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5.5.4Phasor arithmetic
Another detail of phasor math that is both beautiful and practical is the famous expression of Euler’s Relation, the one all math teachers love because it directly relates several fundamental constants in one elegant equation (remember that i and j mean the same thing, just di erent notational conventions for di erent disciplines):
eiπ = −1
This equation is actually a special case of Euler’s Relation, relating imaginary exponents of e to sine and cosine functions:
eiθ = cos θ + i sin θ
What this equation says is really quite amazing: if we raise e to an imaginary exponent of some angle (θ), it is equivalent to the real cosine of that same angle plus the imaginary sine of that same angle. Thus, Euler’s Relation expresses an equivalence between exponential (ex) and trigonometric (sin x, cos x) functions. Specifically, the angle θ describes which way a phasor points on a complex plane, the real and imaginary coordinates of a unit phasor’s23 tip being equal to the cosine and sine values for that angle, respectively.
If we set the angle θ to a value equal to π, we see the general form of Euler’s relation transform into eiπ = −1:
eiθ = cos θ + i sin θ
eiπ = cos π + i sin π
eiπ = −1 + i0
eiπ = −1
23A “unit” phasor is one having a length of 1.
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After seeing this, the natural question to ask is what happens when we set θ equal to other, common angles such as 0, π2 , or 32π (also known as −π2 )? The following examples explore these angles:
Angle (θ) |
Exponential |
Trigonometric |
Rectangular |
Polar |
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0 radians = 0o |
ei0 |
cos 0 + i sin 0 |
1 |
+ i0 |
= 1 |
1 6 |
0o |
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π/2 radians = 90o |
eiπ/2 |
cos 90o + i sin 90o |
0 + i1 = i |
1 6 |
90o |
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π radians = 180o |
eiπ |
cos 180o + i sin 180o |
−1 |
+ i0 |
= −1 |
1 6 |
180o |
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−π/2 radians = −90o |
ei−π/2 |
cos −90o + i sin −90o |
0 − i1 = −i |
1 6 |
−90o |
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We may show all the equivalences on the complex plane, as unit phasors:
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+imaginary |
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eiπ = -1 |
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eiπ/2 = i |
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ei0 = 1 |
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-real |
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+real |
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ei3π/2 = e-iπ/2 = -i
-imaginary
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CHAPTER 5. AC ELECTRICITY |
As we saw previously, the amount of opposition to electrical current o ered by reactive components (i.e. inductors and capacitors) – a quantity known as impedance – may be expressed as functions of j and ω:
ZL = jωL
1
ZC = −j ωC
Knowing that j is equal to ejπ/2 and that −j = e−jπ/2, we may re-write the above expressions for inductive and capacitive impedance as functions of an angle:
ZL = ωLejπ/2
ZC = ωC1 e−jπ/2
Using polar notation as a “shorthand” for the exponential term, the impedances for inductors and capacitors are seen to have fixed angles24:
ZL |
= ωLejπ/2 |
= ωL6 |
π |
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radians |
= |
ωL6 |
90o |
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2 |
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ZC = |
1 |
e−jπ/2 = |
1 |
6 |
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− |
π |
radians |
= |
1 |
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6 |
− 90o |
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ωC |
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ωC |
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2 |
ωC |
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Beginning electronics students will likely find the following expressions of inductive and capacitive impedance more familiar, 2πf being synonymous with ω:
ZL = (2πf L)6 |
90o |
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ZC = |
2πf C |
6 |
− 90o |
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1 |
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24The fact that these impedance phasor quantities have fixed angles in AC circuits where the voltage and current phasors are in constant motion is not a contradiction. Since impedance represents the relationship between voltage and current for a component (Z = V /I), this fixed angle represents a relative phase shift between voltage and current. In other words, the fixed angle of an impedance phasor tells us the voltage and current waveforms will always remain that much out of step with each other despite the fact that the voltage and current phasors themselves are continuously rotating at the system frequency (ω).
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The beauty of complex numbers in AC circuits is that they make AC circuit analysis equivalent to DC circuit analysis. If we represent every voltage and every current and every impedance quantity in an AC circuit as a complex number, all the same25 laws and rules we know from DC circuit analysis will apply to the AC circuit. This means we need to be able to add, subtract, multiply, and divide complex numbers in order to apply Ohm’s Law and Kirchho ’s Laws to AC circuits.
The basic rules of phasor arithmetic are listed here, with phasors having magnitudes of A and B, and angles of M and N , respectively:
AejM + BejN = (A cos M + B cos N ) + j(A sin M + B sin N )
AejM − BejN = (A cos M − B cos N ) + j(A sin M − B sin N )
AejM × BejN = ABej(M +N )
AejM ÷ BejN = BA hej(M −N )i
Addition and subtraction lend themselves readily to the rectangular form of phasor expression, where the real (cosine) and imaginary (sine) terms simply add or subtract. Multiplication and division lend themselves readily to the polar form of phasor expression, where magnitudes multiply or divide and angles add or subtract.
To summarize:
•Voltages and currents in AC circuits may be mathematically represented as phasors, which are imaginary exponential functions (e raised to imaginary powers)
•Phasors are typically written in either rectangular form (real + imaginary) or polar form (magnitude @ angle)
•Ohm’s Law and Kirchho ’s Laws still apply in AC circuits as long as all quantities are in phasor notation
•Addition is best done in rectangular form: add the real parts, and add the imaginary parts
•Subtraction is best done in rectangular form: subtract the real parts, and subtract the imaginary parts
•Multiplication is best done in polar form: multiply the magnitudes, and add the angles
•Division is best done in polar form: divide the magnitudes, and subtract the angles
It should be noted that many electronic calculators possess the ability to perform all these arithmetic functions in complex-number form. If you have access to such a calculator, it will greatly simplify any AC circuit analysis performed with complex numbers!
25With one notable exception: Joule’s Law (P = IV , P = V 2/Z, P = I2Z) for calculating power does not apply in AC circuits because power is not a phasor quantity like voltage and current.