3 Calculate and use Transfer Functions a Find the response of a circuit to an

3 calculate and use transfer functions a find the

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3. Calculate and use Transfer Functions. a. Find the response of a circuit to an arbitrary input. b. Calculate, draw, and use frequency response graphs Bode Plots. Ohm‟s Law and Circuit Elements in s -domain Ohm‟s Law (Time Domain) (1) Ohm‟s Law (Frequency Domain) (2) Complex Impedances Resistor (3)
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52 Complex Impedances Inductor (4) Complex Impedances Capacitor (5) (6) In the Frequency Domain:
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53 Side Note There are actually four theorized passive devices. The fourth, the memristor, was only theorized in the 1970s and implemented in the late 2000s. It is currently being developed into a commercially viable device, and may see extensive use in data storage applications, possibly replacing some current forms of volatile and non-volatile bulk storage. The passive components are related by the underlying processes which allow them to function: Response to Initial Conditions Example RC Circuit with Initial Capacitor Charge Consider the following RC circuit for which the switch is open until . The capacitor is pre-charged to a voltage of . Find the resistor voltage, . + -   C v t 0 t + - R   i t C   v t Equivalent circuit in frequency domain, including initial conditions: 1 sC + - R   V s 0 C v s   I s   C V s
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54 (1) Steady-State Response of Circuits Example 1 RLC Circuit with switch Find the steady-state value of the current through the inductor, . You are given that the switch opens at , with zero initial conditions. You are also given that .   i t 0 t C 25 nF R 625 L 25 mH ( ) L i t Step 1 Draw equivalent circuit at and onward.   I s 1 sC R sL   V s   L I s
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55 Step 2 Find an expression for . (1) Step 3 Find through Ohm‟s Law, and substitute values for R, L, and C. (2)
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56 Step 4 Since all poles of have negative real parts, apply Final Value Theorem. (3) Example 2 RLC Circuit with switch, time-varying current source. For the same circuit as example one, find the steady-state value (if any) of the current through the inductor, . Again, the switch opens at , and initial conditions are zero. You are given that .   i t 0 t C 25 nF R 625 L 25 mH ( ) L i t Step 1 Draw equivalent circuit in frequency domain at time , after switch closes.   I s 1 sC R sL   V s   L I s
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57 Step 2 Take Laplace Transform of input waveform, : (1) Step 3 Find an expression for , apply Ohm‟s Law to get . (2)
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58 Step 4 Check if we can apply Final Value Theorem. If not, we must use inverse Laplace. (3)
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