V

1
V2 AC Circuits
2
Main Topics

• Power in AC Circuits.
• R, L and C in AC Circuits. Impedance.
• Description using Phasors.
• Generalized Ohms Law.
• Serial RC, RL and RLC AC Circuits.
• Parallel RC, RL and RLC AC Circuits.
• The Concept of the Resonance.

3
The Power

• The power at any instant is a product of the
  voltage and current
• P(t) V(t) I(t) V0sin(?t)I0sin(?t ?)
• The mean value of power depends on the phase
  shift between the voltage and the current
• ltPgt VrmsIrmscos?
• The quality cos? is called the power factor.

4
AC Circuit with R Only

• If a current I(t) I0sin?t flows through a
  resistor R Ohms law is valid at any instant. The
  voltage on the resistor will be in-phase
• V(t) RI0sin?t V0sin?t
• V0 RI0
• ltPgt VrmsIrms RIrms2 Vrms2/R
• We define the impedance of the resistor
• XR R

5
AC Circuit with L Only I

• If a current I(t) I0sin?t supplied by some AC
  power-source flows through an inductance L
  Kirchhoffs law is valid in any instant
• V(t) LdI(t)/dt 0
• This gives us the voltage on the inductor
• V(t) ?LI0cos?t V0sin(?t?/2)
• V0 ? LI0

6
AC Circuit with L Only II

• There is a phase-shift between the voltage and
  the current on the inductor. The current is
  delayed by ? ?/2 behind the voltage.
• The mean power now will be zero
• ltPgt VrmsIrms cos? 0
• We define the impedance of the inductance
• XL ?L ? V0 I0 XL

7
AC Circuit with L Only III

• Since the impedance, in this case the inductive
  reactance, is a ratio of the peak (and also rms)
  values of the voltage over current we can regard
  it as a generalization or the resistance.
• Note the dependence on ?! The higher is the
  frequency the higher is the impedance.

8
AC Circuit with C Only I

• If a current I(t) I0sin?t supplied by some AC
  power-source flows through an capacitor C
  Kirchhoffs law is valid in any instant
• V(t) Q(t)/C 0
• This is an integral equation for voltage
• V(t) I0/?C cos?t V0sin(?t ?/2)
• V0 I0/?C

9
AC Circuit with C Only II

• There is a phase-shift between the voltage and
  the current on the inductor. The voltage is
  delayed by ? ?/2 behind the current.
• The mean power now will be again zero
• ltPgt VrmsIrms cos? 0
• We define the impedance of the capacitor
• XC 1/?C ? V0 I0 XC

10
AC Circuit with C Only III

• Since the impedance, in this case the capacitive
  reactance, is a ratio of the peak (and rms)
  values of the voltage over current we can regard
  it again as a generalization or the resistance.
• Note the dependence on ?! Here, the higher is the
  frequency the lower is the impedance.

11
A Loudspeaker Cross-over

• The different frequency behavior of the
  impedances of an inductor and a capacitor can be
  used in filters and for instance to simply
  separate sounds in a loud-speaker.
• high-frequency speaker a tweeter is connected
  is series with an capacitor.
• low-frequency speaker a woofer is connected is
  series with an inductance.

12
General AC Circuits I

• If there are more R, C, L elements in an AC
  circuit we can always, in principle, build
  appropriate differential or integral equations
  and solve them. The only problem is that these
  equations would be very complicated even in very
  simple situations.
• There are, fortunately, several ways how to get
  around this more elegantly.

13
General AC Circuits II

• AC circuits are a two-dimensional problem.
• If we supply any AC circuit by a voltage V0sin?t,
  the time dependence of all the voltages and
  currents in the circuit will also oscillate with
  the same ?t but possibly different phase.
• So it is necessary and sufficient to describe any
  quantity by two parameters its phase and
  magnitude.

14
General AC Circuits III

• There are two mathematical tools commonly used
• Two-dimensional vectors, so called, phasors in a
  coordinate system which rotates with ?t so all
  the phasors, which also rotate, are still
• Complex numbers in Gauss plane. This is preferred
  since more operations (e.g. division, roots) are
  defined for complex numbers.

15
General AC Circuits IV

• The description by both ways is similar The
  magnitude of particular quality (voltage or
  current) is described by a magnitude of a phasor
  (vector) or an absolute value of a complex number
  and the phase is described by the angle with the
  positive x-axis or a real axis.

16
General AC Circuits V

• The complex number approach
• Describe voltages V, currents I, impedances Z and
  admittances Y 1/Z by complex numbers.
• Then a general complex Ohms law is valid
• V ZI or IYV
• Serial combination Zs Z1 Z2
• Parallel combination Yp Y1 Y2
• Kirchhoffs laws are valid for complex I and V

17
General AC Circuits VI

• The table of complex impedances and admitancess
  of ideal elements R, L, C,
• j is the imaginary unit j2 -1
• R ZR R YR 1/R
• L ZL j?L YL -j/?L
• C ZC -j/?C YC j?C

18
RC in Series

• Lets illustrate the complex number approach on a
  serial RC combination
• Let I, common for both R and C, be real.
• Z ZR ZC R j/?C
• Z (ZZ)1/2 (R2 1/?2 C2)1/2
• tg? 1/?RC lt 0 capacity like

19
RL in Series

• Lets have a R and L in series
• Let I, common for both R and L, be real.
• Z ZR ZC R j?L
• Z (ZZ)1/2 (R2 ?2L2)1/2
• tg? ?L/R gt 0 inductance like

20
RC in Parallel

• Lets have a R and L in parallel
• Let V, common for both R and C, be real.
• Y YR YC 1/R j?C
• Y (YY)1/2 (1/R2 ?2C2)1/2
• tg? ?C/R lt 0 again capacity like

21
RLC in Series I

• Lets have a R, L and C in series
• Let again I, common for all R , L, C be real.
• Z ZR ZC ZL R j(?L - 1/?C)
• Z (R2 (?L - 1/?C)2)1/2
• The circuit can be either inductance-like if
• ?L gt 1/?C ? gt 0
• or capacitance-like
• ?L lt 1/?C ? lt 0

22
RLC in Series II

• New effect of resonance takes place when
• ?L 1/?C ? ?2 1/LC
• Then the imaginary parts cancel and the whole
  circuit behaves as a pure resistance
• Z, V have minimum, I maximum
• It can be reached by tuning L, C or f !

23
RLC in Parallel I

• Lets have a R, L and C in parallel
• Let now V, common for all R , L, C be real.
• Y YR YC YL 1/R j(?C - 1/?L)
• Y (1/R2 (?C - 1/?L)2)1/2
• The circuit can be either inductance-like if
• ?L gt 1/?C ? gt 0
• or capacitance-like
• ?L lt 1/?C ? lt 0

24
RLC in Parallel II

• Again the effect of resonance takes place when
  the same condition is fulfilled
• ?L 1/?C ? ?2 1/LC
• Then the imaginary parts cancel and the whole
  circuit behaves as a pure resistance
• Y, I have minimum, Z,V have maximum
• It can be reached by tuning L, C or f !

25
Resonance

• General description of the resonance
• If we need to feed some system capable of
  oscillating on its frequency ?0 then we do it
  most effectively if the frequency our source ?
  matches ?0 and moreover is in phase.
• Good mechanical example is a swing.
• The principle is used in e.g. in tuning circuits
  of receivers.

26
Impedance Matching

• From DC circuits we already know that if we need
  to transfer maximum power between two circuits it
  is necessary that the output resistance of the
  first one matches the input resistance of the
  next one.
• In AC circuits we have to match (complex)
  impedances the same way.
• Unmatched phase may lead to reflection!

27
Homework

• Chapter 31 1, 2, 3, 4, 7, 12, 13, 24, 25, 40.

28
Things to read and learn

• This lecture covers
• The rest of Chapter 31
• Try to understand the physical background and
  ideas. Physics is not just inserting numbers into
  formulas!

29
The Mean Power I

• We choose the representative time interval ? T

30
The Mean Power II

• Since only the first integral in non-zero.

31
AC Circuit with C I

• From definition of the current I dQ/dt and
  relation for a capacitor Vc Q(t)/C

• The capacitor is an integrating device.

32
LC Circuit I

• We use definition of the current I -dQ/dt and
  relation of the charge and voltage on a capacitor
  Vc Q(t)/C

• We take into account that the capacitor is
  discharged by the current. This is homogeneous
  differential equation of the second order. We
  guess the solution.

33
LC Circuit II

• Now we get parameters by substituting into the
  equation

• These are un-dumped oscillations.

34
LC Circuit III

• The current can be obtained from the definition I
  - dQ/dt

• Its behavior in time is harmonic.

35
LC Circuit IV

• The voltage on the capacitor V(t) Q(t)/C

• is also harmonic but note, there is a phase shift
  between the voltage and the current.