Analysis in the Laplace-s domain

working in the laplace-s domain, electrical impedance, dynamic response

resistivity/impedance

resistors, capacitors, inductors act as effective resistive elements each of which affect the current and voltage in specific ways → affects how output voltage varies in time

compounding influence can be thought as overall “effective resistance”

→ hard to analyze complex circuit in time domain → need a better effective method to anaylze circuits + see how elements impede time varying signal they recieve

goal of resistivity/impedance

determine effective resistance, or impedance → Z(s) of our electrical device

because once we have impedance in the s-domain for the device, we can treat the device as a resistor in our circuit analysis → much easier !

time-varying input signal

assuming input signal modeled as cosine function

always convert everything to cosine general form (if sin → add/substract pi/2)

Admittance (Y)

the inverse of impedance -→ funtion of s → in the s-domain

transfer function H(s)

the ratio between output signal and input signal

in the case of RC-circuit → input + output are voltages hence

H(s) = Vo(s)/Vi(s)

describes how system will manipulate/modify the input signal is given

euler’s formula

this identity is the reqson AC signals, phasors and laplace domain expressions can be written using exponential instead of sines/cosines

→ easier differentiation, integration impedance calculation much much easier

in our case x= w*t + omega and we only deal with the REAL PART OF THE SIGNAL

(s-domain is defined as s = j*w)

impedance in the s-domain (resistor)

resistor is the same in time and s-domain → s = jw

impedance in the s-domain (capacitor)

capacitor has an impedance of 1/Cs → s = jw

impedance in s-domain (inductor)

inductor has impedance of Ls

equivalent impedance

response in s-domain

input signal to the system, in s-domain as X(s)

hence we can determine output response of the system Y(s) → Y(s) = H(s)X(s)