2.003 Fall 2003 Complex Exponentials Complex Numbers • Complex numbers have both real and imaginary components. A complex number r may be expressed in Cartesian or Polar forms: r = a + jb (cartesian) = |r|e φ (polar) The following relationships convert from cartesian to polar forms: 2Magnitude |r| = �a2 + b� tan−1 b a > 0 aAngle φ = tan−1 b a < 0 a ± π • Complex numbers can be plotted on the complex plane in either Cartesian or Polar forms Fig.1. Figure 1: Complex plane plots: Cartesian and Polar forms Euler’s Identity Euler’s Identity states that ejφ = cos φ + j sin φ 12.003 Fall 2003 Complex Exponentials This can be shown by taking the series expansion of sin, cos, and e. sin φ = φ − φ3 3! + φ5 5! − φ7 7! + ... cos φ = 1 − φ2 2! + φ4 4! − φ6 6! + ... ejφ = 1 + jφ − φ2 2! − j φ3 3! + φ4 4! + j φ5 5! + ... Combining cos φ + j sin φ = 1 + jφ − (φ)2 2! − j φ3 3! + φ4 4! + j φ5 5! + ... = ejφ Complex Exponentials Consider the case where φ becomes a function of time increasing at a • constant rate ω φ(t) = ωt. then r(t) becomes jωt r(t) = ePlotting r(t) on the complex plane traces out a circle with a constant radius = 1 (Fig. 2 ). Plotting the real and imaginary components of r(t) vs time (Fig. 3 ), we see that the real component is Re{r(t)} = cos ωt while the imaginary component is Im{r(t)} = sin ωt. Consider the variable r(t) which is defined as follows: • st r(t) = e where s is a complex number s = σ + jω 2tr(t et00Im[ r(t) ]=sin ttRe[ r(t) ]=cos tt2.003 Fall 2003 Complex Exponentials Figure 2: Complex plane plots: ) = jωt Figure 3: Real and imaginary components of r(t) vs time • What path does r(t) trace out in the complex plane ? Consider st = e(σ+jω)t = e σt jωt r(t) = e e· One can look at this as a time varying magnitude (eσt) multiplying a point rotating on the unit circle at frequency ω via the function ejωt . Plotting just the magnitude of ejωt vs time shows that there are three distinct regions (Fig. 4 ): 1. σ > 0 where the magnitude grows without bounds. This condition is unstable. 2. σ = 0 where the magnitude remains constant. This condition is 300>0; Unstable=0; Marginally stable<0; StableTimeet2.003 Fall 2003 Complex Exponentials called marginally stable since the magnitude does not grow without bound but does not converge to zero. 3. σ < 0 where the magnitude converges to zero. This condition is termed stable since the system response goes to zero as t → ∞ . Figure 4: Magnitude r(t) for various σ. Effect of Pole Position The stability of a system is determined by the location of the system poles. If a pole is located in the 2nd or 3rd quadrant (which quadrant determines the direction of rotation in the polar plot), the pole is said to be stable. Figure 5 shows the pole position in the complex plane, the trajectory of r(t) in the complex plane, and the real component of the time response for a stable pole. If the pole is located directly on the imaginary axis, the pole is said to be marginally stable. Figure 6 shows the pole position in the complex plane, the trajectory of r(t) in the complex plane, and the real component of the time response for a marginally stable pole. Lastly, if a pole is located in either the 1st or 4th quadrant, the pole is said to be unstable. Figure 7 shows the pole position in the complex plane, the trajectory of r(t) in the complex plane, and the real component of the time response for an unstable pole. 4r(tr(t2.003 Fall 2003 Complex Exponentials Figure 5: Pole position, ), and real time response for stable pole. Figure 6: Pole position, ), and real time response for marginally stable pole. Figure 7: Pole position, r(t), and real time response for unstable pole.