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305221, 226231 Computer Electrical Circuit Analysis การวิเคราะห์วงจรไฟฟ้าทาง คอมพิวเตอร์ 3(2-3-6) ณรงค์ชัย มุ่งแฝงกลาง คมกริช มาเที่ยง สัปดาห์ที่ 10 Frequency Response ( First-Order circuits, Step response )
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Outline Objectives 1 First Order Circuit (RC) 2 First Order Circuit (RL) 3 Step Response of an RC 4 Step Response of an RL 5
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Objectives เพื่อให้นิสิตมีความรู้ความเข้าใจเกี่ยวกับวงจร ( อนุพันธ์ ) อันดับหนึ่ง (First Oder Circuit) A circuit comprising a resistor and capacitor (RC Circuit). A circuit comprising a resistor and an inductor (RL Circuit). เพื่อให้นิสิตมีความรู้ความเข้าใจเกี่ยวกับผลการ ตอบสนองแบบทันทีทันใด (Step Response) Step Response of an RC Circuit. Step Response of an RL Circuit.
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What is first order circuit ? A first-order circuit is characterized by a first-order differential equation.
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Outline Objectives 1 First Order Circuit (RC) 2 First Order Circuit (RL) 3 Step Response of an RC 4 Step Response of an RL 5
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First Order Circuit (RC) A source-free RC circuit occurs when its dc source is suddenly disconnected. The energy already stored in the capacitor is released to the resistors. + - DC
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Example #1
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Source-free RC Circuit 1. Initial condition : 2. Apply KCL: + - VoVo Node
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Source-free RC Circuit (cont.) First-order differential equation : Rearrange the in a terms as : 3. Integrating both sides : Integration Constant
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Source-free RC Circuit (cont.) Rearrange the in a terms as : 4. Taking powers of ‘e’ produces: By the initial conditions,, Hence,
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Natural Response The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
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Time Constant The time constant ( τ )of a circuit is the time required for the response to decay to a factor of 1 / e or 36.8 percent of its initial value. Its implies that t = τ : The voltage any time in the term of time constant ( τ ) :
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Time Constant Table
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Power & Energy Absorb
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Example #2 1. Determine the equivalent resistances (R eq ): 2. Find the time constant ( τ ):
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Example #2 (cont.) 3. Obtain ( v c ) : 4. Obtain ( v x ) : 5. Obtain ( i x ) :
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Outline Objectives 1 First Order Circuit (RC) 2 First Order Circuit (RL) 3 Step Response of an RC 4 Step Response of an RL 5
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First Order Circuit (RL) + - DC A source-free RL circuit occurs when its dc source is suddenly disconnected. The energy already stored in the capacitor is released to the resistors.
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1. Initial condition : 2. Apply KVL: Source-free RL Circuit Or
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Source-free RL Circuit (cont.) 3. Rearrange and integrating gives: Or 4. Taking powers of ‘e’ produces:
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Current Response of RL Circuit Time Constant : The current any time in the term of time constant ( τ ) : Voltage across resistor :
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Power & Energy Absorb
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Example #3
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Example #3 (cont.)
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Singularity/ Switching Functions Singularity functions are functions that either are discontinuous or have discontinuous derivatives.
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Singularity Functions (cont.)
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Outline Objectives 1 First Order Circuit (RC) 2 First Order Circuit (RL) 3 Step Response of an RC 4 Step Response of an RL 5
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Step Response of an RC When the dc source of an RC circuit is suddenly applied, the voltage or current source can be modeled as a step function, and the response is known as “a step response”.
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Step Response of an RC (cont.) 1. Initial voltage ( V 0 ) condition : 2. Apply KCL: Or
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Step Response of an RC (cont.)
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Complete Response
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Complete Response (cont.)
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Example #4
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Example #4 (cont.)
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Outline Objectives 1 First Order Circuit (RC) 2 First Order Circuit (RL) 3 Step Response of an RC 4 Step Response of an RL 5
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1. Let the response be the sum of the transient response and the steady-state response, 2. The transient response is always a decaying exponential, that is, 3. The steady-state response is the value of the current a long time after the switch in Fig. 7.48(a) is closed. Step Response of an RL (cont.)
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Example #5
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Example #5 (cont.)
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