Kevin D. Donohue, University of Kentucky 1Pulse Response - First Order CircuitsUnit Step functions, Pulse Sources, and SPICEKevin D. Donohue, University of Kentucky 2The Unit Step Function The unit step function models the behavior of a switch (i.e. off-on) and can be used to describe more complex circuit sources.><=0for 10for 0)(tttuØMultiplying u(t) by a constant changes the value of u(t) for t > 0.ØSubtracting a positive number from the argument of u(t) shifts the step in the positive t direction.ØAdding step functions together results in piece-wise steps with the resultant values equal to the sum of step values in each t interval.Kevin D. Donohue, University of Kentucky 3ExampleDetermine the unit step response for io(t) the circuit below with input vs(t) = u(t):Show:0.1 mF10 Ω40 Ωio(t)vs(t)()amps40)()1250exp(8.08.0)(steptuttio−−=0 1 2 3 402468101214161820msmaStep ResponseKevin D. Donohue, University of Kentucky 4The Unit Pulse FunctionThe unit pulse function is defined as :It can be expressed in terms of a combination of unit step functions: ><<<= for 00for 1 0for 0);(TtTttTtv()()[]Ttututv−−==)(-10 -5 0 5 10 1500.20.40.60.811.21.4Pulse Function for T = 5TimeVoltsKevin D. Donohue, University of Kentucky 5ExampleDetermine the pulse response for io(t) the circuit below with input vs(t) = 40v(t) V, where pulse duration T is 2 ms:Show:0.1 mF10 Ω40 Ωio(t)vs(t)()()[]amps)m2())m20(125exp(8.08.0)()0125exp(8.08.0)(pulse0−−−−−−−= tuttutti[][]amps)m2(1825.12)()1250exp(8.0)m2()(8.0)(pulse−−−−−−= tututtututio0 1 2 3 4 5 6 7 800.10.20.30.40.50.60.70.8Pulse ResponsemsmaKevin D. Donohue, University of Kentucky 6Plotting Transient ResponsesØ Programs written for Matlab can be created either as functions or scripts. For plotting transient responses it is convenient to write a function that acts like the unit step.Ø A function acts like a subroutine in the Matlab workspace. Arguments must be passed to it and it assigns its output to variables in the workspace. Intermediate variable use by the function are cleared.Kevin D. Donohue, University of Kentucky 7Creating a FunctionØ A function is a text file (can use Notepad or Matlab’seditor. If using a word processor, you must store the as a text file.)Ø The first line begins with the word ‘function’ and is followed by the syntax of the function you wish to create.Ø The following lines can be comments, other Matlab functions, and commands to process the input arguments to create the output arguments.Ø The file must be saved with a based name the same as the function and a ‘.m’ extension in the current or working directory.Kevin D. Donohue, University of Kentucky 8ExampleØ Create a function called ‘unit’ such that for an input array it will output a corresponding array of zeros and ones as the unit step function would do. Must save as text file called unit.mfunction out = unit(t)% This function is the unit step:%% sig= unit(t), where SIG and T are vectors of the same dimension.%% elements of out are either 0 or 1 corresponding to elements of T% being negative or non-negative, respectively.%% written by Kevin D. Donohue 2/93out = zeros(size(t)); % Initialize output array to zerosind_of_positive = find(t >= 0); % Find index array of positive values on t-axisout(ind_of_positive) = 1; % Assign points corresponding to postive values to 1Kevin D. Donohue, University of Kentucky 9Create Plot in Matlab()()[]amps)m2())m20(125exp(8.08.0)()0125exp(8.08.0)(pulse0−−−−−−−= tuttutti>> tau = 1/1250; % Define time constant>> t = [-0.1:(tau/50):(10*tau+2e-3)]; % Create time axis>> i0 = (.8-.8*exp(-t/tau)).*unit(t)-(.8-.8*exp(-(t-2e-3)/tau)).*unit(t-2e-3);>> plot(t,i0)>> xlabel(‘Seconds’)>> ylabel(‘Amps’)>> title(‘Transient Response’)Kevin D. Donohue, University of Kentucky 10SPICE Simulation with Step SourceDo a SPICE Simulation to determine the pulse response for io(t) the Previous circuit with input vs(t) = 40v(t) V, where pulse duration T is 2 ms:Ø To solve you must do a transient analysis, define the voltage source as piece-wise linear, and describe its transient properties. V0R 10 R0 40 C0.0001VAmtranex-Transient-14-TableTIME I(VAM)(s) (Amp)+0.000e+000 +0.000e+000+10.000n +99.999n+10.840n +109.104n: :: :: :+7.285m +972.534u+7.445m +795.710u+7.605m +651.035u+7.765m +532.665u+7.925m +435.817u+8.000m +396.630uI(VAM)Time (s)tranex-Transient-14(Amp)+0.000e+000+500.000m+0.000e+000 +2.000m +4.000m +6.000m +8.000mKevin D. Donohue, University of Kentucky 11SPICE Simulation with SwitchDo a SPICE Simulation to determine the pulse response for io(t) the circuit below with input vs(t) = 40 V, where pulse duration T is 2 ms:Ø To solve you must do a transient analysis, define auxiliary circuits for voltage sources and meters that control the opening and closing of the switches.(Amp)V40R040C0.0001R 10 VAmSwitchV1SwitchV0IVmV00V10IVm0I(VAM)Time (s)tranexsw-Transient-0(Amp)+0.000e+000+200.000m+400.000m+600.000m+0.000e+000 +2.000m +4.000m