Deriving resistor, capacitor & inductor impedance
Derive impedance from the time-domain constitutive laws instead of memorizing a table. Anchor the definition with the resistor, obtain Z_C and Z_L through d/dt ↔ jω, derive admittance, verify units and phase relations, interpret frequency dependence and state the exact limits of ideal-element and steady-state reasoning.
- State the receiver/passive reference convention before using R, C and L constitutive laws.
- Define impedance as the phasor-domain coefficient relating voltage and current for one linear two-terminal element at one frequency.
- Derive Z_R=R from v=Ri.
- Derive Z_C=1/(jωC) and its −90° voltage/current phase relation from i=C dv/dt.
- Derive Z_L=jωL and its +90° voltage/current phase relation from v=L di/dt.
- Derive admittance as Y=1/Z rather than memorizing a second table.
- Use dimensional checks and frequency limits to test the formulas.
- Distinguish ideal sinusoidal-steady-state impedance from DC transients, initial conditions and real-component parasitics/self-resonance.
Where do the familiar formulas ZC=1/(jωC) and ZL=jωL actually come from?
They are not independent rules. They follow directly from the time-domain constitutive laws of the elements once the sinusoidal steady-state mapping d/dt ↔ jω has been justified.
03.1Start from terminal references and constitutive laws
AoN uses the receiver/passive convention when writing the constitutive laws below: positive current enters the terminal defined positive by the voltage reference. Under ADR-0018, if a voltage arrow is explicitly drawn it runs from the negative reference toward the positive reference, so its arrow direction is opposite to the positive current arrow in receiver convention.
No arrow is needed in this chapter’s figure; the algebraic reference convention is stated once here and retained throughout.
These are time-domain element laws. Impedance will be derived from them rather than inserted as a lookup table.
03.2Define impedance only after the phasor relation exists
For a linear two-terminal element in sinusoidal steady state at one declared angular frequency ω, suppose voltage and current have phasors v and i under the same peak-amplitude convention. When their phasor-domain constitutive relation can be written
the complex coefficient Z(ω) is the element impedance at that frequency:
Its SI unit is the ohm (Ω). This definition is a statement about one harmonic component and the declared terminal references; it is not a replacement for the full time-domain law in arbitrary transients.
03.3The resistor is the anchor case
An ideal resistor introduces neither differentiation nor integration, so its voltage and current phasors are in phase under receiver convention. Its ideal impedance is real and independent of frequency.
Real resistors eventually acquire parasitic inductance, capacitance and frequency-dependent losses; Z_R=R is the ideal lumped-element model.
03.4Derive the capacitor impedance
Solve for the voltage/current ratio rather than memorizing a sign:
Therefore the capacitor voltage phasor is 90° behind its current phasor; equivalently, capacitor current leads capacitor voltage by 90°.
03.5Derive the inductor impedance
The inductor voltage phasor is therefore 90° ahead of its current phasor; equivalently, inductor current lags voltage by 90°.
The signs are not mnemonic decorations: both phase relations follow from the same derivative rule d/dt ↔ jω and the two different constitutive-law orientations.
03.6Admittance is the reciprocal description
Admittance has SI unit siemens (S). It is especially natural when currents add at a node.
These are not new independent formulas; they are the reciprocals of the impedances already derived.
03.7Read frequency dependence without overextending the model
Within the ideal model, increasing ω lowers the magnitude of capacitor impedance and raises the magnitude of inductor impedance.
Figure 3.1 — How does frequency change ideal impedance?
The one question is: why does |Z_C| fall while |Z_L| rises with ω?
The plot summarizes ideal sinusoidal-steady-state models only. It does not claim that real components follow these laws to arbitrarily high frequency.
At very high frequency, real components depart from these simple laws because of parasitic fields, losses, lead/package effects, distributed behavior and self-resonance. An ideal-element expression is therefore a model with a validity range, not a universal statement about a physical part at every frequency.
03.8DC limits require careful language
The phasor formulas above assume a nonzero sinusoidal frequency. Their low-frequency limits are useful, but exact DC is more cleanly interpreted from the time-domain laws.
Thus an ideal capacitor behaves as an open circuit for a settled DC state, and an ideal inductor has zero voltage drop for a settled constant current state. Those phrases do not describe a switching transient: initial capacitor voltage and inductor current can matter critically before steady state is reached.
03.9Worked checks at one frequency
Take f=1 kHz, hence ω=2π·1000 rad/s.
If vC=5∠20° V, then iC=vC/ZC≈3.142∠110° mA: the current leads by 90°. If the same voltage is across the 10 mH inductor, the current angle is 20°−90°=−70°.
03.10Mastery exercises
- Starting from i=C dv/dt, derive Z_C without using a memorized table.
- Starting from v=L di/dt, derive Z_L.
- At 10 kHz, calculate the impedance of a 10 nF ideal capacitor.
- At 10 kHz, calculate the impedance of a 1 mH ideal inductor.
- Find the admittance of the capacitor and inductor in exercises 3 and 4.
- Explain from the phasor equations—not a mnemonic—which quantity leads by 90° in C and L.
- Explain why “a capacitor is an open circuit” is incomplete without the words ideal and settled DC.
- A real 100 nF capacitor becomes inductive above its self-resonance. Does that contradict Z_C=1/(jωC)? Explain.
Solutions
1. i=jωCv, hence ZC=v/i=1/(jωC)=−j/(ωC).
2. v=jωLi, hence ZL=jωL.
3. ω=2π·10⁴ rad/s; |ZC|≈1591.5 Ω, so ZC≈−j1.5915 kΩ.
4. |ZL|≈62.83 Ω, so ZL≈j62.83 Ω.
5. YC=jωC≈j0.6283 mS; YL=1/(jωL)≈−j15.915 mS.
6. For C, i=jωCv, so current is +90° relative to voltage. For L, v=jωLi, so voltage is +90° relative to current.
7. During transients a capacitor can carry current while its voltage changes; the open-circuit statement applies to the ideal element after DC has settled.
8. No. The simple formula describes an ideal lumped capacitor. A real component contains parasitic inductance and loss, so its model must be extended near and above self-resonance.
Chapter 04 will use these derived element relations to show that KCL and KVL retain their algebraic form for same-frequency phasors.