Exponential Function in Circuits, Part 2 K. A. Connor Mobile Studio Project Center for Mobile Hands-On STEM SMART LIGHTING Engineering Research Center ECSE Department Rensselaer Polytechnic Institute Intro to ECSE Analysis

The Exponential Function In Part 1 We Saw How to Use Exponential in Damped dynamic systems Circuits Properties of It is its own derivative! Series expansions Small argument expressions Characteristic distances and times

The Exponential Function Can it be used for sinusoidal voltages and currents? Can it incorporate both to handle decaying sinusoids? Phasor Notation Reducing a Differential Equation to an Algebraic Equation Key is Euler’s Equation

The Exponential Function Complex Numbers z = a + jb or a + ib j2 = i2 = -1 What happens when the exponential function has an imaginary argument? This is the power of complex arithmetic – We are not restricted to working with only real numbers.

Exponential Function Series Representation Replace x with jx

Exponential Function Series Representations for sine and cosine

Exponential Function New Representation of Complex Numbers Polar Form

Exponential Function

Phasors Phasor General Form of Voltage as Function of Time Can Also be Written as Polar Form Phasor

Phasors Keeps Track of Phase for Us Simplifies Analysis of C and L Method Convert source to phasor form Write all impedances in complex form Analyze circuit as with resistors Convert solution back to time form

Impedances Inductance

Impedances Inductance

Impedances Inductance – Multiply the expression on both sides by –j and obtain the Imag. expression. Then can use the entire complex expression Drop the Re()

Impedances Inductance – More General Form for V

Impedances Capacitance

Impedances More General Form of Ohm’s Law What Happens at DC and High f? To Think About

RC & RL Filters Low Pass Filters

RC & RL Filters High Pass Filters Note Typo in This Expression: L and R should be interchanged

When Impedances are Equal R and C What Then?

Summary Write Circuit Equations in Phasor Form Solve for Voltages and Currents Multiply by ejωt and then take the real part to convert back to time dependent form Check Against Experiment, if Possible Keeps Everything Algebraic!