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ME 100 Discussion 4 Fall 2021 1 Complex Numbers A complex number is a number of the form: n = a + bj (1) Where a and b are real numbers. Every complex number n has a conjugate, n . The complex conjugate of n is defined to be: n = a - bj (2) Multiplying the complex number by its conjugate, we find: n · n = a 2 + b 2 (3) Complex numbers may also be thought of as vectors that extend from the origin. Using this representation, we may define the phase ( φ ) and magnitude ( | n | ) of a complex number. φ = tan - 1 ( b/a ) (4) | n | = p a 2 + b 2 (5) We also may connect complex numbers to trig functions using Euler’s Identity . Euler’s Identity is expressed: e jθ = cos( θ ) + j sin( θ ) (6) 2 Phasor Analysis To simplify the analysis of AC circuits, we use phasor analysis. Phasor analy- sis involves converting our typical equations for voltage and current (in terms of time) into complex-valued equations. Although this may sound more complex, it actually greatly simplifies our solution process! When using phasor analysis, we wish to convert a signal of the following form (standard form) to a complex function: V ( t ) = V s cos ( ωt + φ ) (7) Note that if we’re provided with a sine function signal, we must convert it to a cosine function with the following formula: sin θ = cos ( θ - 90 ◦ ) (8) The phasor for V ( t ) may be written in the following forms: 1. Exponential Form: ˜ V s = V s e jφ 2. Standard Form: ˜ V s = V s ∠ φ 1
ME 100 Discussion 4 Fall 2021 3. Complex Form: ˜ V s = a + bj (a) a = V s cos φ (b) b = V s sin φ To define the effect of circuit elements on a circuit in phasor form, we use impedance . Impedance is defined through the following equation, which is similar to Ohm’s Law: Z = ˜ V ˜ I (9) Using this formula, and the current-voltage relationships for each circuit ele- ment, we may define the impedance of each component: Component Impedance Units Resistor R Ohms (Ω) Capacitor 1 jωC Ohms (Ω) Inductor jωL Ohms (Ω) Because the equation for impedance is defined in the same way as Ohm’s Law, equivalent impedance is found in the same way as equivalent resistance. Thus, for n components in series: Z eq = Z 1 + Z 2 + ... + Z n (10) For n components in parallel: 1 Z eq = 1 Z 1 + 1 Z 2 + ... + 1 Z n (11) 2
ME 100 Discussion 4 Fall 2021 3 Examples 3.1 Example 1: Phasor Conversion Find the standard form phasor of the voltage V ( t ) = 10 cos (100 t + 45 ◦ ). 3.2 Example 2: Phasor Analysis Solve for the steady state voltage across the 10Ω resistor: 3
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