AC circuits – low-pass filter phasor representation The projection of VC and VR on the x-axis (the real component) represents the measured voltage across that component as a function of time imaginary imaginary imaginary VR VC VR VR real real real VC VC Vin Vout R C VC real VR

AC circuits – low-pass filter real VC VR imaginary

Low-pass filter – break-point frequency The frequency at which the magnitude of the impedances of both components is equal is called the break-point frequency This is often referred to as the 3dB point, since it is the frequency at which the gain has fallen 3 db

High-pass filter Vin Vout R C

LCR circuit What does this circuit do? It behaves like an oscillator, switching energy between kinetic and potential Vin R C L How does a mechanical system behave when it is disturbed away from equilibrium? How does an oscillatory mechanical system behave when it is driven?

LCR circuit First step Apply Ohm’s law Vin R C L a b First step Apply Ohm’s law What looks familiar about the circuit and the formula?

LCR circuit Vin R C L a b

LCR circuit How do we determine the phase relationships? Vin R C L a b real imaginary XC R XL

LCR circuit How do we determine the phase relationships? Vin R C L a b real imaginary XC R XL Z How do we determine the phase relationships?

LCR circuit What else do we want to know about this circuit? Vin R C L

LCR circuit Vin R C L a b The frequency response of this circuit is controlled by the relative values of L and R

LCR circuit Vin R C L a b

LCR circuit

LCR circuit

LCR circuits – Q factor Q is called the quality factor and it dictates the shape of the response curve as a function of frequency Large Q identifies highly resonant circuits

LCR circuits – Q factor Vin R C L a b Q is called the quality factor and it dictates the shape of the response curve as a function of frequency Large Q identifies highly resonant circuits

LCR circuits – Q factor Vin R C L a b A more fundamental definition of the Q factor: the Q of a resonance is equal to 2 times the stored energy divided by the energy loss per cycle For large Q these definitions are operationally equivalent

Resonant circuit – parallel resonance IC IL IR Ip C This circuit is often referred to as a tank or tuned circuit

Resonant circuit – parallel resonance IC IL IR Ip C The series LRC had a maximum current at resonance The parallel resonant circuit has a minimum current and maximum impedance at resonance

Resonant circuit – frequency response Ip = IR IL IC In a parallel resonant circuit large currents oscillate between the energy storing elements These currents may be larger than Ip C R L IC IL IR Ip Adjusting the circuit L or C “tunes” the response to a different frequencies Continuously tuned radios use a variable air or mica capacitor Stepped-tuned radios use a voltage controlled diode, or varactor, as a variable capacitor

Simple circuits - reconsideration vo(t) v2(t) Coupled intego-differential equations

An alternative approach vo(t) v2(t) Our analysis of this simple circuit in the time domain yielded integro-differential equations We know that this circuit acts as a linear operator Therefore, there must be a method by which these equations can be transformed into algebraic relations

An alternative approach vo(t) v2(t) These equations can be easily solved by making a transformation from the variable t (time) to a complex variable s (complex frequency)

Laplace transform - example Im s -a Re s Consider the inverse problem: If the Laplace transform of the outpur has this form then in the time domain the output is a decaying exponential

Laplace transform Given an arbitrary Laplace transform we may determine the real function by means of an inverse transform In general this is done by performing a contour integral in the complex plane For our purposes we can look in tables for the inverse

Laplace transform of an RC circuit vo(t) v2(t)

Laplace transform of an RC circuit vo(t) v2(t)

Laplace transform of an RC circuit vo(t) v2(t)

Laplace transform of an RC circuit Re s zero Im s R C vo(t) v2(t)

Laplace transform – initial conditions We want to find the response of the RC circuit to a step function R C vo(t) v2(t)

Laplace transform – initial conditions We want to find the response of the RC circuit to a step function R C vo(t) v2(t) This is the same result we obtained using differential equations This is a differentiating circuit

Laplace transform of an RL circuit vo(t) v2(t) Using generalized impedances we can immediately obtain the transfer function

Laplace transform of an RL circuit vo(t) v2(t)

Laplace transform – RCL circuit vo(t) v2(t) The transfer function has a zero at the origin and poles in the complex plane where the denominator goes to zero

Laplace transform – RCL circuit vo(t) v2(t)

Laplace transform – RCL circuit vo(t) v2(t)

Laplace transform – RCL circuit vo(t) v2(t) Im s 1 is the frequency of ringing and is the distance from the real axis  is a measure of the damping zero Re s

Laplace transform – RCL circuit vo(t) v2(t) Im s 1 is the frequency of ringing and is the distance from the real axis  is a measure of the damping zero Re s

Laplace transform – RCL circuit vo(t) v2(t) Im s zero Re s