Lumped LC Balun


Balun, stands for balanced to unbalanced, is a single ended to differential converter or viceversa. A lumped LC balun is realized using lumped components, two inductors and two capacitors is shown in Figure 1. It is also called “lattice type” LC balun. Though a lumped LC balun using discrete components on PCBs is very popular low cost solution for narrow band applications, still it had appeared on some RFICs [1].

LC Balun Calculator

Schematic of discrete L-C balun

Figure 1. Schematic of discrete L-C balun

Analysis

To get an insight into circuit and simplify the analysis, the schematic is redrawn as shown in Figure 2. If node Y is grounded, the circuit form single ended to differential converter from V_{in} to V_{out}. Instead if node ‘Z’ is grounded, it form differential ┬áto single ended converter.

Figure 2. Discrete LC balun redrawn with floating (no ground connection)

Figure 2. Discrete LC balun redrawn with floating nodes(no ground connection)

To simply analysis let us assume the circuit is operating at resonant frequency(f_o) where X = \omega_o L = {1 \over \omega_o C}
By KVL around the loop-\textcircled{1},

(1)   \begin{equation*} V_{in} = jX(I_1 + I_L)-jXI_1 \quad\quad \Rightarrow \quad\quad I_L = {V_{in} \over jX} \end{equation*}

By KVL around the loop-\textcircled{2},

(2)   \begin{equation*} jX(I_1 + I_L)+ R_L I_L + (-jX) (-I_1) \quad\quad \Rightarrow \quad\quad I_1 = -{R_L + jX \over 2(jX)} I_L = -{R_L + jX \over 2(jX)^2}V_{in} \end{equation*}

By KCL at node-\textcircled{X},

(3)   \begin{equation*} I_{in} = 2I_1 + I_L \quad\quad\Rightarrow\quad\quad I_{in} = \left(-R_L \over jX \right) {V_{in}\over jX} \end{equation*}

The input impedance of the circuit is

(4)   \begin{eqnarray*} R_{in} &=& {V_{in}\over I_{in}} = -{(jX)^2 \over R_L} = {X^2 \over R_L}\\ \Rightarrow X &=&\sqrt{R_{in}R_L} \end{eqnarray*}

Therefore the characteristic impedance(Z_o) of the LC-balun is given by

(5)   \begin{equation*} \boxed{Z_o =\sqrt{L \over C} = \sqrt{R_{in}R_L}} \end{equation*}

Ouput Voltage
From Figure 2, voltage at \textcircled{+} terminal of load w.r.t node Y is

(6)   \begin{eqnarray*} V_{op,y} &=& -jX(I_1) = - jX\left(-{R_L+jX \over 2jX}\right)I_L = {R_L+jX \over jX}{V_{in}\over 2} \\ &=& {V_{in}\over 2}\sqrt{\left(1+{R^2 \over X^2}\right)}\left/\underline{-\left({\pi\over2}-\tan^{-1}({X\over R})\right)}\right. \end{eqnarray*}

From Figure 2, voltage at \textcircled{-} terminal of load w.r.t node Y is

(7)   \begin{eqnarray*} V_{om,y} &=& jX(I_1 + I_L)= jX\left(-{R_L+jX \over 2jX} I_L + I_L\right) = -{R_L-jX \over jX} {V_{in}\over 2}\\ &=& {V_{in}\over 2}\sqrt{\left(1+{R^2 \over X^2}\right)}\left/\underline{ \left({\pi\over2}-\tan^{-1}({X\over R})\right)}\right. \end{eqnarray*}

From Eq.(6) and Eq.(7) we can state that the voltage signals at \textcircled{+} and \textcircled{-} terminals of the balanced load at not out-of-phase with each other. The signals will be out-of-phase only under the condition R_L\rightarrow\infty. If conversion is required over a frequency band around operating frequency it further degrades. Therefore the circuit has very limited bandwidth.

Differential voltage across load,

(8)   \begin{equation*} V_{out} = V_{op,y} - V_{om,y} = {R_L\over jX}V_{in} \Rightarrow {V_{out}(\omega_o) \over V_{in}(\omega_o)}=\sqrt{R_L\over R_{in}}\left/\underline{~\pi\over ~2}\right. \end{equation*}

Design Equations

Follow the steps below to design a discrete LC balun
1) Since it is a narrow band transformation, know your operating frequency f_o
2) Find the impedance of reactive elements using the equation X=\sqrt{R_s R_L}
3) Compute the values of inductor and capacitor. L = {X \over 2\pi f_o} and C = {1 \over X 2\pi f_o}

Design Example

Example below shows how to calculate the values of L and C for a discrete balun to use as 50\Omega to 200\Omega single ended to differential converter at 900MHz operating frequency.

1) Given fo = 900MHz
2) X = \sqrt{R_s R_L} = \sqrt{50*200} = 100
3) L = X/\omega = 100/(2*\pi*900M) = 17.6nH
4) C = 1/X\omega= 1/(100*2*\pi*900M) = 1.768pF

Simulaton Results

Applications

  1. Single ended to differential conversion or viceversa
  2. Impedance transformation

References

[1] [doi] C. Song, I. Lo, and O. Boric-Lubecke, “2.4 GHz 0.18\mum CMOS Passive Mixer with Integrated Baluns,” in Microwave Symposium Digest, 2009. MTT ’09. IEEE MTT-S International, 2009, pp. 409-412.

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