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 current and voltage associated with a component in
an AC circuit are related by integration or differentiation in time
domain. But if they are represented in phasor form and
in frequency domain, they are related by algebraic equations.
Phasor representation of a sinusoidal variable of angular frequency
is 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.
Impedance and Admittance
As a complex variable, the impedance can be written as:
The magnitude and phase angle of are:
The reciprocal of the impedance is called admittance:
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:
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
.
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.
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.
Solution
Note that is behind
by ,
and is ahead of
by
(``ELI the ICE man''). Also, as and are in phase,
the impedance of the parallel combination of the RL and RC branches
shown below must real introducing no phase shift: