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Lecture 16. Inductors Introduction to first-order circuits RC circuit natural response Related educational modules: Section 2.3, 2.4.1, 2.4.2. Energy storage elements - inductors. Inductors store energy in the form of a magnetic field

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## Lecture 16

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**Lecture 16**Inductors Introduction to first-order circuits RC circuit natural response Related educational modules: Section 2.3, 2.4.1, 2.4.2**Energy storage elements - inductors**• Inductors store energy in the form of a magnetic field • Commonly constructed by coiling a conductive wire around a ferrite core**Inductors**• Circuit symbol: • L is the inductance • Units are Henries (H) • Voltage-current relation:**Inductor voltage-current relations**• Differential form: • Integral form:**Annotate previous slide to show initial current, define**times on integral, sketchy derivation of integration of differential form to get integral form.**Important notes about inductors**• If current is constant, there is no voltage difference across inductor • If nothing in the circuit is changing with time, inductors act as short circuits • Sudden changes in current require infinite voltage • The current through an inductor must be a continuous function of time**Inductor Power and Energy**• Power: • Energy:**Series combinations of inductors**• A series combination of inductors can be represented as a single equivalent inductance**Example**• Determine the equivalent inductance, Leq**First order systems**• First order systems are governed by a first order differential equation • They have a single, first order, derivative term • They have a single (equivalent) energy storage elements • First order electrical circuits have a single (equivalent) capacitor or inductor**First order differential equations**• General form of differential equation: • Initial condition:**Solutions of differential equations – overview**• Solution is of the form: • yh(t) is homogeneous solution • Due to the system’s response to initial conditions • yp(t) is the particular solution • Due to the particular forcing function, u(t), applied to the system**Homogeneous Solution**• Lecture 14: a dynamic system’s response depends upon the system’s state at previous times • The homogeneous solution is the system’s response to its initial conditions only • System response if no input is applied u(t) = 0 • Also called the unforced response, natural response, or zero input response • All physical systems dissipate energy yh(t)0 as t**Particular Solution**• The particular solution is the system’s response to the input only • The form of the particular solution is dictated by the form of the forcing function applied to the system • Also called the forced response or zero state response • Since yh(t)0 as t, and y(t) = yp(t) + yh(t): • y(t) yp(t) as t**Qualitative example: heating frying pan**• Natural response: • Due to pan’s initial temperature; no input • Forced response: • Due to input; if qin is constant, yp(t) is constant • Superimpose to get overall response**On previous slide, note steady-state response (corresponds**to particular solution) and transient response (induced by initial conditions; transition from one steady-state condition to another)**RC circuit natural response – overview**• No power sources • Circuit response is due to energy initially stored in the capacitor v(t=0) = V0 • Capacitor’s initial energy is dissipated through resistor after switch is closed**RC Circuit Natural Response**• Find v(t), for t>0 if the voltage across the capacitor before the switch moves is v(0-) = V0**Derive governing first order differential equation on**previous slide • Talk about initial conditions; emphasize that capacitor voltage cannot change suddenly**Finish derivation on previous slide**• Sketch response on previous slide**RC Circuit Natural Response – summary**• Capacitor voltage: • Exponential function: • Write v(t) in terms of :**Notes:**• R and C set time constant • Increase C => more energy to dissipate • Increase R => energy disspates more slowly**RC circuit natural response – example 1**• Find v(t), t>0**Example 1 – continued**• Equivalent circuit, t>0. v(0) = 6V.

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