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First-Order Circuits (7.1-7.2) Dr. Holbert April 12, 2006 ECE201 Lect-19 1 1st 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 1st order differential equation. ECE201 Lect-19 2 Important Concepts • • • • The differential equation Forced and natural solutions The time constant Transient and steady-state waveforms ECE201 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. ECE201 Lect-19 4 Applications Modeled by a 1st 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. ECE201 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) ECE201 Lect-19 6 Differential Equation(s) t 1 Ri (t ) i ( x)dx vs (t ) C dvs (t ) di (t ) RC i (t ) C dt dt dvs (t ) dvr (t ) RC vr (t ) RC dt dt ECE201 Lect-19 7 What is the differential equation for vc(t)? ECE201 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. ECE201 Lect-19 9 Applications Modeled by a 1st Order LC Circuit • The windings in an electric motor or generator. ECE201 Lect-19 10 The Differential Equation(s) + R is(t) L v(t) – KCL at the top node: t v(t ) 1 v( x)dx is (t ) R L ECE201 Lect-19 11 The Differential Equation dis (t ) L dv(t ) v(t ) L R dt dt ECE201 Lect-19 12 1st Order Differential Equation Voltages and currents in a 1st order circuit satisfy a differential equation of the form dv(t ) a v(t ) f (t ) dt ECE201 Lect-19 13 Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE201 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. ECE201 Lect-19 15 The Complementary Solution The complementary solution has the following form: vc (t ) Ke at Ke t / Initial conditions determine the value of K. ECE201 Lect-19 16 Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE201 Lect-19 17 The Time Constant () • The complementary solution for any 1st order circuit is vc (t ) Ke t / • For an RC circuit, = RC • For an RL circuit, = L/R ECE201 Lect-19 18 What Does vc(t) Look Like? = 10-4 ECE201 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. ECE201 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 ECE201 Lect-19 21 Important Concepts • The differential equation • Forced (particular) and natural (complementary) solutions • The time constant • Transient and steady-state waveforms ECE201 Lect-19 22 Transient Waveforms • The transient portion of the waveform is a decaying exponential: ECE201 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. ECE201 Lect-19 24 LC Characteristics Element V/I Relation DC Steady-State Resistor V(t) = R I(t) V=IR Capacitor I(t) = C dV(t)/dt I=0; open Inductor V=0; short V(t) = L dI(t)/dt ECE201 Lect-19 25 Class Examples • Learning Extension E7.1 • Learning Extension E7.2 ECE201 Lect-19 26