CA2627 Building Science Lecture 04 Single phase AC circuits Instructor: Jiayu Chen Ph.D. Review of Complex Variables ? Real Number vs. Complex Number © Jiayu Chen, Ph.D. 2 Review of Complex Variables © Jiayu Chen, Ph.D. 3 Review of Complex Variables Rectangular, exponential and polar forms The complex number may be represented in 3 forms: • Rectangular coordinates; • Exponential; • Polar form on the complex plane; © Jiayu Chen, Ph.D. 4 Review of Complex Variables © Jiayu Chen, Ph.D. 5 Review of Complex Variables conjugate © Jiayu Chen, Ph.D. 6 Review of Complex Variables © Jiayu Chen, Ph.D. 7 Review of Complex Variables © Jiayu Chen, Ph.D. 8 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 9 Phasors and the Frequency Domain Take a look at these applet demonstrations: Phasor: http://ptolemy.eecs.berkeley.edu/eecs20/berkeley/phasors/demo/phasors.html R, C, L phasor relations: http://www.walter-fendt.de/ph14e/accircuit.htm Phasor: http://ptolemy.eecs.berkeley.edu/eecs20/berkeley/phasors/demo/phasors.html http://www.circuit-magic.com/phasor_diagraPm.htm © Jiayu Chen, Ph.D. 10 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 11 Phasors and the Frequency Domain Normally we work with the real parts (remember, cos and sin are phase shifted by 90 ) The magnitude Im and the phase angle θ, along with knowledge of ω, completely specify the response, and contains the equivalent information as i(t). The phasor concept may be used when the circuit is linear, the steady-state response is sought, and all independent sources are sinusoidal and have the same frequency. © Jiayu Chen, Ph.D. 12 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 13 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 14 Phasors and the Frequency Domain Impedance From the summary table, the relationships in the frequency domain for the phasor current and phasor voltage of a capacitor, inductor, and resistor appear similar to Ohm's law for resistors. The impedance of an element is defined as the ratio of the phasor voltage to the phasor current. The impedance concept is equivalent to stating that capacitors and inductors act as frequency-dependent resistors, that is, as resistors whose resistance is a function of the frequency of the sinusoidal excitation. © Jiayu Chen, Ph.D. 15 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 16 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 17 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 18 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 19 Phasors and the Frequency Domain Kirchhoff Laws using phasors in the frequency domain Kirchhoff voltage law (KVL) holds in the frequency domain with phasor voltages. The sum of the phasor voltages in a closed path is zero. Kirchhoff current law (KCL) holds in the frequency domain with phasor currents. At a node, the sum of the phasor currents is zero. Since both the KVL and the KCL hold in the frequency domain, all the techniques of analysis we developed for resistive circuits hold for phasor currents and voltages. For example, we can use the principle of superposition, source transformations, Thévenin and Norton equivalent circuits, and node voltage and mesh current analysis. All these methods apply as long as the circuit is linear and we only have one frequency. © Jiayu Chen, Ph.D. 20 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 21 Phasors and the Frequency Domain Series and Parallel Impedance © Jiayu Chen, Ph.D. 22 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 23 Phasors and the Frequency Domain Two methods, the node equations and the mesh equations, are also valid. The node equations are a set of simultaneous equations in which the unknowns are the node voltages. 1. Expressing the element voltages and currents in terms of the node voltages. 2. Applying KCL at the nodes of the ac circuit. After writing and solving the node equations, we can determine all of the voltages and currents of the ac circuit using Ohm’s and Kirchhoff’s laws. © Jiayu Chen, Ph.D. 24 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 25 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 26 Phasors and the Frequency Domain The mesh equations are a set of simultaneous equations in which the unknowns are the mesh currents. We write the mesh equations by 1. Expressing the element voltages and currents in terms of the mesh currents. 2. Applying KVL to the meshes of the ac circuit. After writing and solving the mesh equations, we can determine all of the voltages and currents of the ac circuit using Ohm’s and Kirchhoff’s laws. © Jiayu Chen, Ph.D. 27 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 28 Phasors and the Frequency Domain © Jiayu Chen, Ph.D. 29 Phasors and the Frequency Domain Superposition For a linear circuit containing two or more independent sources, any circuit voltage or current may be calculated as the algebraic sum of all the individual currents or voltages caused by each independent source acting alone. The superposition principle is particularly useful if a circuit has two or more sources acting at different frequencies. The circuit will have one set of impedance values at one frequency and a different set of impedance values at another frequency. We can determine the phasor response at each frequency. Then we find the time response corresponding to each phasor response and add them. Note that superposition, in the case of sources operating at different frequencies, applies to time responses only. We cannot superpose the phasor responses. © Jiayu Chen, Ph.D. 30 Phasors and the Frequency Domain Thevenin equivalent and Norton equivalent Thévenin's and Norton's theorems apply to phasor current or voltages and impedances in the same way that they do for resistive circuits. Thévenin theorem states that any circuit can be replaced by an equivalent circuit with an ideal voltage source equal to the open-circuit voltage and an equivalent impedance in series equal to the Thevenin equivalent impedance. Source transformations Transform a voltage source and its associated series impedance to a current source and its associated parallel impedance, or vice versa. © Jiayu Chen, Ph.D. 31 Phasors and the Frequency Domain Phasor diagram Phasors representing the voltage or current of a circuit are time quantities transformed or converted into the frequency domain. Phasors are complex numbers and can be portrayed in a complex plane. The relationship of phasors on a complex plane is called a phasor diagram. A phasor diagram is a graphical representation of phasors and their relationship on the complex plane. © Jiayu Chen, Ph.D. 32 Thank You! © Jiayu Chen, Ph.D. 33