Unformatted text preview: Using Complex Numbers in AC Steady
State Analysis EE 201: Electric Circuit Theory – DePiero / CalPoly AC analysis employs methods similar to the DC case (Ohm’s Law, Mesh and Nodal techniques). However, in order to find the amplitude and phase changes associated with AC steady
state signals, ‘phasor’ techniques are used. These methods are relatively simple as they avoid differential equations. However they do involve complex valued impedances to describe devices (R, L, C) and phasors to describe signals (V, I). To analyze steady
state conditions, the first step is to convert signals to their phasor form, and devices to their impedances. Circuit analysis may then proceed via any appropriate method. The transformation from phasor form back to the time domain is done by inspection, revealing amplitude and phase. When converting to/from the time domain, one trigonometric form (either sine or cosine) should be used consistently. Also note that phasor analysis is done on a per
frequency basis. If multiple sources are present at different frequencies, use superposition for each frequency. Useful Relationships Radian Frequency w = 2πf (Radians / Sec) Impedance of an Inductor Z L = jwL j = −1 Impedance of a Capacitor ZC = 1 jwC Impedance of a Resistor Z R = R €
General Form of an Impedance Z = R + jX = M < φ €
€
M = R 2 + X 2
φ = tan −1 X R € Eular’s Law €
v ( t ) = A cos( wt + φ ) €
€
Z1 + Z 2 € * Z Z1 2
€
€  Z Combine Z’s in parallel, Z1
2
Combine Z’s in series, Z1 + Z2 €
€
Convert between cos() and sin() forms e jθ = cos θ + j sin θ V = A < φ ( R1 + R2 ) + j ( X1 + X 2 ) ( M1 * M 2 ) < (φ1 + φ 2 ) As with resistors in parallel As with resistors in series cos(A) =
sin(A
90) sin (A) = cos (A
90) €
€
Polar to Rectangular Conversion Conversions between polar and rectangular form are commonly needed when working with phasors. Many calculators perform these operations automatically. Nevertheless, knowledge of the basic principles can be useful. In the expression, M = R 2 + X 2 the imaginary, j, of R + jX has no impact on the magnitude. This is consistent with magnitude ˆ
ˆ
ˆ
calculations in vector spaces. Specifically, in a vector space spanned by x and y , the x ˆ
quantity is not squared when finding a magnitude. This is because x specifies a direction, €
not an intensity. Similarly, j defines a direction in the complex plane. €
€
€
Another subtlety exists evaluating the phase angle. When finding tan
1(X/R), the individual signs of X and R are lost after the division. Because of this €mbiguity, tan
1 return results in a
the range +/
90o only. Hence the quadrant of the R + jX should be noted and results of tan
1 should be adjusted as needed. See atan2(y,x) when in a programming environment! ...
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 Spring '08
 ARAKAKI
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