FirstOrder Circuits 7 1 7 2 Dr Holbert
First-Order Circuits (7. 1 -7. 2) Dr. Holbert April 12, 2006 ECE 201 Lect-19 1
1 st Order Circuits • Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 1. • Any voltage or current in such a circuit is the solution to a 1 st order differential equation. ECE 201 Lect-19 2
Important Concepts • • The differential equation Forced and natural solutions The time constant Transient and steady-state waveforms ECE 201 Lect-19 3
A First-Order RC Circuit + vr(t) – R vs(t) + – + vc(t) C – • One capacitor and one resistor • The source and resistor may be equivalent to a circuit with many resistors and sources. ECE 201 Lect-19 4
Applications Modeled by a 1 st Order RC Circuit • Computer RAM – A dynamic RAM stores ones as charge on a capacitor. – The charge leaks out through transistors modeled by large resistances. – The charge must be periodically refreshed. ECE 201 Lect-19 5
The Differential Equation(s) + vr(t) – R vs(t) + – + vc(t) C – KVL around the loop: vr(t) + vc(t) = vs(t) ECE 201 Lect-19 6
Differential Equation(s) ECE 201 Lect-19 7
What is the differential equation for vc(t)? ECE 201 Lect-19 8
A First-Order RL Circuit + is(t) R L v(t) – • One inductor and one resistor • The source and resistor may be equivalent to a circuit with many resistors and sources. ECE 201 Lect-19 9
Applications Modeled by a 1 st Order LC Circuit • The windings in an electric motor or generator. ECE 201 Lect-19 10
The Differential Equation(s) + R is(t) L v(t) – KCL at the top node: ECE 201 Lect-19 11
The Differential Equation ECE 201 Lect-19 12
1 st Order Differential Equation Voltages and currents in a 1 st order circuit satisfy a differential equation of the form ECE 201 Lect-19 13
Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE 201 Lect-19 14
The Particular Solution • The particular solution vp(t) is usually a weighted sum of f(t) and its first derivative. – That is, the particular solution looks like the forcing function • If f(t) is constant, then vp(t) is constant. • If f(t) is sinusoidal, then vp(t) is sinusoidal. ECE 201 Lect-19 15
The Complementary Solution The complementary solution has the following form: Initial conditions determine the value of K. ECE 201 Lect-19 16
Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE 201 Lect-19 17
The Time Constant ( ) • The complementary solution for any 1 st order circuit is • For an RC circuit, = RC • For an RL circuit, = L/R ECE 201 Lect-19 18
What Does vc(t) Look Like? = 10 -4 ECE 201 Lect-19 19
Interpretation of • The time constant, , is the amount of time necessary for an exponential to decay to 36. 7% of its initial value. • -1/ is the initial slope of an exponential with an initial value of 1. ECE 201 Lect-19 20
Implications of the Time Constant • Should the time constant be large or small: – Computer RAM – A sample-and-hold circuit – An electrical motor – A camera flash unit ECE 201 Lect-19 21
Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE 201 Lect-19 22
Transient Waveforms • The transient portion of the waveform is a decaying exponential: ECE 201 Lect-19 23
Steady-State Response • The steady-state response depends on the source(s) in the circuit. – Constant sources give DC (constant) steady-state responses. – Sinusoidal sources give AC (sinusoidal) steady-state responses. ECE 201 Lect-19 24
LC Characteristics ECE 201 Lect-19 25
Class Examples • Learning Extension E 7. 1 • Learning Extension E 7. 2 ECE 201 Lect-19 26
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