1. 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

2. 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

3. 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

4. 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.

5. Exponential Function
Series Representation
Replace x with jx

6. Exponential Function
Series Representations for sine and cosine

7. Exponential Function
New Representation of Complex Numbers
Polar Form

8. Exponential Function

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

10. 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

11. Impedances
Inductance

12. Impedances
Inductance

13. 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()

14. Impedances
Inductance – More General Form for V

15. Impedances
Capacitance

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

17. RC & RL Filters
Low Pass Filters

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

19. When Impedances are Equal
R and C
What Then?

20. 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!