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ECE201 Lect-191 First-Order Circuits ( ) Dr. Holbert April 12, 2006

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ECE201 Lect-192 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.

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ECE201 Lect-193 Important Concepts The differential equation Forced and natural solutions The time constant Transient and steady-state waveforms

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ECE201 Lect-194 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) +–+–

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ECE201 Lect-195 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.

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ECE201 Lect-196 The Differential Equation(s) KVL around the loop: v r (t) + v c (t) = v s (t) R Cv s (t) + – v c (t) +– v r (t) +–+–

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ECE201 Lect-197 Differential Equation(s)

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ECE201 Lect-198 What is the differential equation for v c (t)?

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ECE201 Lect-199 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 + –

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ECE201 Lect-1910 Applications Modeled by a 1st Order LC Circuit The windings in an electric motor or generator.

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ECE201 Lect-1911 The Differential Equation(s) KCL at the top node: v(t) i s (t) RL + –

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ECE201 Lect-1912 The Differential Equation

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ECE201 Lect st Order Differential Equation Voltages and currents in a 1st order circuit satisfy a differential equation of the form

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ECE201 Lect-1914 Important Concepts The differential equation Forced (particular) and natural (complementary) solutions The time constant Transient and steady-state waveforms

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ECE201 Lect-1915 The Particular Solution The particular solution v p (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 v p (t) is constant. If f(t) is sinusoidal, then v p (t) is sinusoidal.

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ECE201 Lect-1916 The Complementary Solution The complementary solution has the following form: Initial conditions determine the value of K.

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ECE201 Lect-1917 Important Concepts The differential equation Forced (particular) and natural (complementary) solutions The time constant Transient and steady-state waveforms

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ECE201 Lect-1918 The Time Constant ( ) The complementary solution for any 1st order circuit is For an RC circuit, = RC For an RL circuit, = L/R

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ECE201 Lect-1919 What Does v c (t) Look Like? = 10 -4

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ECE201 Lect-1920 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.

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ECE201 Lect-1921 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

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ECE201 Lect-1922 Important Concepts The differential equation Forced (particular) and natural (complementary) solutions The time constant Transient and steady-state waveforms

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ECE201 Lect-1923 Transient Waveforms The transient portion of the waveform is a decaying exponential:

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ECE201 Lect-1924 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.

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ECE201 Lect-1925 LC Characteristics

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ECE201 Lect-1926 Class Examples Learning Extension E7.1 Learning Extension E7.2

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