2/15/2005 Amplifiers.doc 1/7 Jim Stiles The Univ. of Kansas Dept. of EECS Amplifiers An ideal amplifier takes an input signal and reproduces it exactly at its output, only with a larger magnitude! Now, let’s express this result using our knowledge of linear circuit theory ! Recall, the output outvt()of a linear device can be determined by convolving its input invt()with the device impulse response ()gt: () ( ) ( )tout invtgttvtdt−∞′′′=−∫ invt()() ()vooutinvtAvt=voAwhere Avo is the open-circuit voltage gain of the amplifier.2/15/2005 Amplifiers.doc 2/7 Jim Stiles The Univ. of Kansas Dept. of EECS The impulse response for the ideal amplifier would therefore be: () ()vogtA tδ= so that: () ( ) ( )()()()tout intvo invo invtgttvtdtAttvtdtAv tδ−∞−∞′′′=−′′′=−=∫∫ We can alternatively represent the ideal amplifier response in the frequency domain, by taking the Fourier Transform of the impulse response: () ()()0jtjtvovoTgte dtAte dtAjωωωδ∞−−∞∞−−∞===+∫∫ 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! ω()Tω voA2/15/2005 Amplifiers.doc 3/7 Jim Stiles The Univ. of Kansas Dept. of EECS Moreover, the ideal amplifier does not alter the relative phase of the sinusoidal signal (i.e., no phase shift). In other words, if: () ( )cosinvttω= then at the output of the ideal amplifier we shall see: ()()() ( )()coscosoutvovtT t TAtωωωω=+∠= 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 !2/15/2005 Amplifiers.doc 4/7 Jim Stiles The Univ. of Kansas Dept. of EECS Let’s look at this second problem first. The ideal amplifier impulse response () ()vogtA tδ= means that the signal at the 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: () ( )vogtA tδτ=− resulting in an amplifier output of: () ( ) ( )()()()tout intvo invo invtgttvtdtAt tvtdtAv tδττ−∞−∞′′′=−′′′=−−=−∫∫ In other words, the output is both an amplified and delayed version of the input. * Ideally, this delay does not distort the signal, as the output will have 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 secondτ ).2/15/2005 Amplifiers.doc 5/7 Jim Stiles The Univ. of Kansas Dept. of EECS * Therefore, propagation delay τ is generally not considered a problem for most amplifier applications. Let’s examine what this delay means in the frequency domain. Evaluating the Fourier Transform of this modified impulse response gives: () ()()() ()cos sinjtjtvovo vojvoTgt e dtAt edtAjAAeωωωτωδτωτωτ∞−−∞∞−−∞==−=+=∫∫ We see that, as with the ideal amplifier, the magnitude ()voTAω= . However, the relative phase is now a linear function of frequency: ()Tωωτ∠= As a result, if ()()cosinvttω= , the output signal will be: ()())() (()coscosoutvAvtT t Ttωωωωωτ==−∠− In other words, the output signal of a real amplifier is phase shifted with respect to the input.2/15/2005 Amplifiers.doc 6/7 Jim Stiles The Univ. of Kansas Dept. of EECS In general, the amplifier phase shift ()Tω∠ will not be a perfectly linear function (i.e., ()Tωωτ∠≠ ), but instead will be a more general function of frequency ω. However, if the phase function ()Tω∠ becomes too “non-linear”, we find that signal dispersion can result—the output signal can be distorted! Now, let’s examine the first 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 ()voTAω=. Note this means that the amplifier gain is Avo for all frequencies 0ω<<∞ (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 ω where the device does not work ! * From the standpoint of an amplifier, “not working” means ()voTAω (i.e., low gain). * Amplifiers will therefore have finite bandwidths.2/15/2005 Amplifiers.doc 7/7 Jim Stiles The Univ. of Kansas Dept. of EECS There is a range of frequencies ω between and LHωω where the gain will (approximately) be Avo. For frequencies outside this range, the gain will typically be small (i.e. ()voTAω): (), vo L Hvo L HATwAωωωωωωω≈<<⎧=⎨<>⎩ The width of this frequency range is called the amplifier bandwidth: LH (radians/sec)f f (cycles/sec)HLBandwidthωω−− One result of having a finite bandwidth is that the amplifier impulse response is not an impulse function ! () ( ) ( )jtvogt T e dt A tωωδτ∞+−∞=≠−∫ The ideal amplifier is not possible! ω()Tω