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Sinusoidal Steady State Response
We have discussed the steady state responses of circuits to driving functions
(eg. Vs) that remained constant or varied as a "square" wave.
Now consider the effects of sinusoidally varying voltage sources such as:
Vs =  = m sin t
where  represents the angular frequency of variation:
 = 2f
(rad/s)
For this RLC circuit, driven by a sinusoidal voltage, what is the steady state
current i in terms of the capacitance, inductance and resistance?
Sinusoidal Steady State Response:
Initial assumption is that all of the elements are lumped rather than distributed
(more appropriate for low than for high frequency driving functions)
The current in the loop is everywhere the same.
For a sinusoidal driving function of:
 = m sin t
The current will also be sinusoidal and fluctuating at the same frequency such
that:
i = im sin (t + )
where: im is the maximum current value and

is the phase angle between the current and the applied .
Before we consider the general case (RLC) let us first consider circuits
composed of each element type individually.
Sinusoidal Steady State Response:
Resistive Circuit:
For the resistive circuit shown below driven by a sinusoidal voltage
our voltage loop equations and the definition of resistance R suggests that:
VR = m sin t = iR
so that i =
m
R
sin t
This suggests that the current through a resistor is in phase
with the voltage across it
and that they both vary with the same frequency
This can also be represented by a phasor diagram.
Sinusoidal Steady State Response:
Phasor Representation:
Phasors represent sinusoidally varying
quantities.
Phasors are conceived to rotate
counterclockwise with angular velocity .
The length of a phasor indicates the
maximum value of the quantity
represented.
Sinusoidal Steady State Response:
Phasor Representation:
Projection of the phasor onto
the vertical axis represents the
value of the quantity at any
particular time.
Relative phase between two quantities is represented by the angle between their
phasors.
Sinusoidal Steady State Response:
Resistive Circuit:
VR,m = iR,m R = m
Current and voltage phasor for the resistive
circuit are shown in a phasor diagram.
Sinusoidal Steady State Response:
Capacitive Circuit:
For the capacitive circuit shown below driven by a sinusoidal voltage
our voltage loop equations and the definition of capacitance C suggests that:
VC = m sin t =
so that q = mC sin t
and iC =
dq
dt
= Cm cos t
VC and iC are 90° out of phase.
q
C
Sinusoidal Steady State Response:
Capacitive Circuit:
Depicted below are the capacitor voltage and current as functions of time.
VC lags the current iC (VC reaches its maximum after iC).
Also shown is the phasor diagram representing this situation.
Can express the capacitor current as:
iC =
 M 


 XC 
cos t
where
1
XC =C
XC is the capacitive reactance and is measured in units of ohms.
The maximum capacitor voltage can be expressed as: VC,m = iC,m XC = m
Sinusoidal Steady State Response:
Inductive Circuit:
For the inductive circuit shown below driven by a sinusoidal voltage
our voltage loop equations and the definition of inductance L suggests that:
VL = m sin t = L
so that diL =
and
iL = -
 M 


 L 
 M 


 L 
sin t dt
cos t
VL and iL are 90° out of phase.
di L
dt
Sinusoidal Steady State Response:
Inductive Circuit:
Depicted below are the inductor voltage and current as functions of time.
VL leads the current iL (VL reaches its maximum before iL).
Also shown is the phasor diagram representing this situation.
Can express the inductor current as:


iL = -  XM  cos t
L
where
XL = L
XL is the inductive reactance and is measured in units of ohms.
The maximum inductor voltage can be expressed as: VL,m = iL,m XL = m
Sinusoidal Steady State Response:
The Single Loop RCL Circuit:
Consider a series RLC circuit with a sinusoidal driving function.
 = m sin t
The current will be of the form:
i = im sin (t + )
where: im and  are determined by the circuit components and m.
Applying the loop theorem we get:
 = VR + VC + VL
VR , VR and VL are all sinusoidally varying quantities with maximum values,
VR,m (= im R), VC,m (= im XC ), and VL,m (= im XL)
This equation is correct for every moment in time but it is not useful for
calculating the current i.
Sinusoidal Steady State Response:
The Single Loop RCL Circuit:
Instead we must rely on vector addition
(see phasor diagrams)
Sinusoidal Steady State Response:
Generalization of Circuit Concepts
Sinusoidally varying voltages and currents can be represented
using phasors which have a magnitude and a phase.
They can be considered as complex numbers ( 2 dimensional vectors).
Complex Impedance:
V ( )
 Z( )  R ( )  jX( )
I ( )
Complex Admittance:
I ( )
 Y( )  G ( )  jB( )
V( )
where  is the angular frequency of the driving function and
j = 1
Complex numbers (like 2 dimensional vectors) can be expressed in
polar (magnitude and phase) and Cartesian (x and y components) forms.
Sinusoidal Steady State Response:
Equivalent Impedances Zeq:
Zeq = ZT = Z1 + Z2 + Z3 + . . . + Zn
Equivalent Admittances Yeq:
Yeq= YT = Y1 + Y2 + Y3 + . . . + Yn
=
1
ZT
Sinusoidal Steady State Response:
For sinusoidal steady state analysis complex impedances can be treated
in a similar manor as resistances are for DC steady state analysis
Complex arithmetic however, must be used.
Example:
For the circuit show below what is Zeq or ZT ?
Sinusoidal Steady State Response:
Sinusoidal Steady State Response:
Kirchoff's Voltage Law:
Example
Sinusoidal Steady State Response:
Kirchoff’s Current Law:
Example
Sinusoidal Steady State Response:
Equivalent Sources:
Sinusoidal Steady State Response:
Thevenin and Norton Equivalent Circuits:
Vt and in vary sinusoidally
Zeq and Yeq are both functions of 
Yeq
1
=
Z eq
Sinusoidal Steady State Response:
Thevenin and Norton Equivalent Circuits:
Example
Driving frequency is such that L = 1 and C = 2
Sinusoidal Steady State Response:
Superposition:
Superposition is based on concepts of linearity.
If a circuit is composed on linear components, superposition still
holds.
Sinusoidal Steady State Response:
Voltage Dividers:
Vo =
Zo
Vi
Z o  Z1
Zo
Z1
Vo
Zo


Zo
Vi
Z o  Z1
1
Z1
Z1can be the equivalent series impedance of any combination of impedances.
Vo , Vi , Zo and Z1 are phasors.
Sinusoidal Steady State Response:
Current Dividers:
Io =
Z1
Ii
Z o  Z1
Io
Z1
1


Z
Ii
Z o  Z1
1 o
Z1
Z1can be the equivalent parallel impedance of any combination of impedances.
Io , Ii , Zo and Z1 are phasors.
Sinusoidal Steady State Response:
Input Impedance:
Impedance looking into the input terminals of a
circuit section.

Assume ZL =
Sinusoidal Steady State Response:
Output Impedance:
Impedance looking into the output terminals of a
circuit section.

Assume Zs =
Sinusoidal Steady State Response:
Loading Effects:
Dependent on ratios of output to input and input to
output impedances respectively.
Frequency dependent!
Sinusoidal Steady State Response:
Frequency Response:
For resistive circuits we defined the voltage transfer function as:
V
T = Vo
i
For sinusoidal steady state analysis the value of a transfer function
is dependent on the frequency  of the driving function.
Therefore a voltage transfer function can be expressed as:
T() =
Vo ( )
Vi ( )
which is a complex function of 
Sinusoidal Steady State Response:
Frequency Response:
The magnitude and phase of T() are called the circuit’s frequency response.
T( ) 
()
Vo ( )
 A v ()
Vi ( )
=
 Im[ T( )]

 Re[ T( )]
tan 1 
Av () is the voltage gain
() is the phase angle
where: Im[T()] is the imaginary part of T()
and
Re[T()] is the real part of T()
Sinusoidal Steady State Response:
Frequency Response:
Av is often expressed in decibels (dB) = 20 log
Vo ( )
Vi ( )
Av is often plotted in dB versus log  and called the magnitude response.
() is often plotted versus log  and called the phase response.
Together these plots are called Bode plots.
Magnitude responses are used the describe circuit functions and bandwidths.
Sinusoidal Steady State Response:
Example Frequency (Amplitude and Phase) Response:
Sinusoidal Steady State Response:
Generalized 1st Order Low Pass Filter:
Sinusoidal Steady State Response:
Generalized 1st Order High Pass Filter:
Sinusoidal Steady State Response:
Sinusoidal Steady State Response:
DF = (Dt/T)*360o
2T
DF = (Dt/T)*2 (rads)
Dt
Sinusoidal Steady State Response:
Sinusoidal Steady State Response:
Sinusoidal Steady State Response:
Sinusoidal Steady State Response: