Impedance Converter
Calculate and convert electrical impedance from resistance and reactance components. Find magnitude, phase angle, admittance, and quality factor.
Impedance Converter
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Formula: |Z| = sqrt(R^2 + X^2) | Phase = atan(X/R) | Y = 1/Z | Q = |X|/R
Worked example โ Impedance = 10 ohms at 36.87 degrees
Formula
|Z| = sqrt(R^2 + X^2) | Phase = atan(X/R) | Y = 1/Z | Q = |X|/R
Impedance Z is a complex quantity with resistance R as the real part and reactance X as the imaginary part. The magnitude is the Pythagorean combination of R and X. The phase angle indicates whether the circuit is inductive (positive) or capacitive (negative). Admittance Y is the inverse of impedance, and quality factor Q measures energy storage efficiency.
Worked Examples
Example 1: Speaker Impedance Calculation
Problem:A speaker has 8 ohms resistance and 6 ohms inductive reactance at 1 kHz. Find the total impedance.
Solution:|Z| = sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 ohms Phase = atan(6/8) = 36.87 degrees Q = 6/8 = 0.75
Result:Impedance = 10 ohms at 36.87 degrees
Example 2: Filter Circuit Impedance
Problem:A circuit has 220 ohms resistance and 150 ohms capacitive reactance. Calculate impedance and admittance.
Solution:|Z| = sqrt(220^2 + 150^2) = sqrt(48400 + 22500) = sqrt(70900) = 266.27 ohms Phase = atan(150/220) = 34.29 degrees Admittance = 1/266.27 = 3.756 mS
Result:Impedance = 266.27 ohms, Admittance = 3.756 mS
Frequently Asked Questions
What is impedance and how does it differ from resistance?
Impedance is the total opposition to alternating current (AC) flow in a circuit, measured in ohms. Unlike resistance, which only opposes DC or the real part of AC, impedance includes both resistance (real component) and reactance (imaginary component). Reactance arises from capacitors and inductors that store and release energy. Impedance is represented as a complex number Z = R + jX, where R is resistance and X is reactance.
How do you calculate impedance magnitude and phase angle?
Impedance magnitude is calculated using the Pythagorean theorem: |Z| = square root of (R squared plus X squared), where R is resistance and X is reactance. The phase angle theta equals arctan(X / R), which gives the angle between voltage and current waveforms. A positive phase angle indicates inductive behavior (current lags voltage), while a negative phase angle indicates capacitive behavior (current leads voltage).
What is admittance and how does it relate to impedance?
Admittance is the reciprocal of impedance, measured in siemens (S). It represents how easily AC current flows through a circuit. Admittance Y = 1/Z = G + jB, where G is conductance (real part) and B is susceptance (imaginary part). Conductance G = R / (R squared + X squared) and susceptance B = -X / (R squared + X squared). Admittance is useful in parallel circuit analysis where impedances are difficult to combine directly.
What is the quality factor Q in impedance?
The quality factor Q is the ratio of reactance to resistance: Q = |X| / R. It measures how much energy is stored versus dissipated per cycle in a reactive component. A high Q indicates low energy loss and sharp frequency selectivity, important in filters and resonant circuits. For example, a Q of 10 means the component stores 10 times more energy than it dissipates per cycle. Typical Q values for inductors range from 10 to 200.
How do inductive and capacitive reactance vary with frequency?
Inductive reactance rises in proportion to frequency, XL = 2 x pi x f x L, so an inductor increasingly blocks higher frequencies. Capacitive reactance falls with frequency, XC = 1 / (2 x pi x f x C), so a capacitor passes high frequencies and blocks DC. A 10 microfarad capacitor at 60 Hz presents XC = 1 / (2 x pi x 60 x 0.00001) = 265 ohms, but only about 2.65 ohms at 6 kHz. Because the two reactances carry opposite signs, they partially cancel whenever both are present.
What is the difference between rectangular and polar form?
Rectangular form writes impedance as Z = R + jX, separating dissipative resistance from energy-storing reactance. Polar form writes the same quantity as a magnitude and an angle. Conversion runs both ways: |Z| = sqrt(R^2 + X^2) and theta = arctan(X / R), while R = |Z| x cos(theta) and X = |Z| x sin(theta). Rectangular form is convenient for adding impedances in series, polar form for multiplying and dividing them, which is why AC circuit analysis moves back and forth between the two.
What happens to impedance at resonance in a series RLC circuit?
At the resonant frequency the inductive and capacitive reactances are equal in magnitude and opposite in sign, so they cancel and total reactance X goes to zero. Impedance collapses to the pure resistance R, its minimum possible value, and the phase angle becomes zero so voltage and current are back in phase. The resonant frequency is f0 = 1 / (2 x pi x sqrt(L x C)). Because impedance is at a minimum, series resonance produces a current peak, which is the basis of band-pass filtering and antenna tuning.
How do you combine impedances in series and in parallel?
In series, impedances add as complex numbers: resistances sum and reactances sum separately, so Z total = (R1 + R2) + j(X1 + X2). In parallel the reciprocals add, which is where admittance earns its keep, since Y total = Y1 + Y2 can be summed in rectangular form and inverted just once at the end. For only two elements the product-over-sum shortcut Z1 x Z2 / (Z1 + Z2) works, but it must be evaluated with complex arithmetic and never with magnitudes alone.
What is characteristic impedance and why do cables specify 50 or 75 ohms?
Characteristic impedance is a property of a transmission line rather than of a load, given approximately by the square root of the line inductance per unit length divided by its capacitance per unit length, and it does not depend on cable length. For coaxial cable the geometry that minimises attenuation and the geometry that maximises power handling fall on either side of roughly 50 ohms, so 50 ohms became the general-purpose compromise for RF and test gear, while 75 ohms, nearer the low-loss optimum, became standard for video and broadcast distribution.
Why does impedance matching matter for power transfer?
The maximum power transfer theorem states that a source delivers the most power to a load when the load impedance is the complex conjugate of the source impedance, meaning equal resistance and equal but opposite reactance. Any mismatch reflects part of the incident wave back toward the source, which appears as standing waves on a transmission line and as wasted power. Maximum power transfer is not the same as maximum efficiency, which is why power distribution deliberately runs mismatched with a source impedance far below the load.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator ยท Editorial policy
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