305221, Computer Electrical Circuit Analysis การวิเคราะห์วงจรไฟฟ้าทาง คอมพิวเตอร์ 3(2-3-6) ณรงค์ชัย มุ่งแฝงกลาง คมกริช มาเที่ยง สัปดาห์ที่ 10 Frequency Response ( First-Order circuits, Step response )

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

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.

What is first order circuit ?  A first-order circuit is characterized by a first-order differential equation.

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

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 : 2. Apply KCL: + - VoVo Node

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,, 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 : The voltage any time in the term of time constant ( τ ) :

Time Constant Table

Power & Energy Absorb

Example #2 1. Determine the equivalent resistances (R eq ): 2. Find the time constant ( τ ):

Example #2 (cont.) 3. Obtain ( v c ) : 4. Obtain ( v x ) : 5. Obtain ( i x ) :

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

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.

1. Initial condition : 2. Apply KVL: Source-free RL Circuit 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 time constant ( τ ) : 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 Objectives 1 First Order Circuit (RC) 2 First Order Circuit (RL) 3 Step Response of an RC 4 Step Response of an RL 5

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 ( V 0 ) condition : 2. Apply KCL: Or

Step Response of an RC (cont.)

Complete Response

Complete Response (cont.)

Example #4

Example #4 (cont.)

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

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.)

Example #5

Example #5 (cont.)