1. 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

2. AC circuits – low-pass filter
real VC VR
imaginary

3. 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

4. High-pass filter
Vin
Vout R C

5. 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?

6. 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?

7. LCR circuit
Vin R C L a b

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

9. 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?

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

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

12. LCR circuit
Vin R C L a b

13. LCR circuit

14. LCR circuit

15. 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

16. 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

17. 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

18. Resonant circuit – parallel resonanceIC IL IR Ip C
This circuit is often referred to as a tank or tuned circuit

19. Resonant circuit – parallel resonanceIC 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

20. 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

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

22. 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

23. 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)

24. 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

25. 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

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

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

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

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

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

31. 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

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

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

34. 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

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

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

37. 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

38. 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

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