1. Basic Electric Circuit Components - Terminal Relationships
Resistors: Ohm’s Law vR(t)=iR(t)R
Convert electric energy to heat
Capacitors: iC(t)=C dvC/dt
Store and release electric energy
Reduce voltage spikes
Inductors: vL(t)=L diL/dt
Store and release magnetic energy
Reduce current spikes

2. Terminal Relationships
For a current source is(t) = Imcos(wt):
Resistors: Ohm’s Law vR(t)=iR(t)R
vR(t) = ImRcos(wt)
Capacitors: iC(t)=C dvC/dt
vC = Imsin(wt)/(wC)… (+ constant set to zero)
Inductors: vL(t)=L diL/dt
vL(t) = -wLImsin(wt)

3. Units summary Im is current magnitude in Amperes (A)
Vm is voltage magnitude in Volts (V)
𝜑 is a phase angle in radians
(or degrees: 180˚=π)
w is the angular frequency (radians/second)
w = 2πf; f is the frequency in Hz
R is the resistance in Ohms (V/A)
C is the capacitance in Farads (A·s/V)
L is the inductance in Henries (V·s/A)

4. Combinations of components
Series combinations – same current, but voltages add
Parallel combinations – same voltage, but currents add

5. Class homework programming project
You will write a program to design a simple electric circuit given the circuit requirements
Your circuit input current is: i(t) = Imcos(wt)
You want the voltage measured across your circuit to be: v(t) = Vmcos(wt+𝜑).
Your design will specify two components (resistors, capacitors, or inductors) that when connected together will produce the correct voltage.

6. Angle addition formula
cos⁡(𝐴+𝐵) =cos(𝐴)cos(𝐵)-sin(𝐴)sin⁡(𝐵)
And…we want our voltage to be:
Vmcos⁡(𝜔𝑡+𝜑) =[Vmcos(𝜑)]cos(𝜔𝑡)-[Vmsin⁡(𝜑)]sin(𝜔𝑡)
If we were to put a resistor and an inductor in series:
v(t)= [ImR]cos(𝜔𝑡) – [wLIm]sin(𝜔𝑡)
Equating the constants in front of cos(𝜔𝑡) and sin(𝜔𝑡) :
R = Vm cos(𝜑) / Im and L = Vm sin(𝜑) / (wIm)

7. Problems with the RL solution:
If cos(𝜑) <0, then R<0, but all values R, L, and C are greater than or equal to zero, so this is not a viable solution!
If sin(𝜑) <0, then L<0, so we need to use a resistor and capacitor in series:
Vmcos⁡(𝜔𝑡+𝜑) =[Vmcos(𝜑)]cos(𝜔𝑡)-[Vmsin⁡(𝜑)]sin(𝜔𝑡)
The RC series pair gives:
v(t)= [ImR]cos(𝜔𝑡) + [Im/(wC)]sin(𝜔𝑡)
Equating the constants in front of cos(𝜔𝑡) and sin(𝜔𝑡) :
R = Vm cos(𝜑) / Im and C = Im / (sin(−𝜑) wVm)

8. Summary of design exercise
Inputs are Vm, Im, w, and 𝜑
There may not be a solution!
If there is a solution, it may be:
□ An RC series pair □ An RL series pair
□ Only a resistor □ Only an inductor
□ Only a capacitor
Alternatively, a parallel combination could be used, but the formalism is left to you, and the solutions are only different, they are not better or worse.