definition the frequencies at which the current or voltage gain drops to 0.707 ( = 1 2 ), or ( ) –3dB ( = 20 log 1 2 ), of the midrange value. All frequencies below fL (ωL) are considered to be in the low frequency range, while those above fH (ωH) are in the high frequency range. Rather than try to include all the effects of all possible capacitances at one time, which is virtually impossible to solve without a computer, we are going to break up the analysis into separate steps involving the low and high frequency regions and input and output circuits. For the low to midfrequency ranges, we can use the transistor models we have been using so far. High frequency analysis will involve the development of transistor models that include the capacitances between each pair of transistor terminals…we’ll get to that later! Before we go further, let’s have a definition review in an attempt to reduce the number of parentheses in the following discussion: ¾ pole: root of the denominator of the system transfer function or, equivalently, a frequency that causes the denominator of the system transfer function to go to zero. ¾ zero: root of the numerator of the system transfer function or, equivalently, a frequency that causes the numerator of the system transfer function to go to zero. ¾ decade: a factor of ten difference in frequency. For example, a decade above 500Hz would be 5kHz and a decade below 500Hz would be 50Hz. ¾ octave: a factor of two difference in frequency. For example, an octave above 500Hz would be 1kHz and an octave below 500Hz would be 250Hz. ¾ dominant pole: a pole that is separated by at least a decade (your author states two or three decades while other sources define a separation of two octaves as sufficient) from its nearest neighbor. A dominant pole for the low frequency response would be at least a decade above its nearest neighbor, while a dominant pole for the high frequency response would be at least a decade below its nearest neighbor. To perform hand analysis (and design) of frequency dependent amplifiers, we are going to want to simplify things as much as possible while still maintaining a reasonable level of realism. Specifically, we will be looking at the location of the poles of the transfer function. If we can define a dominant pole, any other poles would have little or no effect on the frequency response of interest and the frequency of the dominant pole would be the cutoff frequency (or whatever other name you may want to call it). However, if the poles are not widely separated and we cannot define a single dominant pole, or if the circuit is complex and it is not obvious that a dominant pole exists, we can determine the cutoff frequency through the following procedure: 1. Find the time constant associated with each capacitor. First, set all independent sources to zero (short voltage sources and open current sources), then select a capacitor and eliminate all other capacitors in the circuit. Now…how you accomplish this depends on whether you are performing low or high frequency analysis. ¾ For low-frequency analysis, external capacitances are dominant. Capacitances (other than the one of interest) are set to infinity; i.e., short circuits. All internal capacitances are considered opens. ¾ For high-frequency analysis, internal capacitances are dominant. Capacitances (other than the one of interest) are set to zero; i.e., open circuits. All external capacitances are considered shorts. For each capacitance Ci, we are left with a circuit that contains one capacitor and a combination of resistances. The equivalent resistance, Reqi, across the capacitor terminals is calculated and the product of this equivalent resistance and the capacitance yields the time constant associated with the particular capacitance (τi=CiReqi). 2. Determine the cutoff frequencies from the time constants. The reciprocal of each time constant represents the frequency associated with the individual pole (ωi=1/τi). If the frequencies are widely separated, a dominant pole may be defined. If not, the poles will interact and the cutoff may be crudely approximated by ωL ≅ 1 ∑τ i i = i ∑C R i i eqi for the low frequency cutoff, and ωH ⎡ ⎤ ≅ ⎢∑ τ i ⎥ ⎣ i ⎦ −1 = 1 for the high frequency cutoff, ∑ C i Reqi i where each summation is the superposition of the contributions of all relevant capacitances. If poles (and zeroes if necessary) are known, can be determined, or if the crude approximation is not sufficient, a more accurate approximation for the cutoff frequencies may be found through ω L ≈ ω P 1 + ω P22 + L − 2(ω Z2 1 + ω Z2 2 + L) 2 ωH ≅ . 1 1 ω P21 + 1 ω P22 + 1 ω P23 ⎞ ⎛ 1 1 1 + K − 2⎜⎜ 2 + 2 + 2 + K⎟⎟ ⎠ ⎝ ω Z1 ω Z 2 ω Z 3