Lecture 15 Review: Energy storage and dynamic systems Basic time-varying signals Capacitors Related educational modules: –Section 2.2

Dynamic Systems We now consider circuits containing energy storage elements The circuits are dynamic systems They are governed by differential equations We need to be concerned with the input and output of the system as functions of time The system output depends upon the state of the system at previous times

Basic Time-Varying Signals Step functions Exponential functions

Example: Sliding mass with friction

Do forced, natural response; input and output response plots Time constant and effect of mass on time constant Notes: – Mention transient, steady-state – Natural vs. forced response – Homogeneous vs. particular solution

Energy storage elements – capacitors Capacitors store energy as an electric field In general, constructed of two conductive elements separated by a non-conductive material

Capacitors Circuit symbol: C is the capacitance Units are Farads (F) Voltage-charge relation: Recall: So:

Capacitor voltage-current relations Differential form: Integral form:

Annotate previous slide to show initial voltage, define times on integral, sketchy derivation of integration of differential form to get integral form.

Important notes about capacitors 1.If voltage is constant, no current flows through the capacitor If nothing in the circuit is changing with time, capacitors act as open circuits 2.Sudden changes in voltage require infinite current The voltage across a capacitor must be a continuous function of time

Capacitor Power and Energy Power: Energy:

Example The voltage applied to the capacitor by the source is as shown. Plot the power absorbed by the capacitor and the energy stored in the capacitor as functions of time.

Example – continued

Series combinations of capacitors

A series combination of capacitors can be represented as a single equivalent capacitance

Parallel combinations of capacitors

A parallel combination of capacitors can be represented as a single equivalent capacitance

Example Determine the equivalent capacitance, C eq