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Impedance and Generalized Ohm's Law

Impedance of Basic Components

The relationship between the sinusoidal current through and the sinusoidal voltage across a capacitor or an inductor in an AC circuit is described by a differential equation in time domain. However, if we treat such a sinusoidal variable as the projection of a complex exponential, a vector rotating counter-clock wise, onto the real axis:

then the relationship between the voltage and current can be described by an algebraic equation. Specifically, we define the ratio of the complex exponential forms of the voltage and the current associated with a resistor , a capacitor , or an inductor as the impedance of the component:

• Resistor:

The impedance of a resistor is the ratio of the phasor representations of the voltage and current:

The magnitude of phase of the current and voltage are related by:

The resistor introduces no phase shift between the voltage and current, i.e., they are in phase.

• Capacitor:

The impedance of a capacitor is the ratio of the phasor representations of the voltage and current:

The magnitude of phase of the current and voltage are related by:

The phase shift introduced by a capacitor is , i.e., the voltage lags behind the current by , or the current leads the voltage by .

• Inductor:

The impedance of an inductor is the ratio of the phasor representations of the voltage and current:

The magnitude of phase of the current and voltage are related by:

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).

In any of the cases above, the impedance is defined as

This is the generalized version of the Ohm's law for AC circuits.

• 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:

Here is a review of complex arithmetic.

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 2018-04-02