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AC Circuit Basics: i-v Relationships, Impedance, and Admittance
Last Update: 11/11/2025
Introduction
This blog series focuses on the fundamental analysis of Alternating Current (AC) circuits.
We begin by establishing the essential current-voltage (i-v) relationships for pure element AC circuits:
- the resistor (R),
- the inductor (L), and
- the capacitor (C).
To analyze these time-varying signals efficiently, we’ll introduce the powerful concept of phasors.
Phasors are vectors (having magnitude and direction) that allow us to represent sinusoidal voltages and currents as simple rotating vectors.
Equipped with phasors, we’ll analyze combined RLC circuits in both series and parallel configurations.
To handle the combined opposition to current in these circuits, we will introduce and utilize the core concepts of impedance (Z) and admittance (Y).
- Impedance is the total opposition an electrical circuit presents to the flow of an alternating current, and
- admittance measures how easily an alternating current circuit allows current to flow.
You need to understand these basics if you ever hope to understand more complicated circuits.
Pure Resistor AC Circuit
I primarily used this excellent Khan Academy Video to write this section:
A purely resistive AC circuit contains only a resistor (R).
The resistor’s job is to oppose the current flow, converting electrical energy into heat, regardless of whether the current is AC or DC.
Picture: Simple (Pure) Resistor AC Circuit
Note: Since I and V have different units of measure don’t try to compare the magnitude of the curves (apples and oranges).
- We put them together on the same chart to show that over time they move together in sync.
- We cant say anything about their relative magnitudes because they have different units.
Voltage and Current Equations
(1) V = V0sin(ωt) ; Voltage Equation Pure Resistive AC Circuit
- Since the largest value for sin (wt) will be one, V0 is the maximum voltage (the amplitude of the sinusoid)
Substituting V = IR (Ohm’s Law) into equation (1) gives
(2) I =(V0/R)sin(ωt) = I0sin(ωt) ; Current Equation Pure Resistive AC Circuit
Current and Voltage (i-v or I-V) Relationship
The voltage (V) and current (I) are perfectly in phase.
- This means they rise, fall, and cross zero at the exact same moment.
- i.e. the current is oscillating in sync with the voltage (and vice versa of course)
Opposition to AC
The circuit’s opposition to AC in a purely resistance circuit is equal to its resistance (R).
- We could call this resistance impedance as well (Z) but it’s not very useful to do so.
- Impedance is the total opposition to AC, encompassing the overall effect of all elements in a circuit.
- So we’ll come back to it later when we deal with more complicated circuit element configurations. (like R.L.C circuits)
Power
All electrical energy supplied to the resistor is dissipated as heat (real power).
The average power, P, is the actual power that is dissipated by the resistor as heat.
(3a) P = (Vrms)(Irms)
(3b) P = (Irms)2 R
(3c) P= (Vrms)2/R
Where:
- P is the Average Power (or Real Power) in Watts (W).
- Vrms is the Root Mean Square voltage in Volts (V).
- See my blog RMS (Root Mean Square) of a Sinusoid for more on RMS
- Irms is the Root Mean Square current in Amperes (A).
- R is the Resistance in Ohms Omega).
Pure Inductor AC Circuit – Time Domain Analysis
I primarily used these excellent Khan Academy Videos to write this section:
A purely inductive AC circuit contains only an inductor (L), which is an ideal coil with no resistance.
An inductor opposes changes in the current flowing through it.
In an AC circuit, the current is constantly changing, so the inductor generates a back electromotive force (EMF) that opposes the source voltage.
This opposition to AC current flow is called inductive reactance (XL).
This causes the I and V sinusoids to become out of sync.
An inductor does not consume any real power; it simply stores energy in its magnetic field and then releases it.
Picture: Simple (Pure) Inductor AC Circuit
Note: Since I and V have different units of measure don’t try to compare the magnitude of the curves (apples and oranges).
- We put them together on the same chart to show that over time they do not move together in sync.
Voltage and Current Equations
The equation below is a mathematical expression of Faraday’s Law of Induction and Lenz’s Law as applied to a coil (inductor).
(4) VL= L dI/dt ; Inductor Voltage
It describes the instantaneous voltage VL across the inductor in terms of the rate of change of the current I passing through it
- VL = the instantaneous voltage across the inductor (in Volts)
- L = the inductance (a constant property of the coil) in units of Henrys (H).
- dI/dt = the instantaneous rate of change of current with respect to time in Amperes per second (A/s)
In a pure inductor circuit, the source voltage VS is equal to the voltage across the inductor VL.
(5) VL = VS
(6) VL = VS = V = Vosin(ωt) ; Voltage Equation for Pure Inductive AC Circuit
Combing (4) and (6) we get
(7) L dI/dt = V0sin(ωt)
Since integration is the inverse operation of differentiation, integrating both sides of the differential equation (7) is the standard procedure used to recover the original function whose derivative is given.
∫dI = ∫(V0/L)sin(ωt)dt
The solution to this integration is given below (see Example 1 in my blog Integration Definition and Rules for the detailed solution).
I = (V0/L)(-cos(ωt)/ω) + C
If we let V0 = 0, then I = C, but current is zero so I is zero, so C = 0.
- With C = 0 and rearranging the ω, we get
(8) I = (V0/(Lω))(-cos(ωt))
Let
(8b) I0 = (V0/(ωL))
so (8) becomes
(9) I =I0(-cost(ωt)); Current Equation Pure Inductive AC Circuit
Let’s express (9) in terms of the sine function so we can easily compare it to the voltage equation.
From our list of useful cosine and sine equivalences (Geometry and Trigonometry Rules) ,
-cos(θ) = sin(θ – π/2) , so substituting into (9) gives
(10) I = I0 sin(ωt– π/2) ; Current Equation Pure Inductive AC Circuit
So, let’s compare equations (6) and (10) and see if they explain the I and V graphs in the picture above.
(6) V = VS = VL = V0sin(ωt)
Equation (10) compared to (6) says that the current I oscillates 90 degrees (π/2 radians) behind the voltage.
That is, I is lagging behind Voltage (I lags V).
- If you take a look at the picture above, the way to find the leading curve is to start at time zero and see which value gets to zero or maximum first.
- In this case, V is ahead of I so V lead I or I lags V.
- Think of it this way: The inertia of the inductor is preventing I from instantly going up when V goes up.
Inductive Reactance XL
The term Lω in equation 8 is called the Inductive Reactance and is given the symbol XL.
Inductive Reactance XL is the opposition an inductor presents to the flow of Alternating Current (AC).
The value of XL is directly proportional to both the frequency of the AC signal and the inductance of the coil
(11) ωL = (2πf)L = XL = Inductive Reactance
- ω = angular velocity = 2πf
- f is the frequency (Hertz),
- L is the inductance (Henrys).
This relationship means an inductor offers zero opposition to DC (where f = 0) and higher opposition to high-frequency AC
- XL depends on the frequency of voltage source
- R does not depend on the frequency of voltage source
XL is measured in Ohms similar to resistance, but unlike resistance, it does not dissipate energy as heat;
- instead, it causes the current to lag the voltage by 90 degrees and
- stores energy in a magnetic field.
Since (see equation 8b)
(12a) I0 = (V0/(ωL)) ; Pure Inductor AC Circuit Version of Ohm’s Law
we can express this as
(12b) I0 = (V0/XL) ; Pure Inductor AC Circuit Version of Ohm’s Law
This equation is the AC (Alternating Current) version of Ohm’s Law for a circuit that contains only a pure inductor.
Power
Real Power (or True Power) is the power dissipated as heat or converted to useful work.
(13) Preal = VrmsIrmscos(θ)
- P is the Average Power (or Real Power) in Watts (W).
- Vrms is the Root Mean Square voltage in Volts (V).
- See my blog RMS (Root Mean Square) of a Sinusoid for more on RMS
- Irms is the Root Mean Square current in Amperes (A).
For a pure inductor, the phase angle θ (between V and I) is 90 degrees.
(13b) Preal = V I cos(90) = 0 Watts ; Pure Inductive AC Circuit Real Power
The zero real power means the inductor, like the capacitor, does not consume energy permanently.
Instead, it engages in an oscillating energy exchange with the AC source:
- Absorption: During one-quarter of the AC cycle, the source supplies energy to the inductor, which is stored in its magnetic field.
- Return: During the next quarter-cycle, the magnetic field collapses, and the inductor releases an equal amount of energy back to the AC source.
Because the energy is only stored and returned—and not converted to heat or work—the net amount of energy consumed over a full AC cycle is zero, resulting in zero real power
So, the average power consumed by a theoretically pure inductor over a full cycle is zero.
Pure Capacitor AC Circuit – Time Domain Analysis
I primarily used these excellent Khan Academy Videos to write this section:
A pure capacitor AC circuit is an idealized electrical circuit where an alternating current (AC) voltage source is connected to a circuit containing only a capacitor and no other opposition to current flow, such as resistance or inductance.
A capacitor opposes changes in voltage.
In an AC circuit, the voltage is constantly changing, so a current flows as the capacitor continually charges and discharges.
This opposition to AC current flow is called capacitive reactance XC.
This causes the I and V sinusoids to become out of sync.
In a purely capacitive circuit, the capacitor does not consume any real power; it simply stores and releases electrical energy.
Picture: Pure (Simple) Capacitor AC Circuit
Note: Since I and V have different units of measure don’t try to compare the magnitude of the curves (apples and oranges).
- We put them together on the same chart to show that over time they do not move together in sync.
Voltage and Current Equations
In a purely inductive circuit the source voltage VS and capacitor voltage VC are equal
(14) VC = VS
(15) VS = V0sin(ωt) ; Voltage Equation for Pure Capacitive AC Circuit
Voltage in a capacitor is defined as follows (see my blog Capacitance for more)
(16) VC = Q/C ; Capacitor Voltage
Where
- Q = represents the electric charge stored on one plate of the capacitor (measured in Coulombs, C).
- C = Capacitance and has units of Farad.
Substituting (15) and (16) into (14) and solving for Q gives
(17) Q = CV0sin(ωt)
For an AC system (i.e. a continuously oscillating system) the definition of current will be
(18) I = dQ/dt ; AC Current
Substitute (17) into (18)
(19) I = d/dt(CV0sin(ωt))
- See Example 4 (chain rule example) in my blog The Derivative to see how to differentiate this.
- The differential of the right hand side of equation (19) with respect to t is
(20) I = ωCV0cos(ωt) ; Current Equation for a Pure Capacitive AC Circuit
Sine the max value for cosine is 1, that means that I max or I peak, I0 , is limited to ωCV0
(21) I0 = ωCV0
So (20) can be expressed as
(22) I = I0cos(ωt) ; Current Equation for a Pure Capacitive AC Circuit
We want to compare equation (22) with the voltage equation (15) so lets express the cosine function in (22) in terms of sine.
From my blog Geometry and Trigonometry Rules we know that cos(ωt) = sin(ωt + π/2), so (22) can be expressed as
(23) I = I0sin(ωt + π/2) ; Current Equation for a Pure Capacitive AC Circuit
(15) VS = VC = V0sin(ωt)
We see that for a capacitor circuit (see the picture graph above), the current I is oscillating ahead of the voltage.
This is the defining characteristic of a purely capacitive component and is opposite to the behavior of an inductor (where voltage leads current).
- I leads V
- V lags I
- From time = zero, I reaches zero and max before V does
- Think of it this way:
- A change in the voltage of the capacitor means the charge must change.
- To change the charge, a current must first occur, so its going to lead voltage.
Capacitive Reactance XC
Capacitive Reactance (XC) is the opposition a capacitor presents to the flow of Alternating Current (AC).
Unlike a resistor, which dissipates energy as heat, a capacitor’s opposition comes from its ability to store and release energy in an electric field.
This process is continuous in an AC circuit as the capacitor constantly charges and discharges, effectively limiting the AC current.
It is measured in Ohms, just like resistance, but is referred to as “reactance” to indicate it’s an opposition that is dependent on the signal frequency.
Capacitive reactance is calculated using the formula:
(24) XC = 1/(ωC) = 1/(2πfC) ; Capacitive Reactance
where
- XC = Capacitive Reactance in Ohms (Ω)
- ω = angular velocity in radians/s
- f = frequency of AC signal in Hertz (Hz)
- C = Capacitance in Farads (F)
The formula shows that XC is inversely proportional to frequency:
- f ↑ , XC ↓
- f ↓ , XC ↑
At high frequencies, the capacitor has little time to fully charge before the voltage polarity reverses, so it acts almost like a short circuit (low opposition).
At low frequencies the capacitor fully charges and acts like an open circuit (infinite opposition), effectively blocking DC and low-frequency signals.
Let’s come back to our peak current definition (see equation 21).
(21) I0 = ωCV0 ; Pure Capacitor AC Circuit Version of Ohm’s Law
This equation is the AC (Alternating Current) version of Ohm’s Law for a circuit that contains only a pure capacitor.
Where
- I0 = the peak current (the maximum current value in the cycle (Amperes)
- V0 = the peak voltage (the maximum voltage value in the cycle), measured in Volts
- ω is the angular velocity of the AC source, measured in radians per second
- C is the capacitance of the capacitor, measured in Farads (F).
we can express this using the Capacitive Reactance definition.
(25) I0 = (V0/XC) ; Pure Capacitor AC Circuit Version of Ohm’s Law
Power
Capacitive reactance does not consume real power.
In a purely capacitive circuit, the current I leads the voltage V by a phase angle of 90 degrees (π/2 radians).
Real Power (or True Power) is the power dissipated as heat or converted to useful work.
(26a) Preal = Vrms Irms cos(θ)
In a pure capacitor circuit, θ = – 90 degrees so
(26b) Preal = 0 ; Pure Capacitive AC Circuit
- P is the Average Power (or Real Power) in Watts (W).
- Vrms is the Root Mean Square voltage in Volts (V).
- See my blog RMS (Root Mean Square) of a Sinusoid for more on RMS
- Irms is the Root Mean Square current in Amperes (A).
The capacitor
- absorbs energy from the source during one-quarter of the AC cycle (when it’s charging) and
- immediately returns an equal amount of energy to the source during the next quarter-cycle (when it’s discharging).
Because the energy is only stored and returned, and not consumed or dissipated as heat (like in a resistor), the net power consumed over a full cycle is zero.
Pure Inductor and Capacitor Impedance – Frequency Domain Analysis
In this section we are going to use Euler’s formula to transform the sinusoidal time-domain equations for current and voltage into the frequency – domain (phasor) representations.
- i.e. we want to express the “effective resistance or reactance ” of inductors and capacitors in terms of ω and L (Inductance) or ω and C (Capacitance).
Recall Euler’s equation and the expressions for cosine and sine that are based on it (see my post Circuit Analysis Math Basics: Trig, Complex Numbers, and Euler’s Equation for details about them)
(27) ejθ = cosθ + jsinθ; Euler’s Formula
(28) e-jθ = cosθ – jsinθ; Euler’s Formula
(29) cosθ = 1/2 (ejθ + e-jθ) ; cosine definition in terms of exponentials
(30) sinθ = (1/2)(1/j)(ejθ – e-jθ) ; sine definition in terms of exponentials
(31) V = V0cos(ωt + Φ)
(32) I = I0cos(ωt + Φ)
Let θ = ωt + Φ
and substitute into equation (27)
to get
(33) ej(ωt + Φ) = cos(ωt + Φ) + jsin(ωt + Φ)
since xaxb = xa+b,
(34) ejωtejΦ = cos(ωt + Φ) + jsin(ωt + Φ)
Let the Real Part of (ejωtejΦ) = R{ejωtejΦ} = cos(ωt + Φ)
so, (31) becomes
(35) V = V0cos(ωt + Φ) = V0R{ejωtejΦ} = R{V0ejΦejωt} = R{v̂ejωt}
where
(36) v̂ = V0ejΦ = Voltage Phasor
and (32) becomes
(37) I = I0cos(ωt + Φ) = I0R{ejωtejΦ} = R{I0ejΦejωt} = R{îejωt}
where
(38) î = I0ejΦ = Current Phasor
Ideal Inductor Complex Impedance
For an inductor,
(39) V = R{v̂ejωt} = L dI/dt = L d/dt( R{îejωt} ) = L (R{jwîejωt})
(40) R{v̂ejωt} = L (R{jωîejωt}) = R{Ljωîejωt}
Cancelling similar terms from the left hand side and right hand sides, equation (40) simplifies to:
v̂ = Ljωî
rearranging terms, we get,
(41) v̂/î = jωL = ZL ; Complex Impedance of an ideal inductor.
This expression is the complex impedance ZL of an ideal inductor.
It means the opposition to alternating current is purely inductive, with a magnitude XL directly proportional to the signal’s frequency and the inductance.
The presence of the j term ensures that the voltage across the inductor leads the current through it by a 900 phase angle.
We’ve seen the expression ωL before.
(42) = (11) = XL = ωL = Inductive Reactance
Inductive Reactance XL is the magnitude of the inductor’s opposition to current flow, measured in Ohms.
It represents the non-complex, frequency-dependent part of the impedance.
It demonstrates that the inductor acts as a filter:
- its opposition increases (acts more like an open circuit) as the AC frequency increases, and
approaches zero for direct current DC, where ω = 0.
Ideal Capacitor Complex Impedance
For a Capacitor,
(43) I = R{îejωt} = CdV/dt = C d/dt (R{v̂ejωt}) = C (R{jωv̂ejωt})
(44) R{îejωt} = C (R{jωv̂ejωt}) = R{Cjωv̂ejωt}
Cancelling similar terms from the left hand side and right hand sides, equation (44) simplifies to:
î = Cjωv̂
or
(45) v̂/î = 1/(jωC) = ZC ; Complex Impedance of an ideal capacitor
This is the total, complex opposition an ideal capacitor offers to alternating current (AC).
This formula shows that the impedance ZC is inversely proportional to both the signal’s frequency f (where ω = 2πf) and the capacitance (C).
The j in the denominator (equivalent to a -j in the numerator) defines this opposition as purely capacitive and mathematically enforces that the current leads the voltage by a 900 phase angle.
We’ve seen the expression 1/ωC before.
(46) = (24) = XC = 1/ωC = Capacitive Reactance
Capacitive Reactance XC is the magnitude of the capacitor’s opposition to current flow, measured in Ohms.
It represents the non-complex, frequency-dependent part of the impedance, demonstrating that the capacitor acts as a filter:
- its opposition decreases (acts more like a short circuit) as the AC frequency increases, and
increases dramatically at low frequencies.
Now You Know Why Electrical Engineers Prefer the Cosine Version of Voltage and Current
The cosine versions of the current and voltage equations are used primarily because of its direct relationship with Euler’s formula.
As we showed in this section, when representing a sinusoidal voltage V(t) using the complex exponential form, the convention is to define the time-domain signal as the Real Part of the complex exponential.
The Real Part of Euler’s formula is the cosine term.
By choosing the cosine function for the voltage equation, we make it the standard reference that corresponds precisely to the real number results obtained when using complex number methods (like impedance) to analyze AC circuits.
Summary for Pure Element AC Circuits
Here is a high-level summary of pure resistive, inductive, and capacitive AC circuits.
ELI the ICE man
Everyone’s heard of this mnemonic which describes the phase relationship (relative positions of the I and V sinusoid waveforms in time).
Remember that E (EMF) is the same as Voltage (V) here.
- ELI: E (V) ahead of I (leads I) in an inductive L circuit
- ICE: I ahead of E (V) in a capacitive C circuit.
Purely Resistive Circuit R
- Components: Only a Resistor (R) is connected to the AC source.
- Opposition: The opposition to current flow is simply the Resistance (R).
- Phase Relationship: Voltage and Current are in phase.
- They peak and cross zero at the same time.
- Mnemonic: think ohm’s law….V=IR where Voltage and In-phase Resistance.
- Power: Consumes Real Power (P > 0) as energy is dissipated as heat
Purely Inductive Circuit
- Components: Only an Inductor (L) is connected to the AC source.
- Opposition: The opposition is Inductive Reactance: XL
- Phase Relationship: Voltage leads the Current (I) by 90 degrees.
- Mnemonic Check: ELI (Voltage E or V) Leads I (Current)).
- Reactance Relationship: ωL = (2πf)L = XL = Inductive Reactance
- XL is measured in Ohms (Ω).
- XL increases with frequency (f).
- At DC (f=0), XL =0 (short circuit).
- Power: Consumes zero average (real) power
- I0 = (V0/(ωL)) ; Pure Inductor AC Circuit Version of Ohm’s Law
- I0 = (V0/XL) ; Pure Inductor AC Circuit Version of Ohm’s Law
- v̂/î = jωL = jXL = ZL ; Complex Impedance of an ideal inductor.
Where
- I0 = the peak current (the maximum current value in the cycle (Amperes)
- V0 = the peak voltage (the maximum voltage value in the cycle), measured in Volts
- ω is the angular velocity of the AC source, measured in radians per second
- f = frequency in Hz
- L is the Inductance of the inductor, measured in Henrys (H).
- XL is Inductive Reactance measured in Ohms (Ω).
- j = imaginary number = sqrt (-1)
- v̂ = V0ejΦ = Voltage Phasor
- Voltage Phasor: The complex number that represents the amplitude and phase of the time-varying sinusoidal voltage.
- Φ = Phase Angle; The initial phase angle (in radians or degrees) of the sinusoidal wave at t=0.
- î = I0ejΦ = Current Phasor
- Current Phasor: The complex number that represents the amplitude and phase of the time-varying sinusoidal current.
Purely Capacitive Circuit
- Components: Only a Capacitor (C) is connected to the AC source.
- Opposition: The opposition is Capacitive Reactance XC .
- Phase Relationship: Current (I) leads the Voltage (V) by 90 degrees. (or V lags I)
- Mnemonic: ICE: I leads E or E (V) lags I.
- Reactance Relationship: XC = 1/(ωC) = 1/(2πfC) ; Capacitive Reactance
- XC is measured in Ohms (Ω).
- XC decreases with frequency (f)
- At DC (f=0), XC =infinity (open circuit).
- Power: Supplies zero average (real) power.
- I0 = ωCV0 ; Pure Capacitor AC Circuit Version of Ohm’s Law
- I0 = (V0/XC) ; Pure Capacitor AC Circuit Version of Ohm’s Law
- v̂/î = 1/(jωC) = XC/j = -jXC = ZC; Complex Impedance of an ideal capacitor
- note that 1/j = -j
Where
- I0 = the peak current (the maximum current value in the cycle (Amperes)
- V0 = the peak voltage (the maximum voltage value in the cycle), measured in Volts
- ω is the angular velocity of the AC source, measured in radians per second
- f = frequency in Hz
- C is the capacitance of the capacitor, measured in Farads (F).
- XC is Capacitive Reactance measured in Ohms (Ω).
- j = imaginary number = sqrt (-1)
- v̂ = V0ejΦ = Voltage Phasor
- Voltage Phasor: The complex number that represents the amplitude and phase of the time-varying sinusoidal voltage.
- Φ = Phase Angle; The initial phase angle (in radians or degrees) of the sinusoidal wave at t=0.
- î = I0ejΦ = Current Phasor
- Current Phasor: The complex number that represents the amplitude and phase of the time-varying sinusoidal current.
Phasor Basics
See this great introductory video
AC Circuits get interesting (and practical) when we start combining resistors, inductors, and capacitors.
We call these RLC AC circuits and I want to describe some basic relationships between current and voltage in these kinds of systems.
In order to do this we will need to use a tool called the Phasor and apply some basic vector math.
Phasor Characteristics
Consider a vector (something that has direction and magnitude….see my blog on vectors) which has the following characteristics:
- It’s rotating in a counterclockwise direction tracing a circular path.
- The tail of the vector is fixed at the center of the circle.
- The tip of the vector starts on the positive x axis
- in what we typically call quadrant I of the circle.
- The vector’s magnitude is equal to the amplitude (peak value) of a sinusoid wave equation like the ones representing voltage and current
- If V = V0sin(ωt), then vector magnitude = V0
- I = I0sin(ωt), then vector magnitude = I0
Over time, as the vector rotates in the counterclockwise direction it will project a pattern of instantaneous points in time which we can graph.
Picture: A Phasor Projects A Sinusoid Over Time
As the vector rotates , we can plot its y (vertical component) value on a graph of value (voltage or current) vs time.
- The projection of the value from the vector to the graph is shown by the red dotted lines.
- Cool. We get a sine or cosine shape over time which is exactly how we characterize current and voltage in AC circuits.
Assume the vector represents the voltage.
Freeze the vector at a specific position in time as shown in the picture above.
- We can characterize it as a right angle triangle with
- ωt: The angle between the vector and the 0 line is equal to the angular velocity x time (rad/sec x sec = radians).
- y component of vector: V0sin(ωt)
- We just applied the Pythagorean theorem to get this. (sin x = opposite/hypotenuse)
- See my trig blog if you need to remember what that is.
This rotating vector analogy is called a phasor.
Vector Math
We’ll have to add phasors in our study of RLC circuits but this is just vector addition and its pretty straight forward.
To add two vectors (let’s say A + B), you simply
- move “A” such that the tail of “A” connects to the tip of “B” and
- trace a new vector from the tail of “B” to the tip of the moved vector “A”.
- The new vector is “A + B”
Go to my blog Vector Math to get clear on this if you’re still confused.
Voltage and Current Relationships in RLC Circuits
While phasors are actually powerful tools rooted in complex number mathematics (which allows engineers to solve intricate AC circuits using simple algebra instead of calculus), we can simplify the concept for visual analysis.
We don’t need complex math to show how current and voltage either lag or lead each other in RLC circuits.
We can simply use the phasor vectors to represent the peak values and phase angles of currents and voltages and then use simple vector addition to derive the fundamental phase relationships for series and parallel RLC circuit basics.
These simple vector diagrams allow us to instantly see the overall phase relationship between total voltage and total current.
Series R.L.C. AC Circuits
Here are some nice references re RLC Circuits.
- Impedance & phase angle of series L.C.R circuit – Khanacademy
- RLC Circuit: Series and Parallel, Applied circuits
We’ve reviewed pure, singular, resistive, inductive, and capacitive AC circuits and established phase relationships and learned about resistance (R) and reactance (XC and XL ).
Let’s combine these elements in AC circuits and learn a few more things.
A series RLC circuit is one of the most fundamental and crucial circuits in electrical engineering.
It gets its name from its three main components, all connected end-to-end (in series) to an AC voltage source:
- R for Resistor
- L for Inductor
- C for Capacitor
Picture: Series R.L.C. AC Circuits
Series RLC Circuit Characteristics
Common Current
Since the components are in series, the same alternating current (I) flows through every single component at any given instant.
Voltages
By Kirchhoff’s Voltage Law (KVL), the total source voltage VS is the vector sum of the voltages across the resistor VR, the inductor VL, and the capacitor VC.
VS = VR + VL + VC is the vector sum of the voltages across the resistor (VR), the inductor (VL), and the capacitor (VC).
Because of the different phase relationships, this sum is not a simple arithmetic addition (i.e. have to use vector addition).
As we noted, the common current is usually the reference point for phase analysis.
You’ll notice the term Impedance in the picture above.
The impedance is a sort of composite resistance of the combined elements in the circuit.
We’ll address impedance later on, once we do a little vector math.
Series R.L.C. Peak Voltage Calculation
Ok, remember those phasors we mentioned earlier on?
Well, we are going to use them to represent the position and magnitude of current and voltage in a series RLC circuit.
We noted that the current going through the components in a series R.L.C configuration will be the same.
Let’s create a reference phasor (vector) that represents this current I0 and we’ll place it flat on the x axis of a cartesian xy plane as shown in the picture below (see 1. in the picture).
Picture: Series AC R.L.C. Peak Voltage Calculation
We can also define component voltage phasors and define them in terms of Ohm’s Law (in terms of resistance R or reactance X).
Resistor Voltage: VR = I0R
- This is the standard form of Ohm’s Law.
- The peak voltage VR across the resistor is directly proportional to the peak current I0 flowing through it and the resistance R.
- VR is in phase with the current I0
Inductor Voltage: VL = I0XL
- This is the Ohm’s Law equivalent for an inductor.
- The peak voltage VL across the inductor is proportional to the peak current I0 and the Inductive Reactance XL.
- VL leads the current I0 by 90 degrees (π/2 radians): Remember the mnemonic ELI the ICE man where E leads I in an inductor (L)
Capacitor Voltage: VC = I0XC
This is the Ohm’s Law equivalent for a capacitor.
- The peak voltage VC across the capacitor is proportional to the peak current I0 and the Capacitive Reactance XC.
- VC lags the current I0 by 90 degrees (π/2 radians): Remember the mnemonic ELI the ICE man where I leads E in a capacitor (C)
Now, lets place the phasors on the same xy plane and in relation to the current phasor I0.
You can see the voltage phasor (vector) placements as shown in 1. in the picture above where
- VR is in line with I0
- VL is +90 degrees ahead of I0
- Assume XL >XC (so VL >VC) although we could have chosen the opposite.
- VC is 90 degrees behind I0 (i.e. -90 degrees)
Add the Voltage Vectors Up
In order to compute the total voltage we have to add these vectors together.
Lets do VC + VL First.
- I show this in 2. in the picture above.
- Adding these two results in the vector I0(XL –XC)
- Move VC by the tail to the tip of VL
- then draw the resultant vector from the tail of VL to the tip of the moved VC.
- The resultant vector is also vertical but has a tip designated by the grey arrow in the picture.
- We’ re doing basic vector addition here (see my blog Vector Math).
- You can look at (XL –XC) as a net reactance term. You might sometimes see this expressed as Xnet .
Now let’s add the resultant vector I0(XL –XC) to the the remaining vector VR
- See 3. in the picture above.
- We do simple vector addition again (see my blog on vectors to understand how to add vectors)
- and get the resultant vector V0
Compute the Magnitude of V0 Using the Pythagorean Theorem
(V0)2 = (I0R)2 +(I0(XL –XC))2 ; Series AC R.L.C Circuit
V0 = I0 sqrt { (R)2 +(XL –XC)2 } ; Series AC R.L.C Circuit
|Z| = Magnitude of the Impedance = sqrt { (R)2 +(XL –XC)2 } so
V0 = I0|Z| ; Ohm’s Law Equivalent For Series AC RLC Circuit
- It states that the total peak voltage V0 is equal to the peak current I0 multiplied by the total magnitude of the Impedance |Z|.
- This equation ties together the voltage, current, and total opposition (impedance Z) for the entire AC circuit, just as V=IR does for a simple DC circuit.
Opposition
- Impedance (Z) measures how much a circuit impedes (opposes) the flow of electric current.
- Like resistance, impedance is measured in Ohms
- Components: Impedance combines two different types of opposition:
- Resistance (R): The opposition due to resistors, which converts energy into heat.
- Reactance (X): The opposition due to inductors XL and capacitors XC, which stores and releases energy (magnetic or electric).
- Direction (Phase): Unlike simple resistance, impedance is a vector (expressed algebraically as a complex number) because it includes a phase angle.
- This angle tells you whether the total voltage leads or lags the total current in the circuit.
V0 and I0 Phase Angle Φ for Series AC RLC Circuit
Lets solve for the phase angle Φ from 3. in the picture above.
- tan Φ = (XL – XC)/R
- Φ = arctan { (XL – XC)/R }
In the phasor setup we assumed that XL was larger than XC.
This means V will lead I in this series RLC circuit (+ R).
Using this equation, when XC > XL, Φ will be negative; indicating V lags I.
Voltage Triangle Scaled to the Impedance Triangle
I want to circle back to impedance again, to provide further clarity.
By applying the principles of vector addition to the voltages in a series RLC circuit, we established the relationship between the applied V0 and the individual component voltages VR ,VL ,VC.
This vector relationship forms a right triangle, which we can call the Voltage Triangle (as shown in 3. in the series RLC phasor picture above).
Since all these voltages share the same current, we can divide every side of this Voltage Triangle by I0 (using Ohm’s Law for AC circuits: V = IZ).
The result is a new, proportional right triangle (see the picture below) whose sides represent the circuit’s resistive and reactive properties:
- The side representing VR becomes the Resistance (R).
- The side representing the net reactive voltage (VL – VC) becomes the Net Reactance (XL – XC = Xnet).
- The hypotenuse, which was V0, becomes the Impedance (Z): V/I = Z
This is the Impedance Triangle .
It demonstrates the Pythagorean relationship between these circuit elements: Z2 = R2+ Xnet2
The angle (Φ) between the Resistance (R) and the Impedance (Z) in this triangle is the phase angle of the circuit.
Picture: Series AC RLC Voltage Triangle to Impedance Triangle
Series AC RLC Impedance Z and the imaginary number j
The Cartesian vector treatment we used—where the resistance vector (R) lies purely on the horizontal axis and the reactance vector (X) lies purely on the vertical axis—is a seamless precursor to the Complex Plane representation.
In the complex plane, the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.
This is where the mathematical operator j comes in:
- Resistance (R) is a real number and is placed on the real axis.
- Reactance (X) is always associated with a 900 phase shift (either leading for inductive, or lagging for capacitive).
- To represent this rotation mathematically, we multiply the reactance by j.
Thus, the imaginary number jXnet is placed on the imaginary axis.
This allows us to write the Impedance (Z) as a single complex number that incorporates both its magnitude (the hypotenuse) and its phase angle:
Z= R+ jXnet
The utility of j is that it mathematically incorporates the 900 phase angle that inherently exists between resistance and reactance, allowing us to perform all RLC circuit calculations using the standard algebra of complex numbers.
This transition confirms that the geometric vector addition we performed is mathematically equivalent to complex number addition.
- See my blogs on complex numbers: Complex Numbers; Complex Number Math; Circuit Analysis Math Basics
- Check out the picture below as a reminder of how to place a complex number in the complex plane
- Notice the equation for the magnitude of the hypotenuse (in the picture below).
- It will equate to the same |Z|= sqrt(R2+ Xnet2) we derived from the cartesian, phasor, vector analysis.
- Note that the magnitude of the complex number is the sqrt (the squares of the real and imaginary parts…not including j).
- The imaginary number operator j (j = sqrt(-1)) serves as the mathematical representation of that 900 phase separation.
Picture: Complex Numbers on the Complex Plane
Parallel R.L.C. AC Circuits
Here is a nice reference on RLC Circuits:
A parallel RLC circuit is another fundamental configuration in AC analysis.
It features the same three components—a resistor, an inductor, and a capacitor—but they are connected across the same two points, meaning they are arranged in parallel to the AC voltage source.
Picture: Parallel R.L.C. AC Circuits
Parallel RLC AC Circuit Characteristics
Common Voltage
Since the components are in parallel, the same alternating voltage V is applied across every single component at any given instant.
This common voltage is typically the reference point for phase analysis.
Currents
By Kirchhoff’s Current Law (KCL), the total source current IS is the vector sum of the individual currents flowing through the resistor IR, the inductor IL, and the capacitor IC .
IS = IR + IL + IC
Because of the different phase relationships, this sum is not a simple arithmetic addition (must use vector addition).
As we noted, the common current is usually the reference point for phase analysis.
You’ll notice the term Admittance (Y) in the picture above.
While Impedance is opposition to current flow, Admittance is a measure of the ease of current flow.
We’ll address Admittance later on, once we do a little vector math.
Parallel AC R.L.C. Peak Current Calculation
The voltage across the components in a parallel R.L.C configuration will be the same.
Let’s create a reference phasor (vector) that represents this voltage and we’ll place it flat on the x axis of a cartesian xy plane as shown in the picture below (see section 1).
Picture: Parallel AC R.L.C. Peak Current Calculation
We can also define component current phasors and define them in terms of Ohm’s Law (in terms of resistance or reactance):
Resistor Current: IR = V0/R
- This is the standard Ohm’s Law for resistance.
- The peak current is determined solely by the common peak voltage V0 and the Resistance R.
- The current IR is in phase (0 degrees) with V0
Inductor Current : IL = V0/XL
- This is the Ohm’s Law equivalent for the inductor.
- The peak current is limited by the Inductive Reactance XL .
- The current IL lags the common voltage V0 by 90 degrees.
Capacitor Current : IC = V0/XC
- This is the Ohm’s Law equivalent for the capacitor.
- The peak current is limited by the Capacitive Reactance XC.
- The current IC leads the common voltage V by 90 degrees
Now, lets place the phasors on the same xy plane and in relation to the voltage phasor V0.
You can see the voltage phase (vector) placements as shown in section 1 of the picture above where
- IR is in line with V0
- IL is -90 degrees behind V0
- Like for the series RLC, Assume XL > XC so, IL < IC
- IC is 90 degrees ahead of V0 (i,e +90 degrees)
Add the Current Vectors (Phasors) Up
In order to compute the total voltage we have to add these vectors together.
Lets do IC + IL First.
- I show this in section 2 of the picture above.
- Adding these two results in the vector V0(1/XC -1/XL)
- (1/XC -1/XL) is called the net susceptance.
Now lets add the resultant vector V0(1/XC -1/XL) to the the remaining vector V0/R
- See section 3 in the picture above.
- We do simple vector addition again (see my blog on vectors to understand how to add vectors)
- and get the resultant vector I0
Compute the Magnitude of I0 Using the Pythagorean Theorem
(I0)2 = (V0/R)2 +(V0(1/XC -1/XL))2 ; Parallel AC RLC
I0 = V0 sqrt { (1/R)2 + (1/XC -1/XL)2 } ; Parallel AC RLC
Let |Y| = Magnitude of the Admittance = sqrt { (1/R)2 + (1/XC -1/XL)2 } so
I0 = V0|Y| ; Ohm’s Law Equivalent For Parallel AC RLC Circuit
Common Names of Some Terms
- G = 1/R = Conductance
- BC= 1/XC = Capacitive Susceptance
- BL= 1/XL = Inductive Susceptance
- XC = Capacitive Reactance
- XL = Inductive Reactance
Admittance
As noted above, the square root term
sqrt { (1/R)2 + (1/XC -1/XL)2 }
represents the magnitude of the total Admittance |Y| for the circuit.
This term combines the Conductance (1/R) and the Net Susceptance (1/XC -1/XL) through vector addition, defining the circuit’s total ease of current flow.
V0 and I0 Phase Angle Φ for Series AC RLC Circuit
Lets solve for the phase angle Φ from section 3 in the picture above.
- tan Φ = R (1/XC – 1/XL )
- Φ = arctan { R(1/XC –1/XL) }
In the phasor setup we assumed that XL was larger than XC.
This means I will lead V in this parallel AC RLC circuit (+ Φ).
Using this equation, when XC > XL, Φ will be negative; indicating I lags V.
Current Triangle Scaled to the Admittance Triangle
Just as we did before for a series circuit, I want to circle back to the admittance again, to provide further clarity.
By applying the principles of vector addition to the currents in a parallel RLC circuit, we established the relationship between the applied I0 and the individual component currents IR ,IL ,IC.
This vector relationship forms a right triangle, which we can call the Current Triangle (as shown in 3. in the parallel RLC phasor picture above).
Since all these currents share the same voltage, we can divide every side of this Voltage Triangle by V0 (using Ohm’s Law for AC circuits: V = IZ).
The result is a new, proportional right triangle (see the picture below) whose sides represent the circuit’s resistive and reactive properties:
- The side representing IR becomes the 1/R = Conductance
- The side representing the net reactive current (IC – IL) becomes the Net Susceptance (1/XC – 1/XL = Bnet).
- The hypotenuse, which was I0, becomes the Admittance (Y): I/V = Y = 1/Z
This is the Admittance Triangle .
It demonstrates the Pythagorean relationship between these circuit elements: Y2 = (1/R)2+ Bnet2
The angle (Φ) between the Resistance (R) and the Admittance (Y) in this triangle is the phase angle of the circuit.
Parallel AC RLC Admittance Y and the imaginary number j
The Cartesian vector treatment we used—where the conductance vector (G) lies purely on the horizontal axis and the susceptance vector (B) lies purely on the vertical axis—is a seamless precursor to the Complex Plane representation.
In the complex plane, the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.
This is where the mathematical operator j comes in:
- Conductance (G) is a real number and is placed on the real axis.
- Susceptance (B) is always associated with a 900 phase shift (either leading for inductive, or lagging for capacitive).
- To represent this rotation mathematically, we multiply the reactance by j.
Thus, the imaginary number jBnet is placed on the imaginary axis.
This allows us to write the Admittance (Y) as a single complex number that incorporates both its magnitude (the hypotenuse) and its phase angle:
Y= G+ jBnet
The utility of j is that it mathematically incorporates the 900 phase angle that inherently exists between conductance and susceptance, allowing us to perform all RLC circuit calculations using the standard algebra of complex numbers.
This transition confirms that the geometric vector addition we performed is mathematically equivalent to complex number addition.
- See my blogs on complex numbers: Complex Numbers; Complex Number Math; Circuit Analysis Math Basics
- Check out the picture below as a reminder of how to place a complex number in the complex plane
- Notice the equation for the magnitude of the hypotenuse (in the picture below).
- It will equate to the same |Y|= sqrt(G2+ Bnet2) we derived from the cartesian, phasor, vector analysis.
- Note that the magnitude of the complex number is the sqrt (the squares of the real and imaginary parts…not including j).
- The imaginary number operator j (j = sqrt(-1)) serves as the mathematical representation of that 900 phase separation.
Picture: Complex Numbers on the Complex Plane
Appendix 1 – Real Power in an AC Circuit
The power equation P = V Icos(Φ) defines Real Power (P) in an AC circuit, which is the actual power utilized to do useful work, measured in Watts (W).
The term cos(phi), known as the Power Factor (PF), accounts for the phase difference (Φ) between the voltage (V) and current (I).
For pure element AC circuits, this equation simplifies significantly:
- P is maximum (P=VIcos; (Φ)=1) for a purely resistive circuit (Φ=0), and
- zero (P=0 ; cos(Φ)=0) for purely inductive or capacitive circuits (Φ = +-90 degrees),
as these elements only store and return energy.
For series RLC and parallel RLC circuits, the real power P is determined only by the resistance (R) present, as inductors and capacitors cancel out or average to zero power consumption.
Therefore, for all RLC circuits,
- the power factor cos(Φ) is determined by the ratio of resistance to the total impedance, and the real power is P = I2 R, which is mathematically equivalent to
- P = V I cos(Φ)
- where cos(Φ) = R/Z.
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