sign in sign up
<<
Home , Root mean square

Complex Representation of AC Circuits

Kevin Gaughan 15 September 2003

1. Voltage in the Time Domain.

Volts Phase Reference: Sin( ω.t)

Peak = √2.Vrms

φ ωt

v(t)= √2.V rms .Sin( ω.t+ φ)

Time domain equation for a sinusoidal voltage: v(t) = √2.V rms .Sin( ω.t+ φ)

Where: V(t) = voltage at any instant t t = time in seconds Vrms = root voltage ω = in rad/sec ω = 2. π.f where f = frequency in Hz. φ = Angle in radians by which v(t) leads the reference

The graph above uses ω.t for the x-axis rather than t. This is purely for convenience. The units of ωt are in radians rather than seconds and using this as the X axis allows angles such as φ to be shown directly on the graph.

2. Complex Representation of Voltage and Current.

Any sinusoidal voltage or current at constant frequency can be represented by a complex number as follows:

V = Vrms.Cos( φ) + j. Vrms.Sin( φ) (Note that j = √-1)

Where φ is the phase shift of the waveform with respect to a reference waveform. It is normal to pick the main supply voltage as the reference waveform. In a three phase system Phase A is usually chosen as the reference waveform.

Note one way of viewing the expression above is that the voltage V is being represented as the sum of two components. The real part of the above equation is the component that is in phase with the reference voltage and the imaginary part is the component that is 90° out of phase with the reference.

The reason for using this method of representing AC voltages (and currents) is that if voltages and currents are represented in this way and impedances involving , capacitors and inductors are also represented by complex numbers then AC circuits involving Rs, Ls and Cs can be solved using DC circuit theory. All of the methods of DC circuit analysis (Thevenin, Superposition, Kirchoff etc.) can be applied in a straightforward manner to an AC circuit using complex numbers for the voltages and currents. The solutions for V and I will in turn be complex numbers which can be converted back to the time domain if required. If this method were not used then AC circuits involving inductors and capacitors would need to be solved using differential equations.

3. Diagrams

The complex number can also be shown graphically in a phasor diagram:

J Axis V = Voltage Phasor of Length Vrms at angle φ

Vrms.Sin( φ) φ Real Axis Vrms.Cos( φ)

Note by trigonometry that the component of V along the Real axis is V rms .Cos( φ) and the component along the j axis is V rms .Sin( φ) . Note also that Pythagoras’s theorem applies to the right-angled triangle formed by V and its two orthogonal components

Vrms

Vrms.Sin( φ)

φ

Vrms.Cos( φ) Pythagoras:

2 2 2 Vrms = (V rms Cos( φ)) + (V rms .Sin( φ))

This provides us with a convenient method of calculating the rms value of a voltage that is represented in complex form

2 2 If V = a+j.b => V rms = √(a +b ) This is the real world rms value of the voltage.

Note also we can work out the phase shift of a complex number ( φ) using trigonometry:

V = a+j.b => ∠V = φ = Tan -1(b/a) This is the real world phase shift of the voltage.

Please note that by convention Phasor diagrams are drawn with positive angles measured in an anticlockwise direction.

Voltage with positive phase shift ( φ is positive)

φ Phase Reference

Voltage with negative Phase shift 4. Polar Form

The format a+j.b is often referred to as the rectangular form of a complex number. The complex number can also be represented in polar form as a magnitude and phase

V = Vrms ∠φ Meaning V is a voltage of rms V at phase angle φ wrt the reference.

Time Domain Complex Notation Polar form (Rectangular form) V(t) = √2.Vrms.Sin( ω.t+ φ) V = a+j.b V = Vrms ∠φ 2 2 a = Vrms.Cos( φ) Vrms = √(a +b ) b = Vrms.Sin( φ) φ = Tan -1(b/a)

If you remember back to complex number theory you will recall that rectangular notation is more convenient for adding and subtracting complex numbers but polar notation is more convenient for multiplication and addition.

Rectangular form: (a+j.b) + (c+j.d) = (a+c) + j.(b+d) (a+j.b) – (c+j.d) = (a-c) + j. (b-d) Polar Form:

(M ∠φ ). (N ∠θ) = M.N ∠(φ+θ) (M ∠φ )/ (N ∠θ) = M/N ∠(φ-θ)

When solving problems in electric systems it is common to switch between rectangular and polar form at will depending on the nature of the calculation being formed.

5. Complex Representation of Current

Current can also be represented using complex numbers, phasor diagrams and polar form. Again a reference phase needs to be chosen and this is always the same reference phase as is used for voltage.

Time Domain Complex Notation Polar form (Rectangular form) I(t) = √2.Irms.Sin( ω.t+ φ) I = a+j.b I = Irms ∠φ 2 2 a = Irms.Cos( φ) Irms = √(a +b ) b = Irms.Sin( φ) φ = Tan -1(b/a)

A useful point to remember is that when one phasor has an angle φ1 and a second has an angle φ2 then the angle between them is φ2 - φ1 measured from 1 to 2.

Therefore the phase shift between an arbitrary current I ( with angle φi) and an arbitrary voltage V (with angle φv) is φi - φv.

6. Adding and Subtracting Voltage and Current in complex representation

Voltages can be added and subtracted in the complex domain simply by adding and subtracting the complex numbers. Likewise currents.

Phasor Diagrams can be used to graphically show the addition of voltages. If are drawn at the correct angles then the result of addition may be shown graphically by adding the phasors end on end.

V1 = V 1r + j.V 1i

V2 = V 2r +j.V 2i V2 V3

V3 = V 3r +j.V 3i V1

In Complex Representation:

V3 = V1 +V 2

V3r +j.V 3i = (V 1r +V 2r ) + j.(V 1i +V 2i )

Or using Phasor Diagrams:

V3 V2 Phase Reference (Real Axis) V1 7. Complex Representation of Impedance (R, L, C)

Resistors

It can easily be shown that the voltage across a in complex representation is given by

V = R . I

Where V and I are the complex numbers representing the voltage across a resistor and the current through it. Therefore Ohms law for resistors holds in complex representation.

Inductors and Capacitors

In the time domain the Voltage across an inductor is L.dI/dt.

If an AC current flows through an inductor the voltage across the inductor may be calculated as follows:

I(t) = √2.Irms.Sin( ω.t) V = L.dI/dt V = L. √2.Irms. ω.Cos( ω.t) = ω.L. √2.Irms.Sin( ω.t + 90°)

The voltage across an inductor has a magnitude ω.L times the magnitude of the current and has a phase shift 90° ahead of the current through the inductor.

In our introduction to complex representation we said that the “j” component represented a sine wave 90° out of phase with the reference phase. A more general viewpoint is to see “j” as an operator, which shifts any waveform by 90°. Thus in complex representation we could show the 90° phase shift as a multiplication by “j”.

With this in mind we can convert the equation for the voltage across an inductor into complex representation as follows:

Time Domain: V(t) = ω.L . I(t) phase shifted by 90°

Complex Representation: V = ω.L.j. I = j. ω.L. I

We call j. ω.L the impedance of the inductor.

Using Complex notation we have converted differential equation for voltage and current into a simple multiplication.

A similar exercise could be done to show that the impedance of a capacitor in complex representation is –j / ( ω.C)

8. Ohms Law for AC

V = Z.I

V = Complex voltage, I = Complex Current, Z = Complex Impedance.

The full power of complex representation may now be revealed. Any circuit involving R’s L’s and C’s may be represented as a collection of impedances. These impedances may be added in series or parallel just like resistors in a DC circuit as long as complex arithmetic is used throughout. All of the methods of DC circuit analysis (Thevenin, Superposition, Kirchoff etc.) apply to circuits represented with complex impedances.

Please note that a resistor has only a real component and Inductors and capacitors have only imaginary (“j”) components. In a circuit combining resistors inductors and capacitors the overall impedance will have both real and imaginary components.

Z = a+jb a is the real or resistive component b is the imaginary component also known as the reactive component or Reactance.

Note that a positive reactance arises from inductance whereas negative reactance arises from capacitance.

Time Domain Complex Representation Polar Form R R R ∠ 0° L +j. ω.L ω.L ∠ 90° C -j / ω.C 1/( ω.C) ∠-90°

Arbitrary Combination of Z = R+j.X √(R 2+X 2) ∠ Tan -1(X/R) Rs, Ls and Cs Z = Complex Impedance R = Resistance X = Reactance R + L in series R+j. ω.L √(R 2+( ω.L) 2) ∠ Tan -1(ω.L/R) R + C in series R-j / ω.C √(R 2+(1/ ω.C) 2) ∠ Tan -1(-1/( ω.CR))

Note that the preceding table also includes a polar form representation of impedance. Any complex number can be converted to polar by determining the magnitude and phase. Polar form is useful in calculations, which involve multiplication and division. It is also more intuitive than complex notation because the magnitude and phase can be readily related to the real world unlike the real and imaginary parts in rectangular form.

9. The Physical Significance of Polar Form

I

Z V

We know that the complex version of Ohms law holds:

V = Z.I

In Polar Form this can be written:

Vrms ∠φv = |Z| ∠φz . I rms ∠φi

 |Z| ∠φz = (V rms /I rms ) ∠ ( φv-φi)

 |Z| = (V rms /I rms ) φz = φv-φi

Parameter Physical Significance of Physical Significance of Polar Form Magnitude Polar Form Phase Voltage Vrms φv The root Phase shift of voltage with voltage respect to the reference phase Current Irms φi The Phase shift of the current current with respect to the reference phase Impedance |Z| φz The ratio of the rms voltage The phase shift between the across the impedance to the voltage across the rms current through it impedance and the current through it