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5/10/2017 840996991 1/7 Amplifier Frequency Response An ideal amplifier takes an input signal and reproduces it exactly at its output, only with a larger magnitude! vin(t ) vout(t ) Avo vin(t ) Av where Avo is the open-circuit voltage gain of the amplifier. Now, let’s express this result using our linear circuit theory ! Recall, the output vout(t ) of a linear device can be determined by convolving its input vin(t ) with the device impulse response g (t ) : vout (t ) t g (t t )v in (t )dt 5/10/2017 840996991 2/7 The impulse response for the ideal amplifier would therefore be: g (t ) Avo (t ) so that: vout (t ) t g (t t )v in (t )dt t A (t t )v vo in (t )dt Avo vin (t ) We can alternatively represent the ideal amplifier response in the frequency domain, by taking the Fourier Transform of the impulse response: T ( ) g (t )e j t dt j t A ( t ) e dt vo Avo j 0 This result, although simple, has an interesting interpretation. It means that the amplifier exhibits gain of Avo for sinusoidal signals of any and all frequencies! T Avo 5/10/2017 840996991 3/7 Moreover, the ideal amplifier does not alter the relative phase of the sinusoidal signal (i.e., no phase shift). In other words, if: vin (t ) cos (t ) then at the output of the ideal amplifier we shall see: vout (t ) T cos (t T ) Avo cos (t ) BUT, there is one big problem with an ideal amplifier: They are impossible to build !! Q: Why is that ?? A: Two reasons: a) An ideal amplifier has infinite bandwidth. b) An ideal amplifier has zero delay. Not gonna happen ! Let’s look at this first problem first. The ideal amplifier impulse response g (t ) Avo (t ) means that the signal at the 5/10/2017 840996991 4/7 output occurs instantaneously with the signal at the input. This of course cannot happen, as it takes some small, but non-zero amount of time for the signal to propagate through the amplifier. A more realizable amplifier impulse response is: g (t ) Avo (t ) resulting in an amplifier output of: vout (t ) t g (t t ) v in (t )dt t A vo (t t ) vin (t )dt Avo vin (t ) In other words, the output is both an amplified and delayed version of the input. * Note the delay does not distort the signal, as the output has the same form as the input. * Moreover, the delay for electronic devices such as amplifiers is very small in comparison to human time scales (i.e., 1 second ). * Therefore, propagation delay problem for most amplifier applications. is not considered a 5/10/2017 840996991 5/7 Let’s examine what this delay means in the frequency domain. Evaluating the Fourier Transform of this modified impulse response gives: T ( ) g (t ) e j t dt A vo (t ) e j t dt Avo cos ( ) j Avo sin ( ) Avo e j We see that, as with the ideal amplifier, the magnitude T ( ) Avo . However, the relative phase is now: T As a result, if vin t cos (t ) , the output signal will be: vout (t ) T cos (t T ) Av cos (t ) In other words, the output signal of a real amplifier is phase shifted with respect to the input. In general, the amplifier phase shift T will not be a perfectly linear function (i.e., T ), but instead will be a more general function of frequency 5/10/2017 840996991 6/7 Now, let’s examine the second problem with the ideal amplifier. This problem is best discussed in the frequency domain. We discovered that the ideal amplifier has a frequency response of T Avo . Note this means that the amplifier gain is Avo for all frequencies 0 (D.C. to daylight !). The bandwidth of the ideal amplifier is therefore infinite ! * Since every electronic device will exhibit some amount of inductance, capacitance, and resistance, every device will have a finite bandwidth. * In other words, there will be frequencies where the device does not work ! * From the standpoint of an amplifier, “not working” means T Avo (i.e., low gain). * Amplifiers will therefore have finite bandwidths. There is a range of frequencies between L and H where the gain will (approximately) be Avo. For frequencies outside this range, the gain will typically be small (i.e. T Avo ): T w Avo L H Avo L , H 5/10/2017 840996991 7/7 The width of this frequency range is called the amplifier bandwidth: Bandwidth H L (radians/sec) fL fH T (cycles/sec) Avo L H One result of having a finite bandwidth is that the amplifier impulse response is not an impulse function ! g (t ) T ( ) e j t dt Avo (t ) The ideal amplifier is not really possible!