18-First Order Circuits 2

18-First Order Circuits 2 - FOC II: DC Analysis FOCs II: DC...

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OC’s II: DC Analysis FOC s II: DC Analysis Examples, Step Response Big Messy Example & Big Messy Example © Fred Terry Fall, 2008
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Procedure 1)Find Initial/Boundary Conditions 2)Find Natural (homogeneous) Solution 3)Find Forced (particular) Solution For step response problems this is the solution when t 4)Add Forced & Natural to get Complete Solution 5)Match Unknown in Complete Solution to Initial onditions Conditions (1)-(3) can be done in any order, but (4)&(5) must come at the end!!
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DC Conditions • Under DC Conditions, All d/dt Terms Are ZERO –I c =0, V L =0 apacitors Act Like Open Circuits, Inductors Capacitors Act Like Open Circuits, Inductors Act Like Short Circuits asy Issue for FOC’s but Generally True Easy Issue for FOC s, but Generally True for DC Conditions and Very Helpful for igher Order Circuits Higher Order Circuits
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Example 4K Ω 3K Ω Find Initial ondition on + t=0 4V + -30V Condition on Capacitor for t<0 - 5K Ω 24V 500 μ F - V c
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Example 3K Ω Find Initial Condition on apacitor for t<0 + K 24V V c (t<0) + Capacitor for t<0 - 5K Ω - Voltage Divider 5 (0 ) 2 4 1 5 53 c K Vt V V KK Ω <= = Ω+ Ω
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Example 4K Ω 3K Ω Find Forced Solution: + t=0 4V + -30V V Cf =V C (t →∞ ) - 5K Ω 24V 500 μ F - V c
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Example 4K Ω 3K Ω Find Forced Solution: + 4V + -30V V Cf =V C (t →∞ ) - 5K Ω 24V 500 μ F - V c Find Natural dN u Solution
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Example 4K Ω 3K Ω Match Initial Condition +- 5K Ω 24V 500 μ F + - V c -30V
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Example + t=0 4K Ω 3K Ω 4V + 0V : (0) 15 30 c Cf InitialCondition V V VV = =− - 5K Ω 24V 500 μ F - V c -30V ( )( ) / :( ) 45 0 0 2 t cn Natural Solution V t Ae with K F s τ τμ = = /2 ) ) 30 cC f C n ts CompleteSolution V t V V t V A e =+ + () 0 ) 15 30 45 c Vt Match InitialConditionatt V V A A V + = = + = (0) 15 ( ) 30 45 c c V Ve == +→= + ( c i )1 1 . 2 5 c V tC m V e t
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Example +- t=0 4K Ω 5K Ω 3K Ω 24V 00 + -30V 20 5 Vc(t) ic(t) 500 μ F - V c 10 0 - 0 0 - 0 -5 Vc(t) ic(t) mA -20 10 -15 10 -30 -20 -5 0 5 10 15 t (s)
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Unit Step Response 00 () 10 ) t ut t metimes written u t ∀< ⎧⎫ = ⎨⎬ ∀≥ ⎩⎭ 1 1.2 0 0 0 0 1 sometimes written u t tt ut t −= 0.8 u(t) 0.2u(t+1)+0.5u(t-1) Useful way to characterize the time 0 0.4 0.6 u(t) response of linear systems Directly interesting for speed of digital 0.2 systems Can use a series of steps to approximate y signal -0.2 0 - - - any signal 3 2 1 0 1 2 3 t
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Unit Step Response of RC Circuit: atural Response & Initial Conditions Natural Response & Initial Conditions a R + i c + - C V 0 u(t) V c - b
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Unit Step Response of RC Circuit: orced Response Forced Response a R + i c + - C V 0 u(t) V c - b
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Unit Step Response of RC Circuit: otal Response/Match Initial Condition Total Response/Match Initial Condition a R + i c + - C V 0 u(t) V c - b
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18-First Order Circuits 2 - FOC II: DC Analysis FOCs II: DC...

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