INC 112 Basic Circuit Analysis

INC 112 Basic Circuit Analysis Week 9 Force Response of a Sinusoidal Input and Phasor Concept

Forced Response of Sinusoidal Input In this part of the course, we will focus on finding the force response of a sinusoidal input.

Period that have transient • Start oscillate from stop input displacement

Have oscillated for a long time input displacement We will only be interested in this case for force response (not count the transient)

Phase shift Amplitude Input Output

Theory Force response of a sinusoidal input is also a sinusoidal signal with the same frequency but with different amplitude and phase shift. v2(t) Sine wave v1(t) Sine wave Sine wave vL(t) Sine wave

Phase shift What is the relationship between sin(t) and i(t) ? sin(t) i(t)

Find i(t) Note: Only amplitude changes, frequency and phase still remain the same.

Find i(t) from

ωL เรียก ความต้านทานเสมือน (impedance) Phase shift -90

Phasor Diagram of an inductor Phasor Diagram of a resistor v v i i Power = (vi cosθ)/2 = 0 Power = (vi cosθ)/2 = vi/2 Note: No power consumed in inductors i lags v

ความต้านทานเสมือน (impedance) Find i(t) Phase shift +90

Phasor Diagram of a capacitor Phasor Diagram of a resistor i v v i Power = (vi cosθ)/2 = 0 Power = (vi cosθ)/2 = vi/2 Note: No power consumed in capacitors i leads v

Kirchhoff's Law with AC Circuit KCL,KVL still hold. vR i v(t) i vC

This is similar to adding vectors. Therefore, we will represent sine voltage with a vector. 3 5 4

Vector Quantity • Complex numbers can be viewed as vectors where • X-axis represents real parts • Y-axis represents imaginary parts • There are two ways to represent complex numbers. • Cartesian form 3+j4 • Polar form 5∟53o Operation add, subtract, multiply, division?

Complex Number Forms(Rectangular, Polar Form) b r θ a Interchange Rectangular, Polar form

jω s = 4 + j3 3 σ 4 Rectangular form: 4 + j3 Polar form magnitude=5, angle = 37 บวก ลบ คูณ หาร vector ??

Rectangular form Add, Subtraction Polar form Multiplication Division

Note: Impedance depends on frequency and R,L,C values Cartesian form Polar form Example: Find impedance in form of polar value for ω = 1/3 rad/sec

Rules that can be used inPhasor Analysis • Ohm’s law • KVL/KCL • Nodal, Mesh Analysis • Superposition • Thevenin / Norton

Phasor form Example Find i(t), vR(t), vL(t)

In an RLC circuit with sinusoidal voltage/current source, voltages and currents at all points are in sinusoidal wave form too but with different amplitudes and phase shifts.

Summary of Procedures • Change voltage/current sources in to phasor form • Change R, L, C value into phasor form • Use DC circuit analysis techniques normally, but the value of • voltage, current, and resistance can be complex numbers • Change back to the time-domain form if the problem asks.

Example Find i(t), vL(t)

Phasor Diagram V VL I VR Resistor consumes power Inductor consumes no power P = 0

INC 112 Basic Circuit Analysis