Electrical impedance ($Z(\omega )=\hat{V}/\hat{I}$)

Electrical impedance is the ratio of terminal electric potential difference and electric current phasors at one specified positive angular frequency $\omega$. For an electrical circuit modeled as a linear time-invariant system, use the sinusoidal steady state after any decaying transient has settled. With the passive sign convention, positive electric current enters the terminal assigned positive electric potential difference. Define peak complex amplitudes byThe physical $v(t)$ and $i(t)$ are real; the complex numbers encode their amplitudes and wave phases. The unit of electrical impedance is the ohm, $1\Omega =1\mathrm{V}/\mathrm{A}$. Both phasors may instead use root mean square magnitudes, with the same ratio $Z$. A nonlinear diode can generate additional frequencies; a single $Z(\omega )$ then does not describe the whole response.

Write the electrical impedance in Cartesian and polar forms asHere $R$ is the resistive real part, $X$ the signed electrical reactance, and $\varphi$ the voltage-minus-current phase difference, defined modulo a full turn. The Pythagorean theorem gives the modulus from a right triangle in the complex plane. The complex argument gives the rotation from the current phasor to the voltage phasor. Thus $V_{\mathrm{rms}}=|Z|I_{\mathrm{rms}}$, while negative $\varphi$ means electric current leads terminal electric potential difference. For $R>0$, $\varphi =\arctan (X/R)$ lies between $-\pi /2$ and $\pi /2$; outside that case, use the quadrant of the complex number rather than a bare inverse tangent. The geometric magnitude and phase difference are the two pieces of the AC relation.^[ref]

Multiplication by $Z$ acts on phasors, not on instantaneous physical signals. For example, when $i(t)=I_0\cos \omega t$,The second term is in phase quadrature with electric current; it cannot be represented by pointwise multiplication $v(t)=Zi(t)$ when $X\ne 0$. This is the useful compression: one complex multiplication replaces separate amplitude and phase difference calculations, while the real part restores the physical signal.

An ideal resistor, inductor and capacitor have $Z_R=R$, $Z_L=\mathrm{i}\omega L$ and $Z_C=1/(\mathrm{i}\omega C)$. Their electrical impedances add in a series RLC circuit, giving $Z=R+\mathrm{i}(\omega L-1/(\omega C))$. The current amplitude is $V_0/|Z|$, so cancellation of the imaginary parts explains the resonant series RLC response. The component-law derivation and opposite complex exponential convention are in reactance phase convention.

For an ideal capacitor in series with a parallel resistor and inductor, $Z=1/(i\omega C)+(R\parallel i\omega L)$. A finite positive frequency with real input impedance exists precisely when $R^{2}C>L$. At that frequency ${\omega }_{\ast }=[LC-L^2/R^2{]}^{-1/2}$ and $Z_{\ast }=L/(RC)$. The resistor's voltage can exceed the source voltage because the reactive voltage drops cancel in the series sum; all average dissipation is still supplied by the source.

Two parallel branches with complex impedances $R(1+is)$ and $R(1-is)$ have a real input impedance $R(1+s^2)/2$, since their admittances have cancelling imaginary parts. For $s=\sqrt{15}$, adding a series resistance $2R$ gives total impedance $10R$. With source voltage amplitude $V_0$, each branch-resistor drop and the series-resistor drop has amplitude $V_0/5$, despite their different phases.

The impedance magnitude of a linear electrical circuit in sinusoidal steady state is the ratio of voltage amplitude to electric current amplitude. For complex impedance $Z=R+iX$, it is $|Z|=\sqrt{R^2+X^2}$, whereas the dissipative electrical resistance is $\mathrm{Re}Z=R$. Measuring only amplitudes does not separately determine $R$ and $X$; a phase shift measurement supplies the missing information.

Input impedance is the ratio of a small applied input-voltage phasor to the resulting input-current phasor, with the output termination and operating point specified. A finite input impedance loads a voltage source. For an ideal operational amplifier the input impedance is infinite, so neither input terminal draws current.

For a source with internal electrical impedance $Z_s$, conjugate matching $Z_L=Z_s^{\ast }$ maximizes delivered mean power when the source is fixed. For a purely resistive source and equal load electrical resistance $R$, the load sees half the open-circuit voltage. A Johnson-Nyquist noise source with open-circuit mean-square voltage $4k_{B}TRd\nu$ therefore delivers $k_{B}Td\nu$ to the matched load, rather than $4k_{B}Td\nu$.

A load matched to a source receives the maximum available power in the stated circuit model. For a real source resistance $R$ and open-circuit root mean square voltage $V_{\mathrm{oc}}$, a load $R$ receives voltage $V_{\mathrm{oc}}/2$ and power $V_{\mathrm{oc}}^2/(4R)$. For complex impedances, maximum power transfer instead uses conjugate matching under the usual linear-source assumptions.

A Thévenin equivalent voltage source $V_s$ with positive source resistance $R_s$ supplies load power $P_L=V_s^2{R}_L/(R_s+R_L{)}^2$ to a real resistive load. Differentiating gives its maximum at $R_L=R_s$. With reactive impedances and fixed source impedance, conjugate matching maximizes the average delivered power. Maximum delivered power is distinct from maximum efficiency; at a real resistive match half of the source's internally generated power is dissipated in $R_s$.

A lossless intermediate section of positive real characteristic impedance $Z_M$ matches unequal real exterior characteristic impedances $Z_L,Z_R$ when $Z_M^2=Z_L{Z}_R$ and its propagation wave phase is an odd multiple of $\pi /2$. The length is an odd quarter of the internal wavelength. This principle also applies to the characteristic impedance of a stretched string.

For a uniform intermediate string of length $l$, put $\delta =k_{M}l$. Continuity of displacement and transverse force gives the displacement reflection coefficientAt real positive mechanical impedances, the denominator never vanishes. Its numerator vanishes only if $(Z_L-Z_R)\cos \delta =0$ and $(Z_M^2-Z_L{Z}_R)\sin \delta =0$. Thus unequal exterior strings require the quarter-wave condition. Equal exterior strings with a different middle string instead transmit perfectly at integer half-wavelengths; if all three mechanical impedances agree, every frequency is matched.

A series circuit comprising electrical resistance $R>0$ and inductance $L>0$, placed in a parallel circuit with capacitance $C>0$, has electrical admittance $Y=(R+i\omega L{)}^{-1}+i\omega C$. Vanishing imaginary part at nonzero angular frequency gives the displayed condition, requiring $L>R^{2}C$. It defines zero voltage-current phase difference, which need not coincide exactly with the maximum of impedance magnitude when loss is present. At this frequency the electrical impedance is real and equals $L/(RC)$.

For series resistance R and inductance L in parallel with capacitance C, suppose the applied voltage amplitude V is fixed. The DC current gives $R=V/I_{\mathrm{DC}}$. If the high-frequency current is $I\sim \alpha \omega$, then $C=\alpha /V$. At the nonzero zero-phase frequency, current amplitude $I_r=VRC/L$ determines $L=VRC/I_r$. The condition $R^2<L/C$ must hold. The inferred zero-phase frequency is $\sqrt{1/(LC)-R^2/L^2}$; it is not automatically the exact frequency of minimum current magnitude.

The reciprocal of electrical impedance, relating electric current and electric potential difference phasors by $\tilde{I}=Y\tilde{V}$. Its unit is the siemens, $\mathrm{S}={\Omega }^{-1}$. Electrical admittances add for branches in a parallel circuit.

In the $e^{i\omega t}$ phasor convention, electrical admittance is $Y=G+iB$, with conductance $G$ and electrical susceptance $B$. An ideal capacitor has $B=\omega C$ and an ideal inductor has $B=-1/(\omega L)$. electrical susceptances add for parallel branches; their cancellation gives a purely resistive input admittance.

Consider a stipulated series circuit containing an ideal resistor of electrical resistance $R=100\Omega$ and an ideal capacitor of capacitance $C=1\mu \mathrm{F}$. The same electric current passes through both. Their terminal electric potential differences add by Kirchhoff's voltage law, soA quick derivation uses $i=Cd{v}_C/dt$: taking the derivative of the complex exponential multiplies the capacitor voltage phasor by $\mathrm{i}\omega$, hence ${\hat{V}}_C=\hat{I}/(\mathrm{i}\omega C)$. Dimensionally, $1/(\omega C)$ has units $s/F=V/A=\Omega$, matching $R$.

At frequency $f=1000\mathrm{Hz}$, the angular frequency is $\omega =2\pi f$. The signed capacitive electrical reactance, modulus and complex argument of the total electrical impedance areApply $V_{\mathrm{rms}}=1.00\mathrm{V}$ across the complete series circuit. Its root mean square electric current isWith voltage as the zero-phase reference, the current phasor has complex argument $+57.858092^{\circ }$: electric current leads. The phasor drops have moduli $V_{R,\mathrm{rms}}=0.532018\mathrm{V}$ and $V_{C,\mathrm{rms}}=0.846733\mathrm{V}$; these add as complex numbers, not as positive magnitudes. Their squared magnitudes sum to $(1.00\mathrm{V}{)}^2$.

The mean absorbed power uses the root mean square values:The ideal capacitor has zero cycle-averaged absorbed power; all the mean absorbed power becomes Joule heating in the resistor.^[ref] To see why the phase factor appears, average $p(t)=v(t)i(t)$ over a full cycle with $v(t)=\sqrt{2}{V}_{\mathrm{rms}}\cos \omega t$ and $i(t)=\sqrt{2}{I}_{\mathrm{rms}}\cos (\omega t-\varphi )$. The cosine addition formula reduces the average to $V_{\mathrm{rms}}{I}_{\mathrm{rms}}\cos \varphi$; the term at $2\omega$ averages to zero. Independently, $p_C=d(C{v}_C^2/2)/dt$, and the capacitor energy returns to its starting value after each cycle.

At fixed applied $V_{\mathrm{rms}}$, the limit $\omega \to 0^{+}$ gives $|Z|\to \infty$, $\varphi \to -\pi /2$ and $I_{\mathrm{rms}}\to 0$: the ideal capacitor blocks steady electric current. The limit $\omega \to \infty$ gives $Z\to R$, $\varphi \to 0$ and $I_{\mathrm{rms}}\to V_{\mathrm{rms}}/R$. These are limits of the ideal lumped electrical circuit; unwanted lead inductance and component losses can invalidate that model at high frequency. At $\omega RC=1$, the equal legs of the impedance right triangle give $|Z|=\sqrt{2}R$ and $\varphi =-\pi /4$ without numerical evaluation. Measuring both the magnitude and the sign of the phase difference distinguishes this capacitor response from an equally large positive electrical reactance.

An actual teaching experiment turns the phasor definition into readings. Leipzig University's laboratory handout Measurements with an oscilloscope, experiment E 3e, uses a function generator, a two-channel oscilloscope, a known resistor and a capacitor. Figure 5 places the capacitor above the resistor, with a common return below the resistor. Channel 1 measures the applied terminal electric potential difference $v_1$; channel 2 measures $v_R$. The handout asks for peak-to-peak values and relative timing, then comparison of measured and calculated electrical impedances with $R=3\mathrm{k\Omega }$, $C=0.47\mu \mathrm{F}$ at $100$ and $300\mathrm{Hz}$. It also asks students to check the oscilloscope time base against a digitally counted generator frequency. These are apparatus and procedural specifications, not reported measured results.^[ref]

For this arrangement, Ohm's law gives $i(t)=v_R(t)/R$ if probe loading and the sensing resistor's electrical reactance are negligible. Record both voltage magnitudes with the same convention: peak, peak-to-peak, or root mean square. After the transient settles, compare matched rising zero crossings $t_1$ of $v_1$ and $t_R$ of $v_R$, with period $T=1/f$. ThenThe recovered electrical impedance includes the known resistor. For a separate unknown component in series with that resistor, subtract $R$ from the inferred total electrical impedance. A current crossing earlier than the applied voltage crossing gives $\Delta t<0$, hence negative $\varphi$. For the stipulated series RC impedance example at $1000\mathrm{Hz}$, this predicts $\Delta t=-0.160717\mathrm{ms}$; it is a calculated timing, not a lab observation. Timing supplies the complex argument that an amplitude ratio alone cannot recover.

As additional measurement controls, first feed the same sinusoid to both channels to check relative delay and polarity; then replace the reactive chain with a known resistor, expecting nearly zero phase difference. Keep both probe returns at the common return shown in figure 5. A reversed voltage polarity adds half a turn to its inferred wave phase. Match crossings with the same slope and choose $-T/2<\Delta t\le T/2$; adjacent cycles otherwise change $\varphi$ by a full turn. A sweep continuous in frequency helps track that branch. These controls test the signed inference rather than assuming that the scope's delay readout has the desired sign.

Measure the loaded terminal electric potential difference on channel 1 at every frequency. The generator's output impedance can change the applied magnitude; its front-panel setting is not the measured terminal value. The generator's internal electrical impedance is outside the terminal ratio above, while the known sensing resistor is inside it. Probe loading, finite timing resolution, component tolerances and distorted waveforms limit the inference. A repeat at lower drive amplitude tests whether one linear electrical impedance still fits. Electrical reactance bench measurement develops these loading and reference-direction issues further.

Electrical reactance $X$ is the signed imaginary part of electrical impedance $Z=R+\mathrm{i}X$, measured in ohms. It describes the part of the terminal electric potential difference in phase quadrature with the electric current in a sinusoidal steady state. In the $e^{\mathrm{i}\omega t}$ convention, an ideal inductor has $X_L=\omega L>0$ and an ideal capacitor has $X_C=-1/(\omega C)<0$ for $\omega >0$. A large $|X|$ reduces electric current at fixed applied voltage, but an ideal purely reactive component returns the energy it receives over each cycle. This is why electrical reactance and electrical resistance have the same units yet describe different physical behavior. ^[ref].

In a series RLC circuit, electrical impedances add because the same electric current passes through every component. For ideal positive inductance $L$, capacitance $C$, and total electrical resistance $R_{\Sigma }>0$,Here $\varphi$ is voltage wave phase minus current wave phase. Below ${\omega }_0=1/\sqrt{LC}$, $X<0$ and electric current leads terminal voltage; above it, $X>0$ and electric current lags. At ${\omega }_0$, the two signed electrical reactances cancel, leaving $Z=R_{\Sigma }$ and the maximum electric current at fixed terminal voltage. This is the resonant series RLC response. Electrical resistance does not cancel: the toy circuit gives $f_0={\omega }_0/(2\pi )=1591.55\mathrm{Hz}$ and $I_{\mathrm{rms}}=9.091\mathrm{mA}$ there. These numbers follow directly from the stipulated component values. ^[ref].

Cancellation concerns terminal voltage, not the disappearance of stored energy. In the toy circuit at resonance, the individual inductor and capacitor voltages are each $0.9091\mathrm{V}$ in root mean square magnitude, with opposite wave phases, while their sum is zero. Their peak stored magnetic energy and capacitor energy are each $0.8264\mu \mathrm{J}$, attained at different times. With smaller positive $R_{\Sigma }$, either component voltage can exceed the applied voltage. This ability to select a frequency while sustaining internal energy exchange connects electrical reactance to tuning and frequency response.

For an ideal inductor or capacitor, the absorbed instantaneous power equals the rate of change of stored energy:Positive power stores energy; negative power returns it. Over a periodic steady state, each stored energy returns to its initial value, so its cycle-averaged absorbed power is zero. An ideal resistor instead has $p_R=R{i}^2\ge 0$, transferring energy into Joule heating. More generally, the average terminal power of this series RLC circuit isZero average power in an ideal reactive component does not imply zero instantaneous power, zero electric current, or zero stored energy. Real coils and capacitors have losses, so zero ideal reactive loss is a model assumption rather than a claim about every component. ^[ref].

Use complex amplitudes only for signals at one common positive angular frequency $\omega$. Define the real terminal voltage and electric current byThe complex amplitudes here use peak magnitudes; both can instead use root mean square magnitudes without changing their ratio. Choose positive electric current entering the terminal assigned positive voltage, so $p=vi$ is absorbed power. For an ideal linear inductor and capacitor, their terminal laws $v_L=Ldi/dt$ and $i_C=Cd{v}_C/dt$ giveThe derivative multiplies a complex amplitude by $\mathrm{i}\omega$: this single algebraic step explains both signs. Taking $i(t)=I_0\cos \omega t$ as a reference gives $v_L(t)=-\omega L{I}_0\sin \omega t$ and $v_C(t)=I_0\sin \omega t/(\omega C)$ for the sinusoidal response. Thus inductor voltage leads electric current by $\pi /2$, while capacitor voltage lags electric current by $\pi /2$. A constant offset or a free transient is outside this single-frequency representation. ^[ref].

With the alternative $e^{-\mathrm{i}\omega t}$ convention, the same real signals use conjugated complex amplitudes and $Z_L=-\mathrm{i}\omega L$, $Z_C=+\mathrm{i}/(\omega C)$. Their physical leads and lags stay the same; the sign of $\mathrm{Im}Z$ changes. Also, some textbooks call the positive magnitude $1/(\omega C)$ the capacitive reactance and write the series expression as $X_L-|X_C|$. This article consistently uses signed $X_C=-1/(\omega C)$ and hence $X=X_L+X_C$. ^[ref].

The motivating observation is a pair of sinusoids on an oscilloscope: changing frequency changes both their relative amplitudes and their phase difference. Gary Tuttle's EE 201 laboratory uses resistor voltage as the current-phase reference, compares predictions with readings at several frequencies, and explicitly asks students to measure component values and account for coil electrical resistance. The following bench arrangement applies that method to a series RLC circuit; its component values and illustrated traces are our own toy example. ^[ref].

Connect a low-voltage sine generator to a series chain ending in a known sensing resistor $R_s$ at the common return. Use a resistor whose unwanted inductance is negligible over the chosen frequency range. An oscilloscope displays voltage against time: channel A reads the loaded voltage across the whole chain, and channel B reads the voltage $v_s$ across $R_s$. Both channel ground connections and the generator return meet at that same node. Ordinary bench oscilloscope grounds are electrically common; attaching one to an intermediate circuit node would change the circuit. A separate floating component voltage needs a suitable differential measurement, rather than moving a ground clip. ^[ref].

After the free transient decays, record frequency $f$, period $T=1/f$, both voltage amplitudes, and matched rising zero-crossing times $t_A,t_B$. Check that the traces are sinusoidal. Define $\Delta t=t_B-t_A$ on the branch $|\Delta t|<T/2$. With negligible probe loading and negligible sensing-resistor electrical reactance, $v_s=R_{s}i$, so channel B is the electric current reference, and in the reactance phase conventionA later current crossing gives $\Delta t>0$, hence positive electrical reactance; an earlier crossing gives negative electrical reactance. Use the same amplitude convention for both channels. An amplitude-only ratio recovers $|Z|$, so it cannot distinguish the two signs: timing supplies the missing information. The formulas follow from $Z=|Z|e^{\mathrm{i}\varphi }$ and the chosen terminal orientation.

Begin with a known resistor replacing the reactive chain: its expected phase difference is approximately zero, testing channel polarity and relative delay. Measure the same signal with both channels to establish the instrumental delay baseline, then exchange probes to check that a phase offset does not belong to one channel. Next test an inductor and a capacitor separately with $R_s$: after subtracting $R_s$ from the inferred real part, expect approximately $X/f=2\pi L$ and $fX=-1/(2\pi C)$, respectively. Finally combine $L,C,R,R_s$ and sweep through the predicted resonance. These are proposed controls and consequences of the model, not reported observations.

For the toy series RLC circuit, the prediction at $500\mathrm{Hz}$ is $X=-286.894\Omega$, $|Z|=307.259\Omega$, $\varphi =-69.02^{\circ }$, and $V_{s,\mathrm{rms}}=32.55\mathrm{mV}$. At $5000\mathrm{Hz}$, $X=+282.328\Omega$ and $\varphi =+68.71^{\circ }$. Near $1591.55\mathrm{Hz}$ the model predicts zero phase difference and $V_{s,\mathrm{rms}}=90.91\mathrm{mV}$. Keep actual loaded $V_{A,\mathrm{rms}}$ fixed or normalize by it at each frequency; the generator's display alone does not establish that voltage. The $10\Omega$ sensing resistor is part of the $110\Omega$ total and therefore changes the predicted peak electric current.

Real components add unintended electrical impedance: coil wire has electrical resistance, nearby turns add capacitance, and capacitor leads add inductance. These effects can produce a component's own resonance, beyond which the ideal $\omega L$ or $-1/(\omega C)$ approximation fails. Work in a range where that approximation is adequate, measure coil wire electrical resistance, and compare the inferred $R_{\mathrm{chain}}$ with the sensing resistor, intended resistor, and component losses. A measured nonzero real part is evidence of loss in the assembly, not negative or positive electrical reactance becoming heat. ^[ref]; ^[ref].

The probes also add electrical resistance and capacitance to the measured nodes; their loading becomes more important as frequency increases. Use compensated probes with known input electrical impedance and short ground connections, and repeat with a smaller drive voltage to test whether the inferred $X$ depends on amplitude. ^[ref]. Record actual voltage and phase readings separately from calculated predictions, with component tolerances and timing uncertainty. A phase error $\delta \varphi$ alone produces approximately $\delta X=|Z|\cos \varphi \delta \varphi$, so near zero electrical reactance, a small delay can imitate a small residual $X$. Agreement within stated uncertainties tests the model; the smooth toy curves establish no experimental accuracy.

The electrical impedance seen looking into output terminals with independent sources suppressed, as in Thévenin source suppression.