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## Impedance and Generalized Ohm's Law

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:

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.

• 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

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.

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

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:

i.e., or . We therefore get: , , , , and

As , . But as they are apart in phase, we have , and from the vector diagram . We also get 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 2016-12-04