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Chapter 5—Electrical Network 5–6 z 1 + z 2 = x 1 + jy 1 + x 2 + jy 2 =(x 1 + x 2 )+j(y 1 + y 2 ) (5) Multiplication and division of complex numbers are performed in the polar form: z 1 z 2 = |z 1 ||z 2 |/θ 1 + θ 2 (6) z 1 = |z 1| z 2 |z 2 | /θ 1 − θ 2 (7) The impedance of the series RLC circuit shown in Fig. 2 is the complex number Z = R + jωL − j ωC = |Z|̸ θ Ω (8) where ω =2πf is the angular frequency in rad/s, R is the resistance in ohms, L is the inductance in henrys, and C is the capacitance in farads. V R ✎☞✎☞✎☞ L C ∧∧∧ ∨∨∨ ✲ I Figure 2: Series RLC circuit. We define the reactances of the inductance and capacitance as X L = ωL Ω (9) X C = 1 Ω (10) ωC Reactances are always real numbers. The impedance of an inductor, Z = jX L , is imaginary, but X L itself (and X C )isrealandpositive. When a phasor root-mean-square voltage (rms) V = |V |̸ θ is applied to an impedance, the resulting phasor rms current is I = V Z = |V | |Z| / − θ A (11) Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
Chapter 5—Electrical Network 5–7 Example The RL circuit shown in Fig. 3 has R =6Ω,X L =8Ω,andV = 200/0 o . What is the current? 200̸ 0 o V 6Ω ∧∧∧ ∨∨∨ ✲ I ✎☞✎☞✎☞ j8 Ω ✻ ❙ ❙❙❙❙❙✇ 53.13 o ✲ V I Figure 3: Series RL Circuit. First we find the impedance Z. The current is then Z = R + jX L =6+j8 =10/53.13 o I = 200/0o =20/ − 53.13o 10/53.13o AsketchofV and I on the complex plane for this example is also shown in Fig. 3. This sketch is called a phasor diagram. The current in this inductive circuit is said to lag V. In a capacitive circuit the current will lead V. The words “lead” and “lag” always apply to the relationship of the current to the voltage. The phrase “ELI the ICE man” is sometimes used to help beginning students remember these fundamental relationships. The word ELI has the middle letter L (inductance) with E (voltage) before, and I (current) after or lagging the voltage. The word ICE has the middle letter C (capacitance) with E after, and I before or leading the voltage in a capacitive circuit. In addition to the voltage, current, and impedance of a circuit, we are also interested in the power. There are three types of power which are considered in ac circuits, the complex power S, therealpowerP , and the reactive power Q. The relationship among these quantities is S = P + jQ = |S|̸ θ VA (12) The magnitude of the complex power, called the volt-amperes or the apparent power of the circuit is defined as The real power is defined as |S| = |V ||I| VA (13) Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001
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