Sinusoidal Steady-state Analysis

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Sinusoidal Steady-state Analysis

• Complex number reviews
• Phasors and ordinary differential equations
• Complete response and sinusoidal steady-state
  response
• Concepts of impedance and admittance
• Sinusoidal steady-state analysis of simple
  circuits
• Resonance circuit
• Power in sinusoidal steady state
• Impedance and frequency normalization

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Complex number reviews
Complex number
Magnitude
Phase or angle
In polar form
or
The complex number can be of voltage,
current, power, impedance etc.. in any circuit
with sinusoid excitation.
Operations Add, subtract, multiply, divide,
power, root, conjugate
3
Phasors and ordinary differential equations

• A sinusoid of angular frequency is in the
  form

Theorem
The algebraic sum of sinusoids of the same
frequency and of their derivatives is also a
sinusoid of the same frequency
Example 1
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Phasors and ordinary differential equations
phasor
Example 2
phasor form
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Phasors and ordinary differential equations

• Ordinary linear differential equation with
  sinusoid excitation

Lemma Re.. is additive and homogenous
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Phasors and ordinary differential equations

• Application of the phasor to differential
  equation

Let
substitute
in (1) yields
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Phasors and ordinary differential equations
even power
odd power
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Phasors and ordinary differential equations
Example 3
From the circuit in fig1 let the input be a
sinusoidal voltage source and the output is the
voltage across the capacitor.
Fig1
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Phasors and ordinary differential equations
KVL
Particular solution
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Phasors and ordinary differential equations
11
Complete response and sinusoidal steady-state
response

• Complete reponse

sinusoid of the same input frequency (forced
component)
solution of homogeneous equation (natural
component)
(for distinct frequencies)
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Complete response and sinusoidal steady-state
response

• Example 4

For the circuit of fig 1, the sinusoid input
is applied to
the circuit at time . Determine the
complete response of the Capacitor voltage.
C1Farad, L1/2 Henry, R3/2 ohms.
From example 3
Initial conditions
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Complete response and sinusoidal steady-state
response

• Characteristic equation

Natural component
Forced component
From (2)
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Complete response and sinusoidal steady-state
response

• The complete solution is

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Complete response and sinusoidal steady-state
response
The complete solution is
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Complete response and sinusoidal steady-state
response

• Sinusoidal steady-state response

In a linear time invariant circuit driven by a
sinusoid source, the response
Is of the form
Irrespective of initial conditions ,if the
natural frequencies lie in the left-half complex
plane, the natural components converge to zero as
and the response becomes close to
a sinusoid. The sinusoid steady state response
can be calculated by the phasor method.
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Complete response and sinusoidal steady-state
response
Example 5
Let the characteristic polynomial of a
differential a differential equation Be of the
form
The characteristic roots are
and the solution is of the form
In term of cosine
The solution becomes unstable as
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Complete response and sinusoidal steady-state
response
Example 6
Let the characteristic polynomial of a
differential a differential equation Be of the
form
The characteristic roots are
and the solution is of the form
and
The solution is oscillatory at different
frequencies. If the output is
unstable as
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Complete response and sinusoidal steady-state
response

• Superposition in the steady state

If a linear time-invariant circuit is driven by
two or more sinusoidal sources the output
response is the sum of the output from each
source.
Example 7
The circuit of fig1 is applied with two
sinusoidal voltage sources and the output is the
voltage across the capacitor.
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Phasors and ordinary differential equations
KVL
Differential equation for each source
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Phasors and ordinary differential equations
The particular solution is
where
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Complete response and sinusoidal steady-state
response
Summary
A linear time-invariant circuit whose natural
frequencies are all within the open left-half of
the complex frequency plane has a sinusoid steady
state response when driven by a sinusoid input.
If the circuit has Imaginary natural frequencies
that are simple and if these are different from
the angular frequency of the input sinusoid, the
steady-state response also exists. The
sinusoidal steady state response has the same
frequency as the input and can be obtained most
efficiently by the phasor method
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Concepts of impedance and admittance

• Properties of impedances and admittances play
  important roles in
• circuit analyses with sinusoid excitation.

Phasor relation for circuit elements
Fig 2
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Concepts of impedance and admittance
Resistor
The voltage and current phasors are in phase.
Capacitor
The current phasor leads the voltage phasor by 90
degrees.
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Concepts of impedance and admittance
Inductor
The current phasor lags the voltage phasor by 90
degrees.
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Concepts of impedance and admittance

• Definition of impedance and admittance

The driving point impedance of the one port
at the angular frequency is the ratio of the
output voltage phasor V to the input current
phasor I
or
The driving point admittance of the one port
at the angular frequency is the ratio of
the output current phasor I to the input voltage
phasor V
or
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Concepts of impedance and admittance
Angular frequency Z Y
Resistor
Capacitor
Inductor
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Sinusoidal steady-state analysis of simple
circuits
In the sinusoid steady state Kirchhoffs
equations can be written directly in terms o
voltage phasors and current phasors. For example
If each voltage is sinusoid of the same frequency
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Sinusoidal steady-state analysis of simple
circuits
Series parallel connections
In a series sinusoid circuit
Fig 3
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Sinusoidal steady-state analysis of simple
circuits
In a parallel sinusoid circuit
Fig 4
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Sinusoidal steady-state analysis of simple
circuits

• Node and mesh analyses

Node and mesh analysis can be used in a linear
time-invariant circuit to determine the sinusoid
steady state response. KCL, KVL and the
concepts of impedance and admittance are also
important for the analyses.
Example 8
In figure 5 the input is a current source
Determine the sinusoid steady-state voltage at
node 3
Fig 5
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Node and mesh analyses
KCL at node 1
KCL at node 2
KCL at node 3
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Node and mesh analyses
Rearrange the equations
By Crammers Rule
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Node and mesh analyses
Since
Then
and the sinusoid steady-state voltage at node 3 is
Example 9
Solve example 8 using mesh analysis
Fig 6
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Node and mesh analyses
KVL at mesh 1
KVL at mesh 2
KVL at mesh 3
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Node and mesh analyses
Rearrange the equations
By Crammers Rule
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Node and mesh analyses
Since
Then
and the sinusoid steady-state voltage at node 3 is
The solution is exactly the same as from the
node analysis
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Resonance circuit

• Resonance circuits form the basics in electronics
  and communications. It is useful for sinusoidal
  steady-state analysis in complex circuits.

Impedance, Admittance, Phasors
Figure 7 show a simple parallel resonant circuit
driven by a sinusoid source.
Fig 7
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Resonance circuit
The input admittance at the angular frequency
is
The real part of is constant but
the imaginary part varies with frequency
At the frequency
the susceptance is zero. The frequency
is called the resonant frequency.
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Resonance circuit
The admittance of the parallel circuit in Fig 7
is frequency dependant
Fig 8
Susceptance plot
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Resonance circuit
Fig 9
Locus of Y
Locus of Z
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Resonance circuit
The currents in each element are
and
If for example
The admittance of the circuit is
The impedance of the circuit is
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Resonance circuit
The voltage phasor is
Thus
Fig 10
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Resonance circuit
and
Similarly if
The voltage and current phasors are
Note that it is a resonance and
Fig 11
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Resonance circuit
The ratio of the current in the inductor or
capacitor to the input current is the quality
factor or Q-factor of the resonance circuit.
Generally
and the voltages or currents in a resonance
circuit is very large!
Analysis for a series R-L-C resonance is the very
similar
46
Power in sinusoidal steady-state
The instantaneous power enter a one port circuit
is
The energy delivered to the in the interval
is
Fig 12
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Power in sinusoidal steady-state

• Instantaneous, Average and Complex power

In sinusoidal steady-state the power at the port
is
where
If the port current is
where
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Power in sinusoidal steady-state
Then
Fig 13
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Power in sinusoidal steady-state

• Remarks
• The phase difference in power equation is the
  impedance angle
• Pav is the average power over one period and is
  non negative. But p(t) may be negative at some t
• The complex power in a two-port circuit is

• Average power is additive

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Power in sinusoidal steady-state

• Maximum power transfer

The condition for maximum transfer for sinusoid
steady-state is that The load impedance must be
conjugately matched to the source imedance

• Q of a resonance circuit

For a parallel resonance circuit
(Valid for both series and parallel resonance
circuits)
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Impedance and frequency normalization

• In designing a resonance circuit to meet some
  specification component
• values are usually express in normalized form.

From
Let the normalized component values are
Then
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Impedance and frequency normalization

• Popularity of normalized design
• The circuit design can be made at any impedance
  level and center frequency
• Well-known solutions exist

Let
Then
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Impedance and frequency normalization

• Example

Fig. 14 shows a low pass filter whose transfer
impedance
The gain of the filter is 1 at
And at
Design the circuit to have an impedance of 600
ohms at
at 3.5 kHz then
and equal to
and
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Impedance and frequency normalization
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Impedance and frequency normalization
Designed circuit