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Lecture 11 First-order Circuits (1). Hung-yi Lee. Dynamic Circuits. Capacitor, Inductor. (Chapter 5). Time Domain. Frequency Domain. S-Domain. (Chapter 9). (Chapter 6,7). (Chapter 11,13). Abstract. Textbook. First-Order Circuits Chapter 5.3, 9.1. First-Order Circuits.
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Lecture 11First-order Circuits (1) Hung-yi Lee
Dynamic Circuits Capacitor, Inductor (Chapter 5) Time Domain Frequency Domain S-Domain (Chapter 9) (Chapter 6,7) (Chapter 11,13) Abstract
Textbook • First-Order Circuits • Chapter 5.3, 9.1
First-Order Circuits • Containing only one capacitor or inductor The networks excluding capacitor or inductor only contains sources and resistors. Can always be simplified by Thevenin or Norton Theorem
First-Order Circuits RC: RL:
First Order Circuits (this lecture) voc and isc should be dynamic … … … … …
Perspective • Differential Equation • Superposition • State
Zero-Input Response - RC Find vc(t) and ic(t) Capacitor is open circuit
Zero-Input Response - RC Find vc(t) and ic(t) Capacitor is open circuit
Zero-Input Response - RC Find vc(t) and ic(t) ? ?
Zero-Input Response - RC ic(t0) is unknown Voltage on the capacitor should be continuous
Zero-Input Response - RC ic(t0) is unknown Assume
Zero-Input Response … Voltage, Current Voltage of C, Current of L How fast?
Step Response - RC • Solved by differential equation
Step Response - RC vN(t) is general solution vF(t) is special solution vN(t) is the solution of vF(t) is the solution of
Step Response … Voltage, Current Voltage of C, Current of L How fast?
Step Response … Rise time 90% time 10% time
Step Response • Solved by Superposition
Step Response … … = … … Suppress v1, find vc2(t) - Suppress v2, find vc1(t)
Pulse Response Solved by Superposition
Pulse Response = … - …
Pulse Response (If x is small) If
Step Response + Initial Condition Violate Superposition?
Step Response + Initial Condition The initial condition is automatically fulfilled. Do not have to consider the initial condition anymore
Step Response + Initial Condition Zero-Input Response! Step Response (without initial condition)!
Step Response + Initial Condition Differential Equation Superposition Zero-input Response Special solution General solution Step Response The initial condition is considered in the general solution term. The initial condition is automatically fulfilled.
State • The capacitor voltages and inductor currents constitute the state variables of any circuit. (P410) If the circuit does not have any capacitor or inductor The currents or voltages at time t do not depend on their values not at t. Why?
State • The capacitor voltages and inductor currents constitute the state variables of any circuit. (P410) With capacitor or inductor You can not explain the current or voltage at present unless considering the past.
State • The capacitor voltages and inductor currents constitute the state variables of any circuit. (P410) …… If we know the voltage before at t0 We do not care about the current before t0 Capacitor voltages are States
State • The capacitor voltages and inductor currents constitute the state variables of any circuit. (P410) State at t0 Source after t0
State • The capacitor voltages and inductor currents constitute the state variables of any circuit. (P410) The response after t0 From Input after t0 (Ignore state) From state at t0 (Ignore input)
Response y(t): voltage of capacitor or current of inductor y(t) = general solution + special solution = = = natural response +forced response = state response (zero input) + inputresponse (zero state)
Zero-InputResponse … Ignore everything before t0 Considering the circuit from t0: State: vc(t0)=V0 Lead to No input after t0