Reactance, Impendance and Filter circuits

Steady State vs Transient

• steady state is where voltage and current in circuit settle down to state which lasts forever

• if system is suddenly connected to a battery such as closing switch, then system will take time to adapt to the sudden change, this is transient state

transient response

switch from off to on - closing a switch doesnt cause value to instantly jump to final

step response

step function (low voltage to high voltage), known as step response

initial and final values

RC or CRs are first order systems as their behaviours are determined by first order differential equations.

• generalise first-order system transient responses in terms of 2 exponentials

exponentials for initial and final values

v = vf + (vi - vf) x e^(-t/tau), same for e. first tirm is steady-state and second term is transient

impact of time constant on pulse responses

as tau increases, step response becomes more wavy

RL circuit

resistor-inductor, coil of wire

what does time constant control

how fast circuit settles in transient response

sine waves and phase

• described by amplitude, frequency and phase

• voltage: sinusoidal

• current: sinusoidal but leads the voltage by 90 degrees

• phase shift is the reason capacitor analysis is awkward with ordinary resistance

reactance of a capacitor

• V/I size depends on the frequency in a capacitor. This is called reactance

• reactance (X) (capacitor) = 1/(frequency x capacitance

• so reactance is inversely proportional to frequency

• so capacitor blocks slow signals more than fast ones

impedance

current and voltage in a capacitor are not in phase

• we must use impedance instead of plain resistance

• impedance lets us use complex numbers

Impedance formula

Z = 1/frequency x complex number x capacitance

power in capacitor

resistor dissipates energy as heat

• so power always positive

• capacitor stores energy and gives it back later

• so instantaneous power goes pos and neg over a cycle

• zero power dissipation

• capacitors react to energy instead of wasting

frequency response

• feeding a circuit with a sine wave

• output amplitude changes with frequency

• this is called the frequency response

transfer function

output-to-input ratio, written H(jw).

• magnitude response: how much output gets amplified or reduced

• phase response: how much output is shifted

bode plot

graphs of gain and phase versus frequency

dB

decibels, gain is written in decibels

dB straight line approximation

20log(Vout/Vin)

corner frequency

where behaviour changes

RC circuit corner angular frequency and frequency

angular: 1/RC, frequency: 1/(2 pi RC)

low-pass filter vs high pass

RC can act as…

low:

• lets low frequencies through

• attenuates high frequencies

• capacitor connected to ground

high:

• resistor connected to ground

• CR circuit

• circuit blocks low frequencies and lets high ones through

• very low frequencies: output is near zero

-20 decibels per decade

when frequency increases by factor of 10, gain drops by 20 decibels each time

voltage gain in decibels

20log10(Vo/Vi), positive is amplification and negative is attenuation