8.1 Circuit Parameters definition of decibels using decibels transfer functions ▪ impulse response ▪ rise time analysis ▪ Gaussian amplifier transfer function ▪ RC circuit transfer function ▪ analog-to-digital conversion ▪ software transfer functions ▪ averaging transfer function multi-step transfer functions 8.1 : 1/13

Amplification/Attenuation The gain or loss of a circuit is often given in decibels (dB). Pin(f) Pout(f) circuit A positive value of AdB is an amplification, a negative value an attenuation: 10 (W) = 10 dB; 100 (W) = 20 dB; 0.001 (W) = -30 dB; 0.5 (W) = -3 dB. Vin(f) Vout(f) circuit When voltage is measured, the multiplier is 20: 10 (V) = 20 dB, 100 (V) = 40 dB; 0.001 (V) = -60 dB; 0.5 (V) = -6dB. 8.1 : 2/13

Decibels Add Decibels are convenient when evaluating the performance of connected circuits. Pin(f) Pout(f) circuit 1 circuit 2 The ability to add or subtract instead of multiplying and dividing makes it easier to graph the frequency response. -10 -20 A1 log(f) + A2 = AT -10 -10 -20 -20 log(f) log(f) 8.1 : 3/13

Transfer Functions A transfer function is a mathematical or graphic depiction of how a circuit transforms the input into the output. The transfer function can be described in time, F(t), or frequency, F(f). the temporal transfer function is convolved with the input signal Vin(t) circuit F(t) Vout(t) the spectral transfer function is multiplied with the input signal Vin(f) circuit F(f) Vout(f) By expressing the voltages and transfer function in decibels, the output spectrum can be obtained by addition: Aout(f) = Ain(f) + AF(f) 8.1 : 4/13

Impulse Response The temporal transfer function can be obtained directly by sending a voltage impulse, d(t), into the circuit (assume it occurs at t = 0). d(t) F(t) circuit F(t) Since convolution with an impulse is a replication, Vout(t) = F(t). The spectral transfer function is then obtained by a Fourier transform, F(f)  F(t). This approach doesn't always work because of practical limitations. (1) It is often difficult to find a suitable approximation to an impulse function. (2) Circuits often cannot handle impulses much narrower than F(t). This is because stray capacitance will shunt the impulse to ground, and stray inductance will severely attenuate the high frequencies in an impulse. 8.1 : 5/13

Rise Time Analysis The practical difficulties with obtaining and using an impulse input can often be circumvented by using a waveform with a sharp edge. 90% circuit F(t) Vin(t) Vout(t) tr 10% The rise time, tr, of the output edge is due to convolution of Vin(t) with F(t). The rising portion of Vout(t) is the integral of F(t). The functional form of F(t) can be recovered by numeric differentiation of Vout(t). When the functional form of F(t) is known, the rise time can often be used to characterize F(t) and, via Fourier transformation, F(f). A rising edge of Vin(t) that is 3 times faster than F(t) will only cause a ~10% error in the determination of the rise time. An edge that is 10 times faster will cause a ~1% error. 8.1 : 6/13

Gaussian Amplifier (1) Most amplifiers designed to handle pulsed inputs have a Gaussian transfer function. With an impulse input, F(t) can be obtained directly by measuring its FWHM, Gt. F(f) will be Gaussian with Gf = (4 ln2)/pGt. If a rise time analysis is used, the integral of the Gaussian transfer function will be identical to a Gaussian cdf. The 10-90% rise time is obtained from the cdf tabular values, tr = cdf(0.9) - cdf(0.1) = 2.58st 8.1 : 7/13

Gaussian Amplifier (2) The frequency FWHM can be computed using the rise time, and the relationship between G and s, G = 2 (2 ln2)1/2 s Two changes need to be made to the Gf tr relationship: (1) The value of Gf is for a voltage transfer function. We need the FWHM for the power transfer function: Gpower = Gf/21/2. (2) For a Gaussian centered at f = 0, the 3dB frequency is given by Gpower/2. With these changes we can relate the measured rise time to the 3dB frequency: f3dBtr = 0.34 8.1 : 8/13

RC Circuit The transfer function of an RC circuit can be measured with equal ease using either an impulse or step function input. impulse step function Although the frequency voltage transfer function is complex, the power function is a real Lorentzian, where f3dBtRC = 1/2p. By defining a 10-90% rise time, tr = tRCln0.9 - tRCln0.1 = 2.2tRC f3dBtr = 0.35 8.1 : 9/13

Analog-to-Digital Conversion The process of analog-to-digital conversion multiplies Vin(t) by a comb function, comb(Dt). This is the only temporal transfer function we have seen so far that is a multiplication. In the frequency domain, the spectrum of Vout(f) is Vin(f) convolved with comb(Df ). Vin(t)●comb(Dt)  Vin(f)comb(Df ) As long as the frequency span (width) of Vin(f) is less than the comb spacing, the comb acts as a replicator at the fundamental, Df, and every harmonic, nDf . 8.1 : 10/13

Software Modules Increasingly, scientific instrumentation is constructed around an analog-to-digital converter followed by software processing of data. It is often advantageous to think of software modules as having transfer functions. As an example, a digital filter might be used to remove an interference. In this case the transfer function is the "missing one frequency" waveform discussed under Fourier transform examples (slide 7.5-5). Example modules: averaging least-square smoothing digital filters apodization of frequency data curve fitting software autocorrelation software lock-in amplifiers software multi-channel analyzers 8.1 : 11/13

Averaging A running average can be represented by a convolution of the signal with a rectangle. t0 The relationship between t0 and f0 is given by t0f0 = 1. This relationship is true for both power and voltage. To be consistent with other transfer functions, the 3dB frequency with power can be used, where f3dB = 0.44f0. Thus, t0f3dB = 0.44. 8.1 : 12/13

Multiple Step Transfer Functions RC Low Pass Filter Pin Gaussian Amplifier Pout Running Average ADC instrument circuit computer software Pout(t) = [[Pin(t)gauss(t0)exp(t')]●comb(Dt)]  rect(T) Pout(f ) = [[Pin(f)●gauss(f 0)●lorentz(f')]comb(Df )] ● sinc(F) Because of replication by the comb function, the spectrum at each harmonic, nDf, can be given by a sum of decibels. Aout(nDf) = Ain(nDf) + Agauss(nDf) + Alorentz(nDf) + Asinc(nDf) 8.1 : 13/13