class: center, middle # EE-281 # First-Order Circuits with Voltage Sources ## Ozan Keysan [ozan.keysan.me](http://ozan.keysan.me) Office: C-113 <span class="meta">•</span> Tel: 210 7586 --- # First-Order Circuits -- ### First order circuit contains one type of energy storage element (capacitor or inductor), and can be described using a first-order differential equation. -- ## There are two types: - ## RC Circuits - ## RL Circuits --- # RC Circuits <img src="http://sites.bsyse.wsu.edu/pitts/be120/Handouts/EE%20project_files/image043.gif" alt="Drawing" style="width: 800px;"/> --- # RC Circuits  ### Assume at t=0, the capacitor voltage is equal to \\(V_o\\) --- # RC Circuits  ## \\( v(t) = V_oe^{-\dfrac{t}{RC}}\\) --- # RC Circuits ## \\( v(t) = V_oe^{-\dfrac{t}{RC}}\\) -- ## Time Constant: \\(\tau\\) ## \\(\tau=RC\\) -- ## \\( v(t) = V_oe^{-\dfrac{t}{\tau}}\\) --- # RC Circuits ## \\( v(t) = V_oe^{-\dfrac{t}{\tau}}\\)  --- # Exponential Decay - ### Chemical Reactions -- - ### Fluid Dynamics -- - ### Heat Transfer -- - ### Atmospheric Pressure -- - ### Radioactivity --- # Exercise #### If the initial voltage of the capacitor is 60 V, then determine Vc, Vx, io.  --- # RL Circuits  ### Assume at t=0, the inductor current is equal to \\(I_o\\) --- # RL Circuits  ## \\(i(t) = I_o e^{-\dfrac{Rt}{L}}\\) --- # RL Circuits ## \\(i(t) = I_o e^{-\dfrac{Rt}{L}}\\) -- ## Time Constant: \\(\tau = \dfrac{L}{R}\\) -- ## \\(i(t) = I_0 e ^{-\dfrac{t}{\tau}}\\) --- # RL Circuits  --- # RL Circuits ### Initial Energy Stored in Inductor: \\(\dfrac{1}{2}I_0^2L\\) --- #Exercises  --- #Exercises  --- #Exercises  --- #Exercises ### i0 = 12 A  --- # RC Circuits ## \\( v(t) = V_oe^{-\dfrac{t}{RC}}\\)  --- # RL Circuits ## \\(i(t) = I_o e^{-\dfrac{Rt}{L}}\\)  --- #Step Response of RC Circuits ### What happens if we suddenly apply Vs? --  --- #Step Response of RC Circuits ### Assuming the initial voltage of the capacitor is 0: --  --- #Step Response of RC Circuits ### At \\(t=0\\) --> \\(Vc=0\\) ### \\(t=\infty\\) --> \\(Vc=Vs\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RC_step_V.png" alt="Drawing" style="width: 400px;"/> --- #Step Response of RC Circuits ## \\(V_c(t)=V_s (1-e^{-t/RC})\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RC_step_V.png" alt="Drawing" style="width: 500px;"/> --- #Step Response of RC Circuits ### Current Waveform: \\(i_c(t) = C \dfrac{dVc}{dt} = \dfrac{V_s}{R} e^{-t/RC}\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RC_step_I.png" alt="Drawing" style="width: 450px;"/> --- # Step Response of RC Circuits ## With initial voltage \\(V_0\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RC_step_Vo.png" alt="Drawing" style="width: 400px;"/> --- # Step Response of RC Circuits ## \\(V_c(t)=V_s + (V_0 - V_s)e^{-t/RC}\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RC_step_Vo.png" alt="Drawing" style="width: 400px;"/> --- # Step Response of RC Circuits ## \\(V_c(t)=V_s + (V_0 - V_s)e^{-t/RC}\\) ### At \\(t=0\\) --> \\(Vc=V_0\\) ### \\(t=\infty\\) --> \\(Vc=Vs\\) ### Generalized equation: \\( V_c(t) = V\infty + (V_0 - V\infty)e^{-t/RC}\\) --- # Step Response of RC Circuits ### Complete response = Natural Response + Forced Response --- #Example #### Determine v(t) and calculate its value fot t=1 and t=4 seconds.  --- #Example #### Determine v(t) and calculate its value fot t=1 and t=4 seconds.  --- # Unit Functions ## Step Function: \\(u(t)\\) ### u(t) = 0 for t<0 ### u(t) = 1 for t>0 <img src="http://www.pirobot.org/blog/0005/NeuralNetworks1_html_2489bbdb.png" alt="Drawing" style="width: 400px;"/> --- #Example #### Determine the expression for v and i.  --- #Example #### Determine the expression for v and i.  --- # Step Response of RL Circuits  --- #Step Response of RL Circuits ### Assuming the initial current of the inductor is 0: -- ### At \\(t=0\\) --> \\(I_L=0\\) ### \\(t=\infty\\) --> \\(I_L=Vs/R\\) --- #Step Response of RL Circuits ### At \\(t=0\\) --> \\(I_L=0\\) ### \\(t=\infty\\) --> \\(I_L=Vs/R\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RL_step_V.png" alt="Drawing" style="width: 800px;"/> --- #Step Response of RL Circuits ### \\(I_L(t)=\dfrac{V_s}{R} (1-e^{-tR/L})\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RL_step_V.png" alt="Drawing" style="width: 800px;"/> --- #Step Response of RL Circuits ## Generalized equation with \\(I_0\\) ### \\( I_L(t) = I\infty + (I_0 - I\infty)e^{-tR/L}\\) <img src="https://raw.githubusercontent.com/ozank/ee281/master/images/RL_step_V0.png" alt="Drawing" style="width: 300px;"/> --- #Example #### Determine the expression for i.  --- #Example #### Determine the expression for i.  --- #Example #### Determine the expression for i and calculate the value for t=2 s and t=5 s.  --- # Any questions? ## You can download this presentation from: [keysan.me/ee281](http://keysan.me/ee281)