Impedance of Passive Components

Impedance of a component is a complex number denoted “Z” that is the constant of proportionality between voltage across that component and current through it. It is the ratio of complex voltage to complex current. The inverse of impedance, which is the ratio of complex current to complex voltage, is called admittance.

V(s) = Z(s) \cdot I(s)

Time Domain:

Capacitor: X_C(t) = \frac{-1}{\omega C}; Z_C(t) = \frac{-j}{\omega C} = \frac{-j}{2 \pi f C}

Inductor: X_L(t) = \omega L; Z_L(t) = j \omega L = j 2 \pi f L

Resistor: X_R(t) = 0; Z_R(t) = R

Practicing EEs typically use Hertz rather than radians for measuring frequency, so I recommend to memorize the 2πf versions of the impedance equations. It can be helpful to think of impedances as vectors on the complex number plane. The impedance of inductance is zero real portion and purely positive imaginary portion, the impedance of capacitance is zero real portion and purely negative imaginary portion, and the impedance of resistance is purely positive real portion with zero imaginary portion.

Complex Impedance

Frequency Domain:

Capacitor: X_C(s) = \frac{1}{sC}; Z_C(s) = \frac{1}{sC}

Inductor: X_L(s) = sL; Z_L(s) = sL

Resistor: X_R(s) = 0; Z_R(s) = R

Series and Parallel Combinations

Impedances can be combined through series and parallel combinations just as resistances can be.

Impedances add in series: ZEQ = Z1 + Z2

Impedances in parallel reduce: ZEQ = (Z1-1 + Z2-1)-1

Let’s try an example and find the equivalent impedance of the resistor, capacitor, and inductor shown below.

Impedance Combinations

Impedances of the components:

Z_{C_1} = \frac{1}{s C_1}

Z_{L_1} = s L_1

Z_{R_1} = R_1

C1 and R1 are in parallel, combine: ((1/sC1)-1 + (R1)-1)-1:

\frac{1}{s C_1 + \frac{1}{R_1}} = \frac{R_1}{s R_1 C_1 + 1}

This is then in series with the inductor, so combine: Z1 = sL1 + R1/(s*R1*C1+1).

Z_1 = s L_1 + \frac{R_1}{s R_1 C_1 + 1}

Voltage and Current Divider Calculations with Impedance

Voltage and current dividers work the same with impedance in the frequency domain as they do with resistors.

Voltage & Current Dividers

Voltage Divider: V_{OUT}(s) = V_{IN}(s) \cdot \frac{Z_2(s)}{Z_1(s) + Z_2(s)}

Current Divider: I_3(s) = I_{IN} (s) \cdot \frac{Z_4(s)}{Z_3(s) + Z_4(s)}

Current Divider: I_4(s) = I_{IN} (s) \cdot \frac{Z_3(s)}{Z_3(s) + Z_4(s)}

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