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Generalized Ohm's Law
The Ohm's law in DC circuit composed of resistors only can be
generalized to describe the relationship between the sinusoidal voltage
and current associated with a component, a capacitor , an inductor ,
or a resistor . The impedance of the component is defined as the
ratio of the AC voltage and current associated with the component both
represented as complex variables or phasors:
The impedance is in general complex and its magnitude is the ratio of
the magnitudes of the voltage and the current, and its phase is the phase
difference between the voltage and the current:
As phasor representation of a sinusoidal voltage or a current of angular
frequency is actually the only Fourier coefficient in
the frequency domain, the impedance can be considered as the
frequency response function
of the circuit component when the current through it and the voltage across
it, both represented as phasors, are considered as the input and output of
the component, respectively. As there is only one frequency in the signal,
it does not need to be represented.
Impedance of Basic Components
The impedance of a specific component can be obtained according to the
physics of the component.
- Resistor:
As a resistor introduces no phase shift between the voltage and current, its
impedance, same as resistance, is real:
- Capacitor:
The phase shift introduced by a capacitor is
,
i.e., the voltage lags behind the current by .
- Inductor:
The phase shift introduced by a inductor is
,
i.e., the voltage leads the current by .
One way to remember the phase between the voltage and current
associated with capacitor and inductor is ``ELI the ICE man''.
Also, consider two extreme cases:
- When ,
and the capacitor has zero
conductivity due to the insulation between its two plates (open circuit),
and as there is no flux change in the inductor and the resistance
of the coil is ideally zero.
- When
,
and the capacitor
becomes highly conductive, and
as the self-induced
voltage in the coil always acts against any change in the input (Lenz's Law).
Impedance and Admittance
- Impedance
As a complex variable, the impedance can be written as:
- The real part of impedance is called resistance.
- The imaginary part of impedance is called reactance.
Impedance, resistance, and reactance are all measured by the same unit
Ohm ().
The magnitude and phase angle of are:
The impedances
associated with and
are both purely imaginary, i.e., they are both reactance,
indicating these components are reactive and consume no energy.
- Admittance
The reciprocal of the impedance is called admittance:
- The real part of admittance is called conductance:
- The imaginary part of admittance is called susceptance:
Admittance, conductance and susceptance are all measured by the same
unit Siemens ().
The magnitude and phase of complex admittance are
Impedance and admittance are both complex variables. The
real parts and are always positive, but the imaginary
parts and can be either positive or negative. Therefore
and can only be in the 1st or the 4th quadrants of the complex plane.
In particular, the admittances of the three types of elements R, L
and C are
Ohm's law can also be expressed in terms of admittance as well as impedance:
Sometimes it is more convenient in circuit analysis to use admittance
instead of impedance.
- Components parallel:
- Components in series:
Example 1:
Solve the circuit below. The voltage from the generator is
.
First find the impedances and admittances of the components and the two
branches. As , we get
Next express voltage
in phasor form
, and find currents in phasor form:
Verify:
Example 2:
A current
flows through a circuit composed of a resistor , and capacitor
and an inductor
connected in series. Find the resulting voltage across all
three elements.
- Express in phasor:
- Find impedance for each element ():
- Find overall impedance:
- Find voltage across all three elements:
Alternatively, we can find voltage across each element:
In time domain, we have
Adding , , and , we get the total voltage
which is the same as what we got above:
Example 3:
In the circuit below,
with some unknown peak
value ,
, and . The RMS value
of across is 10 V. It is also known that and
are in phase.
- Find .
- Find the RMS values of and .
- Find the RMS values of and
.
- Find the peak value of .
Solution
As and are in phase (zero angular difference),
the admittance of the parallel combination of the RL and RC
branches must not introduce any phase shift, i.e., it is real
with imaginary part equal to zero:
As , we get
, i.e.,
,
,
. The impedance of the
parallel combination of the RL and RC branches is
As
,
. But as
they are apart in phase, we have
.
However, their phase difference is , and
.
The currents through RC and RL branches are:
But their phase difference is , we have
The voltage across is
, and
The peak value is therefore
Next: First Order Systems
Up: Chapter 3: AC Circuit
Previous: Phasor Representation of Sinusoidal
Ruye Wang
2016-07-28