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305221, 226231 Computer Electrical Circuit Analysis การวิเคราะห ์วงจรไฟฟ้าทาง คอมพิวเตอร ์ ณรงค์3(2-3-6) ชยั มุ่งแฝงกลาง คมกริ ช มาเที่ยง สัปดาห์ที่ 10 Frequency Response (First-Order circuits, Step response) Outline 1 Objectives 2 First Order Circuit (RC) 3 First Order Circuit (RL) 4 Step Response of an RC 5 Step Response of an RL Objectives เพื่อให้ นิสิตมีความรู้ ความเข้ าใจเกีย่ วกับวงจร(อนุพนั ธ์ )อันดับหนึ่ง (First Oder Circuit) A circuit comprising a resistor and capacitor (RC Circuit). A circuit comprising a resistor and an inductor (RL Circuit). ่ ่ เพือให้ นิสต ิ มีความรู ้ความเข้าใจเกียวก บ ั ผลการตอบสนองแบบ ทันทีทันใด(Step Response) Step Response of an RC Circuit. Step Response of an RL Circuit. What is first order circuit ? A first-order circuit is characterized by a first-order differential equation. Outline 1 Objectives 2 First Order Circuit (RC) 3 First Order Circuit (RL) 4 Step Response of an RC 5 Step Response of an RL 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 - Example #1 Source-free RC Circuit 1. Initial condition : Node + Vo - 2. Apply KCL: Source-free RC Circuit (cont.) First-order differential equation : Rearrange the in a terms as : 3. Integrating both sides : Integration Constant Source-free RC Circuit (cont.) Rearrange the in a terms as : 4. Taking powers of ‘e’ produces: By the initial conditions, v(0) A Vo , Hence, 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. 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: τ The voltage any time in the term of time constant ( ) : Time Constant Table Power & Energy Absorb Example #2 1. Determine the equivalent resistances (Req): 2. Find the time constant ( τ ): Example #2 (cont.) v 3. Obtain ( c ) : v 4. Obtain ( x ) : i 5. Obtain ( x ) : Outline 1 Objectives 2 First Order Circuit (RC) 3 First Order Circuit (RL) 4 Step Response of an RC 5 Step Response of an RL First Order Circuit (RL) 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. + DC - Source-free RL Circuit 1. Initial condition : 2. Apply KVL: Or Source-free RL Circuit (cont.) 3. Rearrange and integrating gives: Or 4. Taking powers of ‘e’ produces: Current Response of RL Circuit Time Constant : The current any time in the term of τ time constant ( ) : Voltage across resistor : Power & Energy Absorb Example #3 Example #3 (cont.) Singularity/ Switching Functions Singularity functions are functions that either are discontinuous or have discontinuous derivatives. Singularity Functions (cont.) Outline 1 Objectives 2 First Order Circuit (RC) 3 First Order Circuit (RL) 4 Step Response of an RC 5 Step Response of an RL 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”. Step Response of an RC (cont.) 1. Initial voltage (V0) condition : 2. Apply KCL: Or Step Response of an RC (cont.) Step Response of an RC (cont.) Step Response of an RC (cont.) Complete Response Complete Response (cont.) Complete Response (cont.) Example #4 Example #4 (cont.) Example #4 (cont.) Outline 1 Objectives 2 First Order Circuit (RC) 3 First Order Circuit (RL) 4 Step Response of an RC 5 Step Response of an RL Step Response of an RL Step Response of an RL (cont.) 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.) Step Response of an RL (cont.) Step Response of an RL (cont.) Step Response of an RL (cont.) Example #5 Example #5 (cont.) Example #5 (cont.)