Lecture 141 1st Order Circuits

Lecture 142 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.

Lecture 143 Important Concepts The differential equation Forced and natural solutions The time constant Transient and steady state waveforms

Lecture 144 A First Order RC Circuit One capacitor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. R + - Cv s (t) + - v c (t) +- v r (t)

Lecture 145 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.

Lecture 146 More Applications The low-pass filter for an envelope detector in a superhetrodyne AM receiver. A sample-and-hold circuit for a PCM encoder: –The capacitor is charged to the voltage of a waveform to be sampled. –The capacitor holds this voltage until an A/D converter can convert it to bits.

Lecture 147 The Differential Equation(s) KCL around the loop: v r (t) + v c (t) = v s (t) R + - Cv s (t) + - v c (t) +- v r (t)

Lecture 148 Differential Equation(s)

Lecture 149 What is the differential equation for v c (t)?

Lecture 1410 A First Order RL Circuit One inductor and one resistor The source and resistor may be equivalent to a circuit with many resistors and sources. v(t) i s (t) RL + -

Lecture 1411 Applications Modeled by a 1st Order LC Circuit The windings in an electric motor or generator.

Lecture 1412 The Differential Equation(s) KCL at the top node: v(t) i s (t) RL + -

Lecture 1413 The Differential Equation

Lecture st Order Differential Equation Voltages and currents in a 1st order circuit satisfy a differential equation of the form

Lecture 1415 Important Concepts The differential equation Forced (particular) and natural (complementary) solutions The time constant Transient and steady state waveforms

Lecture 1416 The Particular Solution The particular solution v p (t) is usually a weighted sum of f(t) and its first derivative. If f(t) is constant, then v p (t) is constant. If f(t) is sinusoidal, then v p (t) is sinusoidal.

Lecture 1417 The Complementary Solution The complementary solution has the following form: What value must  have to give a solution to

Lecture 1418 Complementary Solution How do I choose the value of K? The initial conditions determine the value of K.

Lecture 1419 Important Concepts The differential equation Forced (particular) and natural (complementary) solutions The time constant Transient and steady state waveforms

Lecture 1420 The Time Constant The complementary solution for any 1st order circuit is For an RC circuit,  = RC For an LC circuit,  = L/R

Lecture 1421 What Does v c (t) Look Like?  = 10 -4

Lecture 1422 Interpretation of   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.

Lecture 1423 Implications of the Time Constant Should the time constant be large or small: –Computer RAM –The low-pass filter for the envelope detector –The sample-and-hold circuit –The electrical motor

Lecture 1424 Important Concepts The differential equation Forced (particular) and natural (complementary) solutions The time constant Transient and steady state waveforms

Lecture 1425 Transient Waveforms The transient portion of the waveform is a decaying exponential:

Lecture 1426 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.

Lecture 1427 Computer RAM Voltage across a memory capacitor may look like this:

Lecture 1428 Low Pass Filter Voltage in the filter may look like this:

Lecture 1429 Sample and Hold The voltage in the sample and hold circuit might look like this: