Phasor Relationships for Circuit Elements (7.4) Dr. Holbert September 4, 2001 ECE201 Lect-5

Phasor Relationships for Circuit Elements Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. A complex exponential is the mathematical tool needed to obtain this relationship. ECE201 Lect-5

I-V Relationship for a Resistor + i(t) v(t) R – Suppose that i(t) is a sinusoid: i(t) = IM ej(wt+q) Find v(t). ECE201 Lect-5

Computing the Voltage ECE201 Lect-5

Class Example Learning Extension E7.5 ECE201 Lect-5

I-V Relationship for a Capacitor v(t) + – i(t) Suppose that v(t) is a sinusoid: v(t) = VM ej(wt+q) Find i(t). ECE201 Lect-5

Computing the Current ECE201 Lect-5

Phasor Relationship Represent v(t) and i(t) as phasors: V = VM  q I = jwC V The derivative in the relationship between v(t) and i(t) becomes a multiplication by jw in the relationship between V and I. ECE201 Lect-5

Example v(t) = 120V cos(377t + 30) C = 2mF What is V? What is I? What is i(t)? ECE201 Lect-5

Class Example Learning Extension E7.7 ECE201 Lect-5

I-V Relationship for an Inductor + i(t) v(t) L – V = jwL I ECE201 Lect-5

Example i(t) = 1mA cos(2p 9.15•107t + 30) L = 1mH What is I? What is V? What is v(t)? ECE201 Lect-5

Class Example Learning Extension E7.6 ECE201 Lect-5

Circuit Element Phasor Relations (ELI and ICE man) ECE201 Lect-5

Phasor Diagrams A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). A phasor diagram helps to visualize the relationships between currents and voltages. ECE201 Lect-5

An Example 2mA  40 + + VC 1mF – V w = 377 + 1kW VR – – ECE201 Lect-5

An Example (cont.) I = 2mA  40 VR = 2V  40 VC = 5.31V  -50 ECE201 Lect-5

Phasor Diagram Imaginary Axis Real Axis V VC VR ECE201 Lect-5

MATLAB Exercise Let’s use MATLAB to plot an ac current and voltage, and then to graphically determine the lead-lag relationship Start MATLAB on your computer We begin by creating a time vector >> t = 0 : 0.0005 : 0.025; Next, we create the voltage and current >> vt = 170 * cos(377*t+10*pi/180); >> it = 100 * cos(377*t-65*pi/180); ECE201 Lect-5

MATLAB Exercise Now we will graph v(t) and i(t) >> plot(t,vt,'b',t, it,'r--'); >> xlabel('Time (sec)'); >> ylabel('Voltage (Volts) or Current (Amps)'); >> title('Household AC Voltage-Current'); >> legend('v(t)=170cos(377t+10)', 'i(t)=100cos(377t-65)'); ECE201 Lect-5

MATLAB Exercise From the graphs created: Determine whether the current leads the voltage, or vice versa Determine the amount of lead by the current or voltage Compare the voltage-current lead-lag relationship obtained by graphical means above to an analytic solution which you should be able to compute ECE201 Lect-5