Circuits II EE221 Unit 4 Instructor: Kevin D. DonohueNode Voltage ExampleTransfer FunctionsSlide 4Transfer Function ExampleSlide 6Evaluating/Plotting TFsResulting PlotsTF UnitsDecibel ScaleSlide 11Bode PlotBode Plot (with Matlab)Matlab ExampleSlide 15Slide 16Slide 17Matlab Example Test ScriptSlide 19Slide 20Plot ResultsSlide 22Plot RangesSlide 24Slide 25Slide 26Slide 27Slide 28Slide 29Complex FrequencyBode SketchesSlide 32Slide 33ResonanceQuality FactorResonant Circuit ExampleSlide 37Circuits IIEE221Unit 4Instructor: Kevin D. DonohueTransfer Function, Complex Frequency, Poles and Zeros, and Bode Plots, Resonant CircuitsNode Voltage ExamplePerform phasor analysis to determine vo(t). Since the frequency of the source is not specified, leave impedances in terms of j = s. Express the phasor of vo(t) in terms of a product between a rational polynomial in s and the phasor of the input. Then substitute  = 2 and solve for vo(t) .Show that Show v0(t) = .33cos(2t-9.46) V , when =2 vs(t) = cos(t) V2 4 0.5 F2 H+vo(t)- 212222233tan90ˆ9)3(ˆ33ˆˆsssoVVsssVVTransfer FunctionsFrom the last example let:Note that for s = j, the above function represents the phasor ratio of output to input for any :Therefore, for any input magnitude, phase, and frequency of i the output can be determined by multiplying the input magnitude by and adding to the input phase. In this sense, describes how to transfer the input value to the output value.33)(ˆ2ssssH21222233tan909)3()(ˆ)(ˆ)(ˆjHjVjVso)(ˆijH)(ˆijH)(ˆsHTransfer FunctionsDefinition: A transfer function (TF) is a complex-valued function of frequency associated with an input-output system, such that for a sinusoidal input and the TF evaluated at the input frequency, the TF magnitude indicates the scaling between the input and output magnitudes, and the TF phase indicates the phase shifting between the input and output phases.The transfer function indicates this relationship for all input frequencies. To find a transfer function, convert to impedance circuit, but leave the impedance values as functions of j=s. Then solve for the ratio of the phasor output divided by the phasor input.Transfer Function ExampleDetermine the transfer function for the circuit below, where the input is vi(t) and the output is io(t).Show transfer function is given by:vi(t)R1R2Cio(t)21212101ˆˆ)(ˆRCRRRsRCRVIsHiTransfer Function Example Determine the transfer function for the circuit below, where the input is vi(t) and the output is vo(t).Show transfer function is given by:-+vi(t)RfR1RhC+vo(t)-CRssRRVVsHhifi11ˆˆ)(ˆ0Evaluating/Plotting TFsThe following Matlab commands can be used to plot the results of the last TF. Let Rf = 50k, Ri = 10k , Rh=1.59k , C = 1FRf = 50e3; Ri = 10e3; Rh=1.59e3; C = 1e-6;f = [0:500]; % generate x-axis pointss = j*2*pi*f;h = (1+Rf/Ri)*s ./ (s + (1/(Rh*C))); % Evaluate TF at every pointfigure(1) % Plot Magnitudeplot(f,abs(h))xlabel('Hertz')ylabel('Magnitude')figure(2) % Plot Phase in Degreesplot(f,phase(h)*180/pi)xlabel('Hertz')ylabel('Degrees')Resulting Plots0 50 100 150 200 250 300 350 400 450 500020406080100HertzDegrees0 50 100 150 200 250 300 350 400 450 5000123456HertzMagnitudeTF UnitsThere are 4 possible ratios of voltages and currents that can be used for a TF. List all possibilities and indicate the units for each one.)(ˆ)(ˆ)(ˆjVjVjHinout)(ˆ)(ˆ)(ˆjVjIjHinout)(ˆ)(ˆ)(ˆjIjIjHinout)(ˆ)(ˆ)(ˆjIjVjHinoutDecibel ScaleA decibel is a logarithmic measure of gain (or attenuation). A power gain between the designated system input and output is denoted in Decibels (or dB) as:Express power in terms of voltage over a load to get another version of the dB formula:inoutdBPPG10log10  outininoutdBoutininoutininoutoutdBRRVVGRRVVRVRVG101022102210log10log20log10//log10Decibel ScaleIf input and output impedances are considered equal (Rin= Rout) the formula reduces to:Linear gain is always positive (between 0 and ). Describe the gain in dB when the system is attenuating. Describe the gain in dB when the linear gain is unity. inoutdBVVG10log20Bode PlotBode plots provide information on the general behavior of circuits over a broad range of frequencies and magnitudes where important features exist.A Bode plot of a TF shows the magnitude in dB and phase in degrees on a frequency scale over the range where all important features exist.Important features of the plot include resonances, nulls, transitions between pass and stop bands, and asymptotic gains (not flat regions that do not change!).Bode Plot (with Matlab)Before building our understanding on how to manipulate TF polynomials for convenient plotting, consider using Matlab to generate the plot from polynomials evaluated over a complex field.Matlab functions behave like a subroutine in a main program. Specify input and output arguments for your function (subroutine). Only the input and output arguments exist in the Matlab workspace or main program. Help information for the function is the first set of comments you include after the function definition. These comments must provide all user needs to know in order to use it properly. At the Matlab prompt, typing >> help “function name” will display these comments. Example: Create a Matlab function that evaluates a TF based on its polynomial coefficients. Generate a script that calls this function to show how it works.Matlab ExampleFor a function, start the first line with the word “function” followed by the function syntax (input and output arguments). Comments immediately following this first line will be used as the “help” for this function and must show the complete syntax on how to use function along with descriptions of input and output. An example of how to call the