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

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)

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)

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

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.

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)

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)

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.